\language=0
%----------------------------------------------------------------------
%|              N. G. Stefanis                                        |
%|              Title: NONPERTURBATIVE CALCULATION OF GREEN AND       |
%|                     VERTEX FUNCTIONS IN TERMS OF PARTICLE CONTOURS |
%|              Institut fuer Theoretische Physik II,        7pp.     |
%|              Ruhr-Universitaet Bochum, D-44780 Bochum, Germany     |
%|              Tel.: ++49-(0)234-700 37 13                           |
%|              FAX : ++49-(0)234-70 94 248                           |
%|              E-mail: nicos@hadron.tp2.ruhr-uni-bochum.de           |
%----------------------------------------------------------------------
%|                       !!!!!!-COMMENTS-!!!!!!                       |
%| This is a self-contained LATeX file which generates a layout       |
%| that follows your specifications.                                  |
%| (No special macros except those included below are used.)          |
%|                                                                    |
%| To include the figures properly the auxiliary file "psbox.tex" has |
%| been incorporated.                                                 |
%| The figures embedded in the paper are supplied as ``eps'' files:   |                          |
%| "smooth.eps"                                                       |
%| "fractal.eps"                                                      |
%| "loop.eps"                                                         |
%| These figures comprise a single picture.                           |
%|                                                                    |
%| To get cross-references and lenght right, compile twice (2)! with  |
%| "latex".                                                           |                                                       |
%|                                                                    |
%| To get enumerated pages, comment out "\pagestyle{empty}" below.    |
%----------------------------------------------------------------------

\documentstyle[12pt]{article}

%                       WORLD SCIENTIFIC STYLE
%----------------NEW ADDITIONS TO EXISTING ARTICLE.STY-----------------

\catcode`\@=11
\long\def\@makefntext#1{
\protect\noindent \hbox to 3.2pt {\hskip-.9pt
$^{{\ninerm\@thefnmark}}$\hfil}#1\hfill}                  %CAN BE USED

\def\thefootnote{\fnsymbol{footnote}}
\def\@makefnmark{\hbox to 0pt{$^{\@thefnmark}$\hss}}      %ORIGINAL

\def\ps@myheadings{\let\@mkboth\@gobbletwo
\def\@oddhead{\hbox{}
\rightmark\hfil\ninerm\thepage}
\def\@oddfoot{}\def\@evenhead{\ninerm\thepage\hfil
\leftmark\hbox{}}\def\@evenfoot{}
\def\sectionmark##1{}\def\subsectionmark##1{}}

%--------------------START OF PROCSLA.STY------------------------------
\newcommand{\symbolfootnote}{\renewcommand{\thefootnote}
{\fnsymbol{footnote}}}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\newcommand{\alphfootnote}
{\setcounter{footnote}{0}
 \renewcommand{\thefootnote}{\sevenrm\alph{footnote}}}

%----------------------------------------------------------------------
%NEWLY-DEFINED SECTION COMMANDS
\newcounter{sectionc}\newcounter{subsectionc}\newcounter{subsubsectionc}
\renewcommand{\section}[1] {\vspace{0.6cm}\addtocounter{sectionc}{1}
\setcounter{subsectionc}{0}\setcounter{subsubsectionc}{0}\noindent
           {\bf\thesectionc. #1}\par\vspace{0.4cm}}
\renewcommand{\subsection}[1]
             {\vspace{0.6cm}\addtocounter{subsectionc}{1}
\setcounter{subsubsectionc}{0}\noindent
     {\it\thesectionc.\thesubsectionc. #1}\par\vspace{0.4cm}}
\renewcommand{\subsubsection}[1]
             {\vspace{0.6cm}\addtocounter{subsubsectionc}{1}
\noindent {\rm\thesectionc.\thesubsectionc.\thesubsubsectionc.
     #1}\par\vspace{0.4cm}}
\newcommand{\nonumsection}[1] {\vspace{0.6cm}\noindent{\bf #1}
     \par\vspace{0.4cm}}

%NEW MACRO TO HANDLE APPENDICES
\newcounter{appendixc}
\newcounter{subappendixc}[appendixc]
\newcounter{subsubappendixc}[subappendixc]
\renewcommand{\thesubappendixc}{\Alph{appendixc}.\arabic{subappendixc}}
\renewcommand{\thesubsubappendixc}
     {\Alph{appendixc}.\arabic{subappendixc}.\arabic{subsubappendixc}}

\renewcommand{\appendix}[1] {\vspace{0.6cm}
       \refstepcounter{appendixc}
       \setcounter{figure}{0}
       \setcounter{table}{0}
       \setcounter{equation}{0}
       \renewcommand{\thefigure}{\Alph{appendixc}.\arabic{figure}}
       \renewcommand{\thetable}{\Alph{appendixc}.\arabic{table}}
       \renewcommand{\theappendixc}{\Alph{appendixc}}
       \renewcommand{\theequation}{\Alph{appendixc}.\arabic{equation}}
%      \noindent{\bf Appendix \theappendixc. #1}\par\vspace{0.4cm}}
       \noindent{\bf Appendix \theappendixc #1}\par\vspace{0.4cm}}
\newcommand{\subappendix}[1] {\vspace{0.6cm}
       \refstepcounter{subappendixc}
       \noindent{\bf Appendix \thesubappendixc. #1}\par\vspace{0.4cm}}
\newcommand{\subsubappendix}[1] {\vspace{0.6cm}
       \refstepcounter{subsubappendixc}
       \noindent{\it Appendix \thesubsubappendixc. #1}
     \par\vspace{0.4cm}}

%----------------------------------------------------------------------
%MACRO FOR ABSTRACT BLOCK
\def\abstracts#1{{
     \centering{\begin{minipage}{30pc}\tenrm\baselineskip=10pt\noindent
     \centerline{\tenrm ABSTRACT}\vspace{0.3cm}
     \parindent=0pt #1
     \end{minipage}}\par}}

%----------------------------------------------------------------------
%NEW MACRO FOR BIBLIOGRAPHY
\newcommand{\bibit}{\it}
\newcommand{\bibbf}{\bf}
\renewenvironment{thebibliography}[1]
    {\begin{list}{[\arabic{enumi}]}
    {\usecounter{enumi}\setlength{\parsep}{0pt}
%1.25cm IS STRICTLY FOR PROCSLA.TEX ONLY
\setlength{\leftmargin 0.63cm}{\rightmargin 0pt}
%0.52cm IS FOR NEW DATA FILES
%\setlength{\leftmargin 0.52cm}{\rightmargin 0pt}
      \setlength{\itemsep}{0pt} \settowidth
      {\labelwidth}{#1.}\sloppy}}{\end{list}}

%----------------------------------------------------------------------
%FOLLOWING THREE COMMANDS ARE FOR `LIST' COMMAND.
\topsep=0in\parsep=0in\itemsep=0in
\parindent=1.5pc

%----------------------------------------------------------------------
%LIST ENVIRONMENTS
\newcounter{itemlistc}
\newcounter{romanlistc}
\newcounter{alphlistc}
\newcounter{arabiclistc}
\newenvironment{itemlist}
      {\setcounter{itemlistc}{0}
      \begin{list}{$\bullet$}
      {\usecounter{itemlistc}
      \setlength{\parsep}{0pt}
      \setlength{\itemsep}{0pt}}}{\end{list}}

\newenvironment{romanlist}
      {\setcounter{romanlistc}{0}
      \begin{list}{$($\roman{romanlistc}$)$}
      {\usecounter{romanlistc}
      \setlength{\parsep}{0pt}
      \setlength{\itemsep}{0pt}}}{\end{list}}

\newenvironment{alphlist}
     {\setcounter{alphlistc}{0}
      \begin{list}{$($\alph{alphlistc}$)$}
     {\usecounter{alphlistc}
      \setlength{\parsep}{0pt}
      \setlength{\itemsep}{0pt}}}{\end{list}}

\newenvironment{arabiclist}
     {\setcounter{arabiclistc}{0}
      \begin{list}{\arabic{arabiclistc}}
     {\usecounter{arabiclistc}
      \setlength{\parsep}{0pt}
      \setlength{\itemsep}{0pt}}}{\end{list}}

%----------------------------------------------------------------------
%FIGURE CAPTION
\newcommand{\fcaption}[1]{
        \refstepcounter{figure}
        \setbox\@tempboxa = \hbox{\tenrm Fig.~\thefigure. #1}
        \ifdim \wd\@tempboxa > 6in
          {\begin{center}
        \parbox{6in}{\tenrm\baselineskip=12pt Fig.~\thefigure. #1}
            \end{center}}
        \else
             {\begin{center}
             {\tenrm Fig.~\thefigure. #1}
              \end{center}}
        \fi}

%TABLE CAPTION
\newcommand{\tcaption}[1]{
        \refstepcounter{table}
        \setbox\@tempboxa = \hbox{\tenrm Table~\thetable. #1}
        \ifdim \wd\@tempboxa > 6in
           {\begin{center}
        \parbox{6in}{\tenrm\baselineskip=12pt Table~\thetable. #1}
            \end{center}}
        \else
             {\begin{center}
             {\tenrm Table~\thetable. #1}
              \end{center}}
        \fi}

%----------------------------------------------------------------------
%ACKNOWLEDGEMENT: this portion is from John Hershberger
\def\@citex[#1]#2{\if@filesw\immediate\write\@auxout
     {\string\citation{#2}}\fi
\def\@citea{}\@cite{\@for\@citeb:=#2\do
     {\@citea\def\@citea{,}\@ifundefined
     {b@\@citeb}{{\bf ?}\@warning
     {Citation `\@citeb' on page \thepage \space undefined}}
     {\csname b@\@citeb\endcsname}}}{#1}}

%\newif\if@cghi
%\def\cite{\@cghitrue\@ifnextchar [{\@tempswatrue
%     \@citex}{\@tempswafalse\@citex[]}}
%\def\citelow{\@cghifalse\@ifnextchar [{\@tempswatrue
%     \@citex}{\@tempswafalse\@citex[]}}
%\def\@cite#1#2{{$\null^{#1}$\if@tempswa\typeout
%     {IJCGA warning: optional citation argument
%     ignored: `#2'} \fi}}
%\newcommand{\citeup}{\cite}

%----------------------------------------------------------------------
%FOR FNSYMBOL FOOTNOTE AND ALPH{FOOTNOTE}
\def\fnm#1{$^{\mbox{\scriptsize #1}}$}
\def\fnt#1#2{\footnotetext{\kern-.3em
     {$^{\mbox{\sevenrm #1}}$}{#2}}}

%----------------------------------------------------------------------
\font\twelvebf=cmbx10 scaled\magstep 1
\font\twelverm=cmr10  scaled\magstep 1
\font\twelveit=cmti10 scaled\magstep 1
\font\elevenbfit=cmbxti10 scaled\magstephalf
\font\elevenbf=cmbx10     scaled\magstephalf
\font\elevenrm=cmr10      scaled\magstephalf
\font\elevenit=cmti10     scaled\magstephalf
\font\bfit=cmbxti10
\font\tenbf=cmbx10
\font\tenrm=cmr10
\font\tenit=cmti10
\font\ninebf=cmbx9
\font\ninerm=cmr9
\font\nineit=cmti9
\font\eightbf=cmbx8
\font\eightrm=cmr8
\font\eightit=cmti8

%--------------------END OF PROCSLA.STY--------------------------------

%--------------------START OF DATA FILE--------------------------------
\textwidth 15.0 cm
\textheight 20.5 cm
%\pagestyle{empty}
\topmargin -0.25truein
\oddsidemargin 0.30truein
\evensidemargin 0.30truein
\raggedbottom
\parindent=1.5pc
\baselineskip=14pt

\begin{document}
\hfill RUB-TPII-08/96

\centerline{\tenbf NONPERTURBATIVE CALCULATION OF GREEN AND VERTEX}
\centerline{\tenbf FUNCTIONS IN TERMS OF PARTICLE CONTOURS}
%\footnote{Invited talk presented at the International Conference
%          {\it Problems of Quantum Field Theory},
%          May 13 - 18, 1996,
%          Alushta, Crimea, Ukraine}
%\baselineskip=22pt
\vspace{0.3cm}
\centerline{\tenrm N. G. STEFANIS}
\baselineskip=13pt
\centerline{\tenit Institut f\"ur Theoretische Physik II,
                   Ruhr-Universit\"at Bochum}
\baselineskip=12pt
\centerline{\tenit D-44780 Bochum, Germany}
\vspace{0.5cm}
\abstracts{\tenrm
The infrared regime of fermionic Green and vertex functions is
studied analytically within a geometric approach which simulates
soft interactions by an {\it effective} theory of contours.
Expanding the particle path integral in terms of dominant contours
at large distances, all-order results in the coupling constant are
obtained for the renormalized fermion propagator and a universal
vertex function with physical characteristics close to those
associated with the Isgur-Wise function in the weak decays of heavy
mesons. The extension to the ultraviolet regime is scetched.}
%\vfil
%\vspace{0.8cm}
%\rm\baselineskip=12pt
\section{Conceptual Skeleton}
\label{sec:conskel}

This article presents some applications of a geometric formalism,
relying on (Euclidean) particle path integrals, which emulates soft
interactions in fermionic Green and vertex functions by means of
dominant contours in the relevant regimes of
dynamics~\cite{SKK94,KKS95}.
Expanding in terms of contours enables the evaluation of all-order
expressions in the coupling constant, hence transcending perturbation
theory. As a result, typical problems encountered in resumming
Feynman graphs are avoided.

The method discussed below represents, in some aspects, an
alternative concept to recent superparticle approaches (see,
e.g.,~\cite{Str92}) and to calculations within the heavy quark
effective theory (see, e.g.,~\cite{Neu94}).
All results are derived in the framework of an {\it effective}
Abelian gauge theory (QED), in which the infraparticle~\cite{Sch63}
(i.e., fermion) mass $m$ serves as a dividing line between the hard
(``heavy'') and the soft (``light'') degrees of freedom
(see Table~\ref{table:geometry}).
This enables one to trade infrared (IR) divergences in the full
theory for ultraviolet (UV) divergences in the low-energy effective
theory.

\begin{table}[h]
\tcaption{
          Connection between geometrical approach and
          conventional quantum field theory
         }
\vspace{0.1cm}
\begin{center}
\begin{tabular}{ll}
\hline
Geometry (Contours)  & Physics (Feynman graphs) \\
\hline
Open smooth contour  & Infrapropagator (self-energy and vertex
                       corrections) \cr
Fractal contour      & Fermion Green function in UV regime
                       (conjecture) \cr
Self-intersecting
contour              & Infraparticle vertex function with soft boson
                       ``cloud''\cr
with infinitesimal loop   & transported intact through the
                            interaction point \cr
                          & ({\it exclusive} form factor) \cr
Contour with cusp
where                     & Infraparticle vertex giving rise to
                            bremsstrahlung \cr
four-velocity jumps
                      & ({\it semi-inclusive} form factor) \cr
Contour end-points    & Multiplicatively renormalizable singularities
                        \cr
Contour cusps         & Angle-dependent anomalous dimensions \cr
\end{tabular}
\end{center}
\label{table:geometry}
\end{table}

\section{Contour Representation of Fermionic Systems}
\label{sec:contrep}

The main ingredients of the present approach can be summarized as
follows~\cite{KS89,KK92}:
\begin{itemize}
\item Recast a quantum field-theoretical system into particle-based
      language, i.e., convert path integrals over fields into those
      over particle contours (using a Euclidean metric) within a
      Feynman-Schwinger framework.
\item Incorporate a spacetime {\it built-in} resolution scale
      $\alpha$ in the initial field theoretical casting.
      Discretized copies of Euclidean manifolds $I\!\!R^{d}$ along
      contours in \hbox{$I\!\!R\otimes I\!\!R^{d}$} are related by an
      averaging operator with a rapidly decreasing kernel within the
      discretization range (i.e., nonlocality is confined within the
      volume $\alpha ^{d}$).
      The point to notice is that discretization emerges as
      {\it averaging} over cells (in analogy to Kadanoff's block-spin
      renormalization~\cite{Kad77}) rather than as ``latticization''
      of the space-time continuum. Hence, typical problems of lattice
      formulations, like fermion dubbling, etc., are not encountered,
      and the continuum limit can be taken at every stage of the
      calculation.
\item Use the {\it geometry} of contours as guiding principle in
      emulating quantum field interactions by appropriate contour
      configurations (see, Fig.~\ref{fig:contours}).
\end{itemize}

Following~\cite{KK92}, the action of the averaging operator is
defined by
\begin{equation}
  \left(\not\!\! D^{D} + m^{D}\right)\, \psi (n\alpha )
=
  -\int_{}^{} d^{d}y\,
  \left[(\not\!\partial - m)\, f(|y|)\right]
  U\left(L_{n\alpha,n\alpha + y}\right)\, \psi (n\alpha + y) \; ,
\label{eq:deriv}
\end{equation}
%Eq (1) Action of the covariant derivative
where $m$ is the bare fermion mass and $f(|y|)$ is the averaging
kernel with
$
 f(|y|)
\buildrel \alpha \to 0 \over \longrightarrow
 \delta (y),
$
which is, for instance, satisfied  by a Gaussian distribution.
Gauge invariance is manifest due to the non-integrable phase factor
\begin{equation}
  U(L_{x,x+y})
\equiv
  {\cal P} \exp\left[-ig\int_{L_{x,x+y}}^{}
  A_{\mu}(z)\, dz_{\mu}\right] \; .
\label{eq:nipf}
\end{equation}
%Eq (2) Non-integrable phase factor

Closed-form expressions for the generating functional or Green and
vertex functions involve matrix elements of the evolution operator
$
 U(T)
=
 {\rm e}^{-iHT}
=
 {\rm e}^{-(\not\! D + m)T}
$
expressed in terms of particle eigenstates with respect to Fock's
``fifth parameter'' $T$ (e.g., Schwinger's proper time).
Then the quantum mechanical Green function for spinor propagation in
discretized form is given by
\begin{equation}
  G^{D}(n\alpha,m\alpha |A)
=
  \left\langle
              n\alpha \left|
              \left(\not\!\! D^{D} + m^{D}\right)^{-1}
              \right| m\alpha
  \right\rangle
=
  \lim_{c\to 0} \int_{c}^{\infty} dT\,
  \left\langle
              n\alpha \left|
              e^{-iH\,T}
              \right| m\alpha
  \right\rangle
\label{eq:Schwinger}
\end{equation}
%Eq (3) Proper-time expression for the matrix element
which can be converted into a path-integral expression to read
\begin{eqnarray}
  G(x,y|A)
= \!\!\!
& \phantom{} & \!\!\!\!\!\!\!\!\!\!
  \lim_{c\to 0} \int_{c}^{\infty}dT \,
  \int_{{{x(0)=x}\atop {x(T)=y}}}^{}
  \left[dx(\tau )\right] \left[dp(\tau )\right]
  \exp
      \left[
            i\int_{0}^{T} d\tau p(\tau ) \cdot {\dot{x}}(\tau )
      \right]
\nonumber  \\
& \times & \!\!\!\!
  {\cal P}
  \exp
      \left\{
             -\int_{0}^{T}d\tau
              \left[i \gamma \!\cdot\! p(\tau ) + m\right]
      \right\}
  \exp
      \left\{
             ig\int_{0}^{T}d\tau {\dot{x}}(\tau )\!\cdot\!
              A\left[x(\tau )\right]
      \right\} .
\label{eq:Greenext}
\end{eqnarray}
%Eq (4) Particle path integral for G(x,y|A)
As a result, the full fermion Green function in terms of particle
path integrals reads
\begin{eqnarray}
  G(x,y) \!\!\!
& = & \!\!\!
  \int_{c}^{\infty}dT
  \int_{ {x(0)=x\atop x(T)=y} }^{}
  \left[dx(\tau )\right]\! \left[dp(\tau )\right]
  \exp
      \left[
            i\int_{0}^{T}d\tau\, p(\tau )\cdot {\dot{x}}(\tau )
      \right]
\nonumber  \\
& \times & \!\!\!\!
  {\cal P}
  \exp
      \left\{
             -\int_{0}^{T}\!d\tau \!\left[i\gamma \cdot p(\tau )
                                 + m \right]
      \right\}
  \left\langle
              \exp\left\{
                         ig\!\int_{0}^{T}\! d\tau \,
                         {\dot{x}}(\tau )\!\cdot\!
                         A\left[x(\tau )\right]
                   \right\}
  \right\rangle _{A} ,
\label{eq:fullGreen}
\end{eqnarray}
%Eq (5) Full one-fermion Green function in particle-based formalism
where the expectation value of the boson (Wilson) line exponential is
\begin{eqnarray}
  \Biggl\langle
  \exp
& \phantom{} & \!\!\!\!\!\!\!\!\!\!\!\!\!
      \left\{
             ig\int_{0}^{T}d\tau {\dot{x}}(\tau )\cdot
              A\left[x(\tau )\right]
      \right\}
  \Biggr\rangle _{A} \!\!\!
 = \!
  \int_{}^{}\left[dA_{\mu}(x)\right]{\rm det}
  \left[G^{-1}(x,y|A)\right]
\nonumber  \\
& \times & \!\!\!\!\!
  \exp
      \left\{
             ig \int_{0}^{T}d\tau\, {\dot{x}}(\tau )\cdot
              A\left[x(\tau )\right]
      \right\}
  \exp
      \left\{
             - \frac{1}{2}\!\int_{}^{}\!d^{d}z A_{\mu}(z)
               {\cal D}_{\mu\nu}^{-1}(\lambda )A_{\nu}(z)
      \right\} .
\label{eq:expA}
\end{eqnarray}
%Eq (6) Expectation value of the gauge field in the one-fermion case
Eq.~(\ref{eq:fullGreen}) is the particle-based version of the
one-fermion Green function in field theory:
\begin{equation}
  G(x,y)
=
  \int_{}^{}\left[dA_{\mu}\right]
           \exp
               \left\{
                      - \frac{1}{2}\int_{}^{}\!\!d^{d}z\, A_{\mu}(z)
                        {\cal D}^{-1}_{\mu\nu}(\lambda )
                        A_{\nu}(z)
               \right\} \!
           {\rm det} \left[G_{D}^{-1}(x,y |A)\right] \!
           G^{D}(x,y | A) ,
\label{eq:oneFGreen}
\end{equation}
%Eq (7) Full one-fermion Green function in field formalism
where the Grassmann integrations over fermion field variables have
been carried out.

%********************************************************************
%*                         F I G U R E  1                           *
%********************************************************************
%
%\input psbox.tex
%
%       %%%%%%%    %%%%%        %%%%%%    %%%%%   %     %
%       %      %  %             %     %  %     %   %   %
%       %      %  %             %     %  %     %    % %
%       %%%%%%%    %%%%%        %%%%%%   %     %     %
%       %               %       %     %  %     %    % %
%       %               %       %     %  %     %   %   %
%       %         %%%%%%        %%%%%%    %%%%%   %     %
%
%       By Jean Orloff
%       Comments & suggestions by e-mail: ORLOFF@surya11.cern.ch
%       No modification of this file allowed if not e-sent to me.
%
% WHAT IS IT:
% psbox is a set of machine-independent TeX macros to
% 1) allow (Encapsulated) PostScript figure inclusion in all versions
%    of TeX (Plain, LaTeX) on all machines using a PostScript printer
% 2) facilitate the communication (e-mail, ftp, ...) of all the files
%    (text, macros, figs) needed to reproduce a TeX document by grouping
%    them together into a single, TeXable file.
% For more info, get the file pub/TeX/psbox/PSBOXALL.TEX by anonymous
%     ftp from cs.nyu.edu(=128.122.140.24)
%
% History:
%  1.34  \readfilename=final fix for all filename scans; try \psforptips
%  1.33: corrects \psnewinput for LaTeX (still fails if fname=a{b}c)
%  1.32: corrects \psfordvialw and adds .TEX to PSBOXALL(!)
%  1.31: adds \psfordvialw(?)
%  1.30: adds \splitfile & \joinfiles for multi-file management
%  1.24: fix error handling & add \psonlyboxes
%  1.22: makes \drawingBox \global for use in Phyzzx
%  1.21: accepts %%BoundingBox: (atend)
%  1.20: tries to add \psfordvitps for the TeXPS package.
%  1.10: adds \psforoztex, error handling...
%2345678 1 2345678 2 2345678 3 2345678 4 2345678 5 2345678 6 2345678 7 23456789
%
% Checking version no to avoid multiple loadings
\def\temp{1.34}%
\let\tempp=\relax
\expandafter\ifx\csname psboxversion\endcsname\relax
  \message{PSBOX(\temp) loading}%
\else
    \ifdim\temp cm>\psboxversion cm
      \message{PSBOX(\temp) loading}%
    \else
      \message{PSBOX(\psboxversion) is already loaded: I won't load
        PSBOX(\temp)!}%
      \let\temp=\psboxversion
      \let\tempp=\endinput
    \fi
\fi
\tempp
\let\psboxversion=\temp
\catcode`\@=11
% Every macro likes a little privacy...
%
%Trying to tame the variety of \special commands for Postscript: the
%  universal internal command \PSspeci@l##1##2 takes ##1 to be the
%  filename and ##2 to be the integer scale factor*1000 (as for usual
%   TeX \scale commands)
%
\def\psfortextures{%     For TeXtures on the Macintosh
%-----------------
\def\PSspeci@l##1##2{%
\special{illustration ##1\space scaled ##2}%
}}%
\def\psfordvitops{%      For the DVItoPS converter on IBM mainframes
%----------------
\def\PSspeci@l##1##2{%
\special{dvitops: import ##1\space \the\drawingwd \the\drawinght}%
}}%
\def\psfordvips{%      For DVIPS converter on VAX, UNIX and PC's
%--------------
\def\PSspeci@l##1##2{%
%    \special{/@scaleunit 1000 def}% never read dox without trying!
\d@my=0.1bp \d@mx=\drawingwd \divide\d@mx by\d@my% BUG! for large \drawingwd
\special{PSfile=##1\space llx=\psllx\space lly=\pslly\space%
urx=\psurx\space ury=\psury\space rwi=\number\d@mx
}}}%
\def\psforoztex{%        For the OzTeX shareware on the Macintosh
%--------------
\def\PSspeci@l##1##2{%
\special{##1 \space
      ##2 1000 div dup scale
      \number-\psllx\space \number-\pslly\space translate
}}}%
\def\psfordvitps{%       From the UNIX TeXPS package, vers.>3.12
%---------------
% Convert a dimension into the number \psn@sp (in scaled points)
\def\psdimt@n@sp##1{\d@mx=##1\relax\edef\psn@sp{\number\d@mx}}
\def\PSspeci@l##1##2{%
% psfig.psr contains the def of "startTexFig": if you can locate it
% and include the correct pathname, it should work
\special{dvitps: Include0 "psfig.psr"}% contains def of "startTexFig"
\psdimt@n@sp{\drawingwd}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\drawinght}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\psllx bp}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\pslly bp}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\psurx bp}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\psury bp}
\special{dvitps: Literal "\psn@sp\space startTexFig\space"}
\special{dvitps: Include1 "##1"}
\special{dvitps: Literal "endTexFig\space"}
}}%
\def\psfordvialw{%   Try for dvialw, a UNIX public domain
%---------------
\def\PSspeci@l##1##2{
\special{language "PostScript",
position = "bottom left",
literal "  \psllx\space \pslly\space translate
  ##2 1000 div dup scale
  -\psllx\space -\pslly\space translate",
include "##1"}
}}%
\def\psforptips{%   For MS-DOS; LUOMA@brandeis.bitnet
%---------------
\def\PSspeci@l##1##2{{
\d@mx=\psurx bp
\advance \d@mx by -\psllx bp
\divide \d@mx by 1000\multiply\d@mx by \xscale
\incm{\d@mx}
\let\tmpx\dimincm
\d@my=\psury bp
\advance \d@my by -\pslly bp
\divide \d@my by 1000\multiply\d@my by \xscale
\incm{\d@my}
\let\tmpy\dimincm
\d@mx=-\psllx bp
\divide \d@mx by 1000\multiply\d@mx by \xscale
\d@my=-\pslly bp
\divide \d@my by 1000\multiply\d@my by \xscale
\at(\d@mx;\d@my){\special{ps:##1 x=\tmpx, y=\tmpy}}
}}}%
\def\psonlyboxes{%     Draft-like behaviour if none of the others works
%---------------
\def\PSspeci@l##1##2{%
\at(0cm;0cm){\boxit{\vbox to\drawinght
  {\vss\hbox to\drawingwd{\at(0cm;0cm){\hbox{({\tt##1})}}\hss}}}}
}}%
\def\psloc@lerr#1{%
\let\savedPSspeci@l=\PSspeci@l%
\def\PSspeci@l##1##2{%
\at(0cm;0cm){\boxit{\vbox to\drawinght
  {\vss\hbox to\drawingwd{\at(0cm;0cm){\hbox{({\tt##1}) #1}}\hss}}}}
\let\PSspeci@l=\savedPSspeci@l% restore normal output for other figs!
}}%
%\def\psfor...  add your own!
%
% Some common defs
%
\newread\pst@mpin
\newdimen\drawinght\newdimen\drawingwd
\newdimen\psxoffset\newdimen\psyoffset
\newbox\drawingBox
\newcount\xscale \newcount\yscale \newdimen\pscm\pscm=1cm
\newdimen\d@mx \newdimen\d@my
\newdimen\pswdincr \newdimen\pshtincr
\let\ps@nnotation=\relax
{\catcode`\|=0 |catcode`|\=12 |catcode`|%=12 |catcode`~=12
|catcode`#=12 |catcode`*=14
|xdef|backslashother{\}*
|xdef|percentother{%}*
|xdef|tildeother{~}*
|xdef|sharpother{#}*
}%
% useful to display special chars in \tt; fails for \,#,%
\def\R@moveMeaningHeader#1:->{}%
\def\uncatcode#1{%
\edef#1{\expandafter\R@moveMeaningHeader\meaning#1}}%
%
\def\execute#1{#1}% NOT stupid: cs in #1 are then identified BEFORE execution
\def\psm@keother#1{\catcode`#112\relax}% borrowed from latex
\def\executeinspecs#1{%
\execute{\begingroup\let\do\psm@keother\dospecials\catcode`\^^M=9#1\endgroup}}%
\def\@mpty{}%
% \if\matchin#1#2<=> \iftrue if #1 contains #2, <=>\iffalse otherwise:
% \if\matchexpin: idem, but #1 & #2 are first fully expanded (no \if
% inside!)
% \tmpa & \tmpb contain what's before and after the occurence of #2
\def\matchexpin#1#2{
  \fi%
%\message{(#1>#2)}
  \edef\tmpb{{#2}}%
  \expandafter\makem@tchtmp\tmpb%
  \edef\tmpa{#1}\edef\tmpb{#2}%
  \expandafter\expandafter\expandafter\m@tchtmp\expandafter\tmpa\tmpb\endm@tch%
  \if\match%
}%
\def\matchin#1#2{%
  \fi%
  \makem@tchtmp{#2}%
  \m@tchtmp#1#2\endm@tch%
  \if\match%
}%
\def\makem@tchtmp#1{\def\m@tchtmp##1#1##2\endm@tch{%
  \def\tmpa{##1}\def\tmpb{##2}\let\m@tchtmp=\relax%
  \ifx\tmpb\@mpty\def\match{YN}%
  \else\def\match{YY}\fi%
}}%
% converts any dimen in cm, with 1E-4 cm precision
\def\incm#1{{\psxoffset=1cm\d@my=#1
 \d@mx=\d@my
  \divide\d@mx by \psxoffset
  \xdef\dimincm{\number\d@mx.}
  \advance\d@my by -\number\d@mx cm
  \multiply\d@my by 100
 \d@mx=\d@my
  \divide\d@mx by \psxoffset
  \edef\dimincm{\dimincm\number\d@mx}
  \advance\d@my by -\number\d@mx cm
  \multiply\d@my by 100
 \d@mx=\d@my
  \divide\d@mx by \psxoffset
  \xdef\dimincm{\dimincm\number\d@mx}
}}%
%
%  \ReadPSize{PSfilename} reads the dimensions of a PostScript drawing
%      and stores it in \drawinght(wd)
\newif\ifNotB@undingBox
\newhelp\PShelp{Proceed: you'll have a 5cm square blank box instead of
your graphics (Jean Orloff).}%
\def\s@tsize#1 #2 #3 #4\@ndsize{
  \def\psllx{#1}\def\pslly{#2}%
  \def\psurx{#3}\def\psury{#4}%  needed by a crazyness of dvips!
  \ifx\psurx\@mpty\NotB@undingBoxtrue% this is not a valid one!
  \else
    \drawinght=#4bp\advance\drawinght by-#2bp
    \drawingwd=#3bp\advance\drawingwd by-#1bp
%  !Units related by crazy factors as bp/pt=72.27/72 should be BANNED!
  \fi
  }%
\def\sc@nBBline#1:#2\@ndBBline{\edef\p@rameter{#1}\edef\v@lue{#2}}%
\def\g@bblefirstblank#1#2:{\ifx#1 \else#1\fi#2}%
{\catcode`\%=12
\xdef\B@undingBox{%%BoundingBox}}%
%% is not a true comment in PostScript, even if % is!
\def\ReadPSize#1{
 \readfilename#1\relax
 \let\PSfilename=\lastreadfilename
 \openin\pst@mpin=#1\relax
 \ifeof\pst@mpin \errhelp=\PShelp
   \errmessage{I haven't found your postscript file (\PSfilename)}%
   \psloc@lerr{was not found}%
   \s@tsize 0 0 142 142\@ndsize
   \closein\pst@mpin
 \else
% each entry in \GlobalInputList should be unique
   \if\matchexpin{\GlobalInputList}{, \lastreadfilename}%
   \else\xdef\GlobalInputList{\GlobalInputList, \lastreadfilename}%
     \immediate\write\psbj@inaux{\lastreadfilename,}%
   \fi%
   \loop
     \executeinspecs{\catcode`\ =10\global\read\pst@mpin to\n@xtline}%
     \ifeof\pst@mpin
       \errhelp=\PShelp
       \errmessage{(\PSfilename) is not an Encapsulated PostScript File:
           I could not find any \B@undingBox: line.}%
       \edef\v@lue{0 0 142 142:}%
       \psloc@lerr{is not an EPSFile}%
       \NotB@undingBoxfalse
     \else
       \expandafter\sc@nBBline\n@xtline:\@ndBBline
       \ifx\p@rameter\B@undingBox\NotB@undingBoxfalse
         \edef\t@mp{%
           \expandafter\g@bblefirstblank\v@lue\space\space\space}%
         \expandafter\s@tsize\t@mp\@ndsize
       \else\NotB@undingBoxtrue
       \fi
     \fi
   \ifNotB@undingBox\repeat
   \closein\pst@mpin
 \fi
\message{#1}%
}%
%
% \psboxto(xdim;ydim){psfilename}: you specify the dimensions and
%    TeX uniformly scales to fit the largest one. If xdim=0pt, the
%    scale is fully determined by ydim and vice versa.
%    Notice: psboxes are a real vboxes; couldn't take hbox otherwise all
%    indentation and all cr's would be interpreted as spaces (hugh!).
%
\def\psboxto(#1;#2)#3{\vbox{%
   \ReadPSize{#3}%
   \advance\pswdincr by \drawingwd
   \advance\pshtincr by \drawinght
   \divide\pswdincr by 1000
   \divide\pshtincr by 1000
   \d@mx=#1
   \ifdim\d@mx=0pt\xscale=1000
         \else \xscale=\d@mx \divide \xscale by \pswdincr\fi
   \d@my=#2
   \ifdim\d@my=0pt\yscale=1000
         \else \yscale=\d@my \divide \yscale by \pshtincr\fi
   \ifnum\yscale=1000
         \else\ifnum\xscale=1000\xscale=\yscale
                    \else\ifnum\yscale<\xscale\xscale=\yscale\fi
              \fi
   \fi
   \divide\drawingwd by1000 \multiply\drawingwd by\xscale
   \divide\drawinght by1000 \multiply\drawinght by\xscale
   \divide\psxoffset by1000 \multiply\psxoffset by\xscale
   \divide\psyoffset by1000 \multiply\psyoffset by\xscale
   \global\divide\pscm by 1000
   \global\multiply\pscm by\xscale
   \multiply\pswdincr by\xscale \multiply\pshtincr by\xscale
   \ifdim\d@mx=0pt\d@mx=\pswdincr\fi
   \ifdim\d@my=0pt\d@my=\pshtincr\fi
   \message{scaled \the\xscale}%
 \hbox to\d@mx{\hss\vbox to\d@my{\vss
   \global\setbox\drawingBox=\hbox to 0pt{\kern\psxoffset\vbox to 0pt{%
      \kern-\psyoffset
      \PSspeci@l{\PSfilename}{\the\xscale}%
      \vss}\hss\ps@nnotation}%
   \global\wd\drawingBox=\the\pswdincr
   \global\ht\drawingBox=\the\pshtincr
   \global\drawingwd=\pswdincr
   \global\drawinght=\pshtincr
   \baselineskip=0pt
   \copy\drawingBox
 \vss}\hss}%
  \global\psxoffset=0pt
  \global\psyoffset=0pt
  \global\pswdincr=0pt
  \global\pshtincr=0pt % These are local to one figure
  \global\pscm=1cm %should not be necessary
}}%
%
% \psboxscaled{scalefactor*1000}{PSfilename} allows to bypass the
%   rounding errors of TeX integer divisions for situations where the
%   TeX box should fit the original BoundingBox with a precision
%   better
%   than 1/1000.
%
\def\psboxscaled#1#2{\vbox{%
  \ReadPSize{#2}%
  \xscale=#1
  \message{scaled \the\xscale}%
  \divide\pswdincr by 1000 \multiply\pswdincr by \xscale
  \divide\pshtincr by 1000 \multiply\pshtincr by \xscale
  \divide\psxoffset by1000 \multiply\psxoffset by\xscale
  \divide\psyoffset by1000 \multiply\psyoffset by\xscale
  \divide\drawingwd by1000 \multiply\drawingwd by\xscale
  \divide\drawinght by1000 \multiply\drawinght by\xscale
  \global\divide\pscm by 1000
  \global\multiply\pscm by\xscale
  \global\setbox\drawingBox=\hbox to 0pt{\kern\psxoffset\vbox to 0pt{%
     \kern-\psyoffset
     \PSspeci@l{\PSfilename}{\the\xscale}%
     \vss}\hss\ps@nnotation}%
  \advance\pswdincr by \drawingwd
  \advance\pshtincr by \drawinght
  \global\wd\drawingBox=\the\pswdincr
  \global\ht\drawingBox=\the\pshtincr
  \global\drawingwd=\pswdincr
  \global\drawinght=\pshtincr
  \baselineskip=0pt
  \copy\drawingBox
  \global\psxoffset=0pt
  \global\psyoffset=0pt
  \global\pswdincr=0pt
  \global\pshtincr=0pt % These are local to one figure
  \global\pscm=1cm
}}%
%
%  \psbox{PSfilename} makes a TeX box having the minimal size to
%      enclose the picture
\def\psbox#1{\psboxscaled{1000}{#1}}%
%------------------------------------------------------
%  \joinfiles file1, file2, ...n \into joinedfilename .
%     makes one file out of many
%  \splitfile joinedfilename
%     the opposite
\newif\ifn@teof\n@teoftrue
\newif\ifc@ntrolline
\newif\ifmatch
\newread\j@insplitin
\newwrite\j@insplitout
\newwrite\psbj@inaux
\immediate\openout\psbj@inaux=psbjoin.aux
\immediate\write\psbj@inaux{\string\joinfiles}%
\immediate\write\psbj@inaux{\jobname,}%
%
% INPUT REDEFINITION
%
% works if #1 is a single character
\def\toother#1{\ifcat\relax#1\else\expandafter%
  \toother@ux\meaning#1\endtoother@ux\fi}%
\def\toother@ux#1 #2#3\endtoother@ux{\def\tmp{#3}%
  \ifx\tmp\@mpty\def\tmp{#2}\let\next=\relax%
  \else\def\next{\toother@ux#2#3\endtoother@ux}\fi%
\next}%
%
% \readfilename defs:
%
\let\readfilenamehook=\relax
\def\re@d{\expandafter\re@daux}% spares typing 10 \expandafter's...
\def\re@daux{\futurelet\nextchar\stopre@dtest}%
\def\re@dnext{\xdef\lastreadfilename{\lastreadfilename\nextchar}%
  \afterassignment\re@d\let\nextchar}%
\def\stopre@d{\egroup\readfilenamehook}%
\def\stopre@dtest{%
  \ifcat\nextchar\relax\let\nextread\stopre@d
  \else
    \ifcat\nextchar\space\def\nextread{%
      \afterassignment\stopre@d\chardef\nextchar=`}%
    \else\let\nextread=\re@dnext
      \toother\nextchar
      \edef\nextchar{\tmp}%
    \fi
  \fi\nextread}%
\def\readfilename{\bgroup%
  \let\\=\backslashother \let\%=\percentother \let\~=\tildeother
  \let\#=\sharpother \xdef\lastreadfilename{}%
  \re@d}%
%
% redefines \input using \readfilename
%
\xdef\GlobalInputList{\jobname}%
\def\psnewinput{%
  \def\readfilenamehook{% each entry in \GlobalInputList should be unique
    \if\matchexpin{\GlobalInputList}{, \lastreadfilename}%
    \else\xdef\GlobalInputList{\GlobalInputList, \lastreadfilename}%
      \immediate\write\psbj@inaux{\lastreadfilename,}%
    \fi%
    \ps@ldinput\lastreadfilename\relax%
    \let\readfilenamehook=\relax%
  }\readfilename%
}%
\expandafter\ifx\csname @@input\endcsname\relax    % then Plain
  \immediate\let\ps@ldinput=\input\def\input{\psnewinput}%
\else
  \immediate\let\ps@ldinput=\@@input
  \def\@@input{\psnewinput}%
\fi%
%
\def\nowarnopenout{%
 \def\warnopenout##1##2{%
   \readfilename##2\relax
   \message{\lastreadfilename}%
   \immediate\openout##1=\lastreadfilename\relax}}%
\def\warnopenout#1#2{%
 \readfilename#2\relax
 \def\t@mp{TrashMe,psbjoin.aux,psbjoint.tex,}\uncatcode\t@mp
 \if\matchexpin{\t@mp}{\lastreadfilename,}%
 \else
   \immediate\openin\pst@mpin=\lastreadfilename\relax
   \ifeof\pst@mpin
     \else
     \errhelp{If the content of this file is so precious to you, abort (ie
press x or e) and rename it before retrying.}%
     \errmessage{I'm just about to replace your file named \lastreadfilename}%
   \fi
   \immediate\closein\pst@mpin
 \fi
 \message{\lastreadfilename}%
 \immediate\openout#1=\lastreadfilename\relax}%
% % will have an unusual catcode below; use * instead
%\vbox
{\catcode`\%=12\catcode`\*=14
\gdef\splitfile#1{*
 \readfilename#1\relax
 \immediate\openin\j@insplitin=\lastreadfilename\relax
 \ifeof\j@insplitin
   \message{! I couldn't find and split \lastreadfilename!}*
 \else
   \immediate\openout\j@insplitout=TrashMe
   \message{< Splitting \lastreadfilename\space into}*
   \loop
     \ifeof\j@insplitin
       \immediate\closein\j@insplitin\n@teoffalse
     \else
       \n@teoftrue
       \executeinspecs{\global\read\j@insplitin to\spl@tinline\expandafter
         \ch@ckbeginnewfile\spl@tinline%Beginning-Of-File-Named:%\endcheck}*
       \ifc@ntrolline
       \else
         \toks0=\expandafter{\spl@tinline}*
         \immediate\write\j@insplitout{\the\toks0}*
       \fi
     \fi
   \ifn@teof\repeat
   \immediate\closeout\j@insplitout
 \fi\message{>}*
}*
\gdef\ch@ckbeginnewfile#1%Beginning-Of-File-Named:#2%#3\endcheck{*
 \def\t@mp{#1}*
 \ifx\@mpty\t@mp
   \def\t@mp{#3}*
   \ifx\@mpty\t@mp
     \global\c@ntrollinefalse
   \else
     \immediate\closeout\j@insplitout
     \warnopenout\j@insplitout{#2}*
     \global\c@ntrollinetrue
   \fi
 \else
   \global\c@ntrollinefalse
 \fi}*
\gdef\joinfiles#1\into#2{*
 \message{< Joining following files into}*
 \warnopenout\j@insplitout{#2}*
 \message{:}*
 {*
 \edef\w@##1{\immediate\write\j@insplitout{##1}}*
\w@{% This collection of files was produced with CERN psbox package}*
\w@{% To decompose and tex it:}*
\w@{%-save this with a filename CONTAINING ONLY LETTERS and a .TEX}*
\w@{% extension (say, JOINTFIL.TEX), in some uncrowded directory;}*
\w@{%-make sure you can \string\input\space psbox.tex (version>=1.3);}*
\w@{%  (else ftp cs.nyu.edu(=128.122.140.24):pub/TeX/psbox/, then get}*
\w@{%  and tex the file psboxall.tex; more info in psbREAD.ME)}*
\w@{%-tex JOINTFIL.TEX using Plain, or LaTeX, or whatever is needed by}*
\w@{%  the first file in the joining (after splitting JOINTFIL.TEX into}*
\w@{%  it's constituents, TeX will try to process it as it stands).}*
\w@{\string\input\space psbox.tex}*
\w@{\string\splitfile{\string\jobname}}*
\w@{\string\let\string\autojoin=\string\relax}*
}*
 \expandafter\tre@tfilelist#1, \endtre@t
 \immediate\closeout\j@insplitout
 \message{>}*
}*
\gdef\tre@tfilelist#1, #2\endtre@t{*
 \readfilename#1\relax
 \ifx\@mpty\lastreadfilename
 \else
   \immediate\openin\j@insplitin=\lastreadfilename\relax
   \ifeof\j@insplitin
     \errmessage{I couldn't find file \lastreadfilename}*
   \else
     \message{\lastreadfilename}*
     \immediate\write\j@insplitout{%Beginning-Of-File-Named:\lastreadfilename}*
     \executeinspecs{\global\read\j@insplitin to\oldj@ininline}*
     \loop
       \ifeof\j@insplitin\immediate\closein\j@insplitin\n@teoffalse
       \else\n@teoftrue
         \executeinspecs{\global\read\j@insplitin to\j@ininline}*
         \toks0=\expandafter{\oldj@ininline}*
         \let\oldj@ininline=\j@ininline
         \immediate\write\j@insplitout{\the\toks0}*
       \fi
     \ifn@teof
     \repeat
   \immediate\closein\j@insplitin
   \fi
   \tre@tfilelist#2, \endtre@t
 \fi}*
}%
% To be put at the end of a file, for making a tar-like file containing
%   everything it used.
\def\autojoin{%
 \immediate\write\psbj@inaux{\string\into{psbjoint.tex}}%
 \immediate\closeout\psbj@inaux
 \expandafter\joinfiles\GlobalInputList\into{psbjoint.tex}%
}%
%----------------------------------------------------------------
%  Annotations & Captions etc...
%
%
% \centinsert{anybox} is just a centered \midinsert, but is included as
%    people barely use the original inserts from TeX.
%
\def\centinsert#1{\midinsert\line{\hss#1\hss}\endinsert}%
\def\psannotate#1#2{\vbox{%
  \def\ps@nnotation{#2\global\let\ps@nnotation=\relax}#1}}%
\def\pscaption#1#2{\vbox{%
   \setbox\drawingBox=#1
   \copy\drawingBox
   \vskip\baselineskip
   \vbox{\hsize=\wd\drawingBox\setbox0=\hbox{#2}%
     \ifdim\wd0>\hsize
       \noindent\unhbox0\tolerance=5000
    \else\centerline{\box0}%
    \fi
}}}%
% for compatibility with older versions, but \psfig is a bad name!
%\def\psfig#1#2#3{\pscaption{\psannotate{#1}{#2}}{#3}}
%\def\psfigurebox#1#2#3{\pscaption{\psannotate{\psbox{#1}}{#2}}{#3}}
%
% \at(#1;#2)#3 puts #3 at #1-higher and #2-right of the current
%    position without moving it (to be used in annotations).
\def\at(#1;#2)#3{\setbox0=\hbox{#3}\ht0=0pt\dp0=0pt
  \rlap{\kern#1\vbox to0pt{\kern-#2\box0\vss}}}%
%
% \gridfill(ht;wd) makes a 1cm*1cm grid of ht by wd whose lower-left
%   corner is the current point
\newdimen\gridht \newdimen\gridwd
\def\gridfill(#1;#2){%
  \setbox0=\hbox to 1\pscm
  {\vrule height1\pscm width.4pt\leaders\hrule\hfill}%
  \gridht=#1
  \divide\gridht by \ht0
  \multiply\gridht by \ht0
  \gridwd=#2
  \divide\gridwd by \wd0
  \multiply\gridwd by \wd0
  \advance \gridwd by \wd0
  \vbox to \gridht{\leaders\hbox to\gridwd{\leaders\box0\hfill}\vfill}}%
%
% Useful to measure where to put annotations
\def\fillinggrid{\at(0cm;0cm){\vbox{%
  \gridfill(\drawinght;\drawingwd)}}}%
%
% \textleftof\anybox: Sample text\endtext
%   inserts "Sample text" on the left of \anybox ie \vbox, \psbox.
%   \textrightof is the symmetric (not documented, too uggly)
% Welcome any suggestion about clean wraparound macros from
%   TeXhackers reading this
%
\def\textleftof#1:{%
  \setbox1=#1
  \setbox0=\vbox\bgroup
    \advance\hsize by -\wd1 \advance\hsize by -2em}%
\def\textrightof#1:{%
  \setbox0=#1
  \setbox1=\vbox\bgroup
    \advance\hsize by -\wd0 \advance\hsize by -2em}%
\def\endtext{%
  \egroup
  \hbox to \hsize{\valign{\vfil##\vfil\cr%
\box0\cr%
\noalign{\hss}\box1\cr}}}%
%
% \frameit{\thick}{\skip}{\anybox}
%    draws with thickness \thick a box around \anybox, leaving \skip of
%    blank around it. eg \frameit{0.5pt}{1pt}{\hbox{hello}}
% \boxit{\anybox} is a shortcut.
\def\frameit#1#2#3{\hbox{\vrule width#1\vbox{%
  \hrule height#1\vskip#2\hbox{\hskip#2\vbox{#3}\hskip#2}%
        \vskip#2\hrule height#1}\vrule width#1}}%
\def\boxit#1{\frameit{0.4pt}{0pt}{#1}}%
%
%
\catcode`\@=12 % cs containing @ are unreachable
%
% CUSTOMIZE YOUR DEFAULT DRIVER:
%    Uncomment the line corresponding to your TeX system:
%\psfortextures%     For TeXtures on the Macintosh
%\psforoztex   %     For OzTeX shareware on the Macintosh
%\psfordvitops %     For the DVItoPS converter for TeX on IBM mainframes
 \psfordvips   %     For DVIPS converter on VAX and UNIX
%\psfordvitps  %     For dvitps from TeXPS package under UNIX
%\psfordvialw  %     For dvialw, UNIX public domain
%\psonlyboxes  %     Blank Boxes (when all else fails).
\begin{figure}
%\unitlength 1mm
\begin{picture}(0,40)
  \put(35,-50){\psboxscaled{680}{smooth.eps}}
\end{picture}
\begin{picture}(0,40)
  \put(150,-60){\psboxscaled{360}{fractal.eps}}
\end{picture}
\begin{picture}(0,40)
  \put(250,-82){\psboxscaled{600}{loop.eps}}
\end{picture}
\vspace{1.7 true cm}
\caption[fig:mech]
{\tenrm  Contours relevant for different regimes of dynamics.
         The left picture shows a smooth contour which accounts for
         the fermion Green function in the IR regime.
         The fractal curve in the center illustrates quantum
         mechanical particle motion in the background of violent
         field fluctuations (UV regime).
         The right picture shows a self-intersecting contour with an
         infinitesimal loop that gives rise to a universal
         infraparticle vertex function.}
\label{fig:contours}
\end{figure}
%F I G U R E   1
%
%********************************************************************

\section{IR Structure of the Fermion Propagator}
\label{irprop}

As we will be interested in the evaluation of
Eq.~(\ref{eq:fullGreen}) in the IR regime, it is sufficient to
restrict attention to uniformly curved smooth contours subject to
the condition that points close in configuration space are also close
in parameter space, i.e.,
$\tau \approx \tau '$ if $x(\tau )\approx x(\tau ')$
(see Fig.~\ref{fig:contours}).
Physically, this means that all bosonic modes with wavelengths larger
than $m^{-1}$ (large scale of the system) are absorbed into the
boson-fermion composite effective field and are invisible.
Correspondingly, in the limit $(k/m)\to 0$ ($k$ being the boson
virtuality, i.e., the small scale of the system), changes in the
fermion four-velocity $\propto k/m$ vanish and fluctuations of the
fermion momentum around the mass shell are of order $k\to 0$, so that
the (composite) fermion acquires a ``heaviness'' and the
particle-based description becomes justified.

The extension of this approach to the UV regime ($(k/m)\to \infty$)
is seemingly at odds with the common assumption that if a contour is
very complex, it can hardly be associated with a particle's
trajectory -- a semiclassical notion.
However, one may associate the motion of a charged fermion in the
background of violent field fluctuations to a {\it fractal} contour
(see Fig.~\ref{fig:contours})
that is nowhere differentiable ($|{\dot x}_{\mu}|\to \infty$).
Such contours yield additional divergences the renormalization of
which introduces angle-dependent anomalous
dimensions~\cite{Pol79,CD81}.
In addition, pair-creation/annihilation effects must be accommodated
in the formalism. In our approach, local contour discontinuities
caused by such effects can be viewed as taking place at time scales
smaller than the evolution pace $\epsilon$ that links successive
discretized copies of $I\!\!R^{d}$. Then, by appropriately gauging
the spacetime resolution scales $\alpha$ and $\epsilon$
($\alpha /\epsilon\to 1$ for $\alpha ,\epsilon\to 0$),
and by readjusting the coupling constant (and mass), discontinuous
contours are out of harm's way at distances larger than the
discretization constants.
This smoothing procedure is somewhat formal because the burden is now
placed on properly defining the new effective action emerging from
this renormalization-group (RG) type transformation. It is understood
that vacuum polarization effects will be accounted for in the fermion
determinant.

Implementing the smoothness criterion, the boson line exponential
yields \\
$
  \exp
      \{- g^{2}\int_{0}^{T}d\tau \int_{\tau}^{T}d\tau '
          {\dot x}_{\mu}(\tau ){\dot x}_{\nu}(\tau ')
          \langle
                 0 |
                     A_{\mu}[x(\tau )]A_{\nu}[x(\tau ')]
                   | 0
          \rangle
      \}
$.
The boson correlator contains short-distance singularities
$\propto |x(\tau ') - x(\tau )|^{-n}$ with $n=2,4$ when the two
contours $x(\tau )$ and $x(\tau ')$ approach each other. To render
the integrals finite, the ``ribbon regularization''
technique~\cite{Wit84} is employed, which replaces contours by the
edges of an untwisted and unwrinkled stretched-out ribbon, the width
$b$ of which plays the role of an UV-regulator:
$
  x_{\mu}(\tau )
\to
  x_{\mu}(\tau )+b n_{\mu}(\tau )
$,
$n_{\mu}(\tau )$ being a unit vector normal to the contour
direction and lying on the ribbon's surface. Note that the divergent
parts of the correlator do not depend on derivative terms, so that
they can be absorbed into an overall multiplicative renormalization
constant. Taylor-expanding contours, yields
$
  x_{\mu}(\tau ')
\simeq
  x_{\mu}(\tau ) + (\tau ' - \tau ){\dot x}_{\mu}(\tau )
$
(with $|\dot x|=1$), and the result for the boson line exponential
reads
\begin{equation}
  \left\langle
              \exp
                  \left\{
                         ig\int_{0}^{T}d\tau\, {\dot{x}}_{\mu}(\tau )
                          A_{\mu}\left[x(\tau )\right]
                  \right\}
  \right\rangle _{A}^{reg}
=
  \exp
      \left(
            - \alpha \, \frac{T}{2b} + a\ln \frac{T}{b}
            + {\rm finite} \;\, {\rm terms}
      \right)
\label{eq:gaugefin}
\end{equation}
%Eq (8) Final result for boson line exponential
from which we infer how the physical mass $m_{ren}$ varies with the
change in couplings and cutoffs:
$
  m_{r}
\buildrel b\to 0 \over =
  m(b) + \frac{\alpha}{2b}
$.
The logarithmic term entering Eq.~(\ref{eq:gaugefin}), furnishes a
wave function renormalization constant $Z_{2}$, which gives rise to
an anomalous dimension $\gamma_{F}$ for the fermion field.
Introducing a renormalization mass scale $\mu$ by
$
  \ln \frac{T}{b}
  = \ln (T\mu)  -  \ln (b\mu )
$,
we find the following all-order expressions (in the fine coupling
constant $\alpha$) which are independent of the contour:
\begin{equation}
  Z_{2}
=
  e^{-a\ln (\mu b)} \;\;\;\;\;\;  {\rm and} \;\;\;\;\;\;
  \gamma_{F}(\alpha , \lambda )
=
  \frac{\mu}{Z_{2}}\;\frac{\partial Z_{2}}{\partial \mu}
  =
  -\,a
=
  -\, \frac{\alpha}{2\pi}\,(3-\lambda ) \; .
\label{eq:wfrenconst}
\end{equation}
%Eq (9) Wave function renormalization constant Z_2 and fermion
%       anomalous dimension gamma_F
Accordingly, the full, renormalized one-fermion Green function
is~\cite{KS89,KKS92}
\begin{equation}
  G_{ren}(x,y)
=
  \mu ^{a}
  \int_{}^{}\frac{d^{4}p}{(2\pi)^{4}}\,
  e^{-ip\cdot (x-y)}\;
  \frac{\Gamma (1+a)}{\left(i\!\!\not\!p + m_{r}\right)^{1+a}} \; ,
\label{eq:Euclprop}
\end{equation}
%Eq (10) Renormalized full infrapropagator in Euclidean space
which makes apparent the infraparticle structure of the physical
fermion~\cite{BN37}.

\section{Universal Infraparticle Vertex Function}
\label{softvert}
In the light of the preceeding discussions it seems useful to
consider a nontrivial contour with an obstruction, notably a
self-intersecting loop at some point $z$. This contour corresponds
to the following generic process: an infraparticle on route along a
smooth contour with four-velocity $u_{1}(\tau )$ is suddenly derailed
at $z$ and then proceeds to propagate again on another smooth contour
characterized by four-velocity $u_{2}(\tau )$. For the sake of
simplicity and with no loss of generality, both smooth contours can
be identified with straight lines (which are the RG asymptotes).
Such a process can be adequately described by an appropriate
three-point function $\Gamma (x,y,z)$ which accounts for the
four-velocity change at $z$, and which labels initial and final
states by four-velocities.
The purpose of the infinitesimal self-intersecting loop is to
{\it simulate} the overlap of the soft boson ``clouds'' accompanying
the charged fermion during the four-velocity transition process.
This is a universal process and, under certain circumstances, it may
not depend on the particular group character of the soft
interactions. Being an effective geometric approach, relying on
contours, this treatment can only provide a {\it global} formulation
of soft interactions, while specific {\it local} features are left
out, limiting the validity range of the method. It is, however,
realistic to follow this assessment and calculate a vertex function
which enables the emulation of the Isgur-Wise form factor~\cite{IW89}
in the physically accessible kinematic
regime~\cite{SKK94,KKS95,KKS93}.

To this end, we define the three-point function
\begin{equation}
  \Gamma (x,y,z)
\buildrel c\to 0 \over =
  \int_{c}^{\infty}dT
  \int_{0}^{T}ds
  \int_{}^{}[dp(\tau )]\, {\rm tr} {\cal P}
  \exp
      \left\{
             \int_{0}^{T}d\tau \,
             \left[ip(\tau )\!\cdot \!\gamma + m \right]
      \right\}
  G(x,y,z) \; ,
\label{eq:3pf}
\end{equation}
%Eq. (11) Three-point function in particle path-integral form
where
\begin{equation}
  G(x,y,z)
\!=\!
  \int_{{x(0)=x\atop x(s)=z}\atop x(T)=y}^{} [dx(\tau )]
  \exp
      \left[
            i\!\int_{0}^{T} \!d\tau\, p(\tau )\!\cdot {\dot x}(\tau )
      \right] \!
  \left\langle
              \exp
                  \left\{
                         ig\!\int_{0}^{T}\!d\tau {\dot x}(\tau )\!
                         \cdot A[x(\tau )]
                  \right\}
  \right\rangle _{A} .
\label{eq:red3pf}
\end{equation}
%Eq. (12) Reduced three-point function in particle path-integral form
The next step is to split the interaction point according to
$
 x(s) \to [x(s_{1}), x(s_{2}]
$,
hence generating an entrance and an exit point at the vertex $z$, at
mutual distance of the order of the ribbon width $b$, and
compensating for the doubling induced on the integration measure,
i.e.,
$
 d^{4}x(s) \to d^{4}x(s_{1})d^{4}x(s_{2})
$
via
$
 \delta \left[ x(s_{1}) - x(s_{2}) \right]
$.
The net effect of the integration constraints, implemented through
the product of delta functions
$
 \delta \left[x(0) - x \right]
 \delta \left[x(T) - y \right]
 \delta \left[x(s_{1}) - x(s_{2}) \right]
 \delta \left[x(s_{1}) - z \right] ,
$
is to rearrange the contour involved in the evaluation of the
three-point function into three disjoint branches, one of which forms
a self-intersecting loop $C_{zz}$ at $z$.

Under these contour specifications, the boson correlator acquires the
form \\
$
  \left\langle
              \exp
                  \left[
                         ig\int_{L_{x,y}^{z}}^{}dw_{\mu}\;A_{\mu}(w)
                  \right]
  \right\rangle _{A}
  \left\langle
              \exp
                  \left\{
                         ig\int_{s_{1}}^{s_{2}}d\tau\,
                         {\dot x}(\tau )\cdot A[x(\tau )]
                  \right\}
  \right\rangle _{A}
$.
This is done by attaching the disjoint contour segments $L_{zx}$ and
$L_{zy}$, corresponding to four-velocities $u_{1}$ and $u_{2}$, at
the interaction point $z\pm 0^{+}$, where the four-velocity becomes
discontinuous. An appropriate coordinate rendering of $L_{x,y}^{z}$
in the interval $[0,1]$, with the interaction point $z$ removed, is
\begin{equation}
  x^{L}(t)
=
  \left\{\begin{array}{ll}
        x\left[t(T-\tilde \tau )\right] \; ,
  \;\;\;\;\;\;\;\;\;\; & 0\leq t < \tilde t \\
        x\left[t(T-\tilde \tau) + \tilde \tau \right]
  \;\;\;\;\;\;\;\;\;\; & \tilde t < t \leq T \; ,
\end{array}
\right.
\label{eq:discpar}
\end{equation}
%Eq. (13) Coordinate rendering of the broken infraparticle contour
where
$
 \tilde t
=
 s_{1}/\left(T-\tilde \tau \right)
=
  \left(s_{2}-\tilde \tau\right)/\left(T-\tilde \tau\right)
$
with
$
 \tilde \tau
=
 s_{2}-s_{1}
$
and
$
 x^{L}(\tilde t)
=
 z
$,
the interim point $\tilde t$ beeing excised from the interval
$[0,1]$.

Because the boson correlator factorizes, the three-point function
becomes the product of a purely kinematical factor $G(x,y,z)$ which
describes coexisting in and out four-velocity states (see for
details~\cite{KKS95}), and a factor $G^{(i)}(z,z)$ which encodes
information how the dynamical ``pinch'' at $z$ derails the
infraparticle and causes a discontinuity in the four-velocity, i.e.,
$
  G^{(i)}(x,y,z)
=
  G(x,y|z)\,G^{(i)}(z,z)
$.
The zeroth order term ($i=0)$ constitutes an overall kinematical
factor,
$
 G^{(0)}(z,z)=1
$,
associated to a loop that shrinks to zero upon the identification
$s_{1}=s_{2}$. The first nontrivial contribution comes from the
factor $G^{(1)}(z,z)$ which corresponds to a self-crossing loop
(see Fig.~\ref{fig:contours}) in the particle's contour to be entered
and exited {\it smoothly} with four-velocities $u_{1}$ and $u_{2}$,
respectively.
It thus appears that this term is responsible, in a geometric sense,
for the transport of an intact soft boson ``cloud'' through the
interaction point.

A uniform parametrization of the loop integral
$
 \oint_{{C}_{zz}^{(1)}}^{}dx_{\mu}A_{\mu}(x)
$
is provided by
$
  \tau
\to
  t
=
  \tau - \left(s_{1} + s_{2}\right)/2
$
accompanied by the coordinate readjustment
$
  x(\tau )
\to
  x^{c}(t)
=
  x
   \left[ + \left(s_{1} + s_{2}\right)/2\right]
$,
where $t$ is restricted in the interval
$[-\frac{\tilde \tau}{2},\frac{\tilde \tau}{2}]$ and
$\tilde \tau = s_{2} - s_{1}$.
Then, all gauge-dependent terms vanish identically, as expected, and
employing again the ribbon regularization, the final result for the
boson line exponential is~\cite{KKS95}
\begin{equation}
  I(\theta )
\simeq
 -\;
     \left(g^{2}/4\pi ^{2}\right) \,
     \ln \left({\tilde \tau}/2b\right) \,
     {\pi\theta}/\sqrt{1-\theta^{2}} \; ,
\label{eq:Itheta}
\end{equation}
%Eq (14) Nontrivial loop contribution as a function of theta in
%        ribbon regularization
where $\theta = u_{1}\!\cdot u_{2}$.
In order to disentangle IR from UV effects, we make a RG scale
readjustment to tune the coupling constant to values relevant
in the IR domain:
$
  g^{2}(b^{2})\,\ln ({\tilde \tau}/2b)
=
  g^{2}(\mu ^{2})\,\ln ({\tilde \tau}\mu )
=
  g^{2}(\mu ^{2})\,\ln k
$.
One may view $k$ as the parameter which relates the scales
characterizing long-  and short-distance regimes.
The result for the three-point function after renormalization is
\begin{equation}
  \Gamma _{ren}^{(1)}(x,y,z)
=
  -\exp
       \left(
             -\frac{g^{2}}{4\pi}\,\frac{\theta}{\sqrt{1-\theta ^{2}}}
              \, \ln k
       \right)
  \Gamma _{ren}^{(0)}(x,y,z) \; .
\label{eq:Gammaren}
\end{equation}
%Eq. (15) Renormalized three-point function with loop contribution
The nonperturbative part of the vertex function is obtained by
removing kinematical contributions according to
\begin{equation}
  F(x,y,z)
=
  \frac{\Gamma_{ren}^{(0)}(x,y,z)
       +\Gamma_{ren}^{(1)}(x,y,z)}
  {\Gamma_{ren}^{(0)}(x,y,z)}
\label{eq:nonpertvertex}
\end{equation}
%Eq. (16) Removal of kinematics in the vertex function
to get the following {\it universal} (i.e., mass- and
contour-independent) expression
\begin{equation}
  F(x,y,z)
=
  F(\theta )
=
  1 - \exp
          \left(
                -\frac{g^{2}}{4\pi}\;
                 \frac{\theta}{\sqrt{1-\theta^{2}}}\;\ln k
          \right) \; ,
\label{eq:finvertex}
\end{equation}
%Eq (17) Vertex function in terms of the invariant velocity change
which can be analytically continued to Minkowski space by replacing
$
  \theta
\to
  w
\equiv
      {\tilde u}_{\mu}^{1}\,{\tilde u}^{2\mu}
  = \frac{1}{\theta} \; ,
$
%Eq (17) Continuation of the four-velocity change to Minkowski space
where ${\tilde u^{1}}$ and ${\tilde u^{2}}$ are the Minkowski
counterparts of $u^{1}$ and $u^{2}$, respectively~\cite{KKS95}.
The connection to the leading-order Isgur-Wise form factor is
established by replacing $e^{2}$ by $(4/3)g^{2}_{s}$ and adapting the
parameter $k$ to scales involved in heavy-meson decays by setting
$
  k
=
  \frac{m_{Q}}{\bar\Lambda}
$,
where $m_{Q}$ is the heavy-quark mass, and ${\bar\Lambda}$
characterizes the energy scale of the light degrees of freedom.
Then the final result reads
\begin{equation}
  \xi (w)
=
  1\;-\;\exp
            \left(
                  -\;\frac{4}{3}\;\alpha _{s}\;
                   \frac{1}{\sqrt{w^{2}-1}}\;\ln k
            \right) \; .
\label{eq:WISGUR}
\end{equation}
%Eq (18) Final result for the vertex function (Isgur-Wise form
%        factor)
In the kinematic region accessible to semileptonic decays
($1\le w\le 1.6$), and using rather typical values of the involved
parameters~\cite{Neu94}, {\it viz.},
$k\approx 8$ and $\alpha _{s}\simeq 0.21$,
this first-principles vertex function yields remarkable agreement with
recent data of different experimental groups
(for an explicit comparison, see~\cite{SKK94,KKS95}).

\section{Conclusions}
\label{concl}

This article tries to draw a unifying view on {\it universal}
features of soft interactions using an approach which lends itself
by construction to the renormalization group and relies upon particle
path integrals. Arguments are given how to pave the way for
generalizations of the framework and extensions to the UV regime.
\itemsep0pt
\parsep10mm
\bigskip

\centerline{\bf References}
\begin{thebibliography}{99}
\bibitem{SKK94}N. G. Stefanis, A. I. Karanikas, C. N. Ktorides,
               in {\it Proceedings of the International Conference on
               Hadron Structure '94}, Ko\v sice, Slovakia, 1994,
               edited by J.~Urb\'an and J.~Vrl\'akov\'a
               (Ko\v sice, Slovakia, 1994), p. 134.
%[1]
\bibitem{KKS95}A. I. Karanikas, C. N. Ktorides, N. G. Stefanis,
               {\it Phys. Rev.} {\bf D52} (1995) 5898.
%[2]
\bibitem{Str92}M. J. Strassler,
               {\it Nucl. Phys.} {\bf B385} (1992) 145.
%[3]
\bibitem{Neu94}M. Neubert,
               {\it Phys. Reports} {\bf 245} (1994) 259.
%[4]
\bibitem{Sch63}B. Schroer,
               {\it Fortschr. Phys.} {\bf 11} (1963) 1.
%[5]
\bibitem{KS89}A. Kernemann, N. G. Stefanis,
              {\it Phys. Rev.} {\bf D40} (1989) 2103.
%[6]
\bibitem{KK92}A. I. Karanikas, C. N. Ktorides,
              {\it Phys. Lett.} {\bf B275} (1992) 403;
              {\it Int. J. Mod. Phys.} {\bf A7} (1992) 5563;
              {\it Phys. Rev.} {\bf D52} (1995) 5883.
%[7]
\bibitem{Kad77}L. P. Kadanoff,
               {\it Rev. Mod. Phys.} {\bf 49} (1977) 267.
%[8]
\bibitem{Pol79}A. M. Polyakov,
               {\it Nucl. Phys.} {\bf B164} (1979) 171.
%[9]
\bibitem{CD81}N. S. Craigie, H. Dorn,
              {\it Nucl. Phys.} {\bf B185} (1981) 204.
%[10]
\bibitem{Wit84}E. Witten,
               {\it Commun. Math. Phys.} {\bf 92} (1984) 451.
%[11]
\bibitem{KKS92}A. I. Karanikas, C. N. Ktorides, N. G. Stefanis,
               {\it Phys. Lett.} {\bf B289} (1992) 176.
%[12]
\bibitem{BN37}F. Bloch, A. Nordsieck,
              {\it Phys. Rev.} {\bf 52} (1937) 54.
%[13]
\bibitem{IW89}N. Isgur, M. Wise,
              {\it Phys. Lett.} {\bf B232} (1989) 113;
              {\bf B237} (1990) 527.
%[14]
\bibitem{KKS93}A. I. Karanikas, C. N. Ktorides, N. G. Stefanis,
               {\it Phys. Lett.} {\bf B301} (1993) 397.
%[15]
\end{thebibliography}
\end{document}

