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%|              N. G. Stefanis et al.                               |
%|              Title: ``WORLDLINE DESCRIPTION OF THE ISGUR-WISE    |
%|                       FUNCTION''                                 |
%|              Institut fuer Theoretische Physik II,        5pp.   |
%|              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         |
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\begin{document}
\hfill RUB-TPII-11/96
\bigskip
\bigskip
\bigskip

\centerline{\tenbf WORLDLINE DESCRIPTION OF THE ISGUR-WISE FUNCTION
\footnote{Invited talk presented by the first author at the
          International Workshop
          {\it Quark Confinement and the Hadron Spectrum~II},
          June 26-29, 1996, Villa Olmo, Como, Italy.}}
\baselineskip=22pt
\vspace{0.8cm}
\centerline{\tenrm N. G. STEFANIS
\footnote{E-mail: nicos@hadron.tp2.ruhr-uni-bochum.de}
           }
\baselineskip=13pt
\centerline{\tenit Institut f\"ur Theoretische Physik II,
                   Ruhr-Universit\"at Bochum}
\baselineskip=12pt
\centerline{\tenit D-44780 Bochum, Germany}
\vspace{0.3cm}
%\centerline{\tenrm and}
\vspace{0.3cm}
\centerline{\ninerm A. I. KARANIKAS
\footnote{E-mail: akaranik@atlas.uoa.gr}$\;$
and C. N. KTORIDES
\footnote{E-mail: cktorid@atlas.uoa.gr}
           }
\baselineskip=13pt
\centerline{\nineit University of Athens, Department of Physics,
                    NPPS, Panepistimiopolis}
\baselineskip=12pt
\centerline{\nineit GR-15771 Athens, Greece}
\vspace{0.9cm}
\abstracts{\eightrm
An effective field theoretic description of the Isgur-Wise
function in heavy-meson transitions is presented which emulates
soft interactions by a fermion worldline with an infinitesimal
self-intersecting loop.
A point-splitting regularization technique is used, which replaces
pointlike worldlines by ``ribbons'' in the sense of Witten.
The calculated vertex function is correctly normalized, does not
depend on the heavy-quark mass, and complies with different sets
of recent experimental data of the ARGUS and CLEO collaborations.
          }
\newpage

\vfil
%\vspace{0.8cm}
\rm\baselineskip=14pt
\tenrm
\section{\indent {\tenbf Introduction}}
\label{sec:INTRO}
In recent years a whole body of knowledge has been developed about
the weak decays of hadrons containing a single heavy quark
(see, e.g., Ref.~1). %\cite{Neu94}
The strong interactions of such a heavy quark with light quarks and
gluons can be described by an effective field theory which is
invariant under changes of the flavor and the spin of the heavy quark
and involves a single universal vertex function (form factor),
introduced by Isgur and Wise,~\cite{IW89} which contains the strong
interactions of quarks and gluons.
This function is independent of the heavy quark mass and depends only
on (the invariant product of) the four-velocities of the initial and
final state mesons. Lacking an analytical understanding of the
confinement dynamics, the Isgur-Wise function cannot be calculated
from first principles within Quantum Chromodynamics (QCD).
Current nonperturbative calculations utilize QCD sum rules or lattice
simulations, employing heavy-quark symmetry.

In this paper we report on an alternative computation~\cite{KKS93}
which attempts at emulating the Isgur-Wise form factor by a universal
vertex function, responsible for the elastic scattering of an
infraparticle.~\cite{Sch63}
Technically, this is achieved by recasting the field theoretical
system into particle-based language,~\cite{KS89,KK92} and employing
in the evaluation of the particle path integral a self-intersecting
worldline (contour) with an infinitesimal loop. The latter serves to
simulate the transport of the intact soft boson ``cloud'' through the
interaction point, and gives rise to an {\tenit exclusive} form
factor.
An introduction to the basics of this approach has been given
in Ref.~7. %\cite{SKK94}
Full accounts of the formalism were published in Ref.~8; %\cite{KKS95}
more recent developments are discussed in Ref.~9. %\cite{Ste96}
%\newpage

\section{\indent {\tenbf Worldline Approach to the Isgur-Wise
                         Function}}
\label{sec:WISGUR}
A way as to make possible a first-principles calculation of the
Isgur-Wise function relies in the present report on the
assumption~\cite{Ste96} that {\tenit global} characteristics of the
soft QCD dynamics may be treated within an Abelian setup of
infraparticle worldlines.
{\tenit Local} aspects that are directly connected to the specific
features of $SU(3)_{c}$ are left out, limiting the validity range
of the method.
On the other hand, this approach avoids typical problems encountered
by {\tenit explicitly} resumming Feynman graphs and obtains results
by {\tenit implicitly} taking into account an infinite number of such
contributions within a low-energy {\tenit effective}
theory.~\cite{KKS95,Ste96}

Let us now turn to the following generic process:
an infraparticle propagates from point $x$ along a smooth worldline,
say a straight line, with four-velocity $u_{1}(\tau )$ to point $z$,
where it is suddenly derailed by a dynamical ``pinch'', and then
proceeds to propagate on another smooth (straight) worldline,
characterized by a four-velocity $u_{2}(\tau )$, until it reaches
point $y$.
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:
\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 (1) 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\{
                         i g \!\int_{0}^{T}\!d\tau {\dot x}(\tau )\!
                               \cdot A[x(\tau )]
                  \right\}
  \right\rangle _{A} .
\label{eq:red3pf}
\end{equation}
%Eq (2) Reduced three-point function in particle path-integral form

Dissecting the particle's worldline into three disjoint branches
$L_{xz^{-}}$, $C_{zz}$, and $L_{z^{+}y}$ (where $z^{\pm}=z\pm 0^{+}$
to denote that the crossing point $z$ has been excised), the
expectation value of the Wilson line acquires the form
$
  \left\langle
              \exp
                  \left[
                        i g\int_{L_{x,y}^{z}}^{}dw_{\mu}\;A_{\mu}(w)
                  \right]
  \right\rangle _{A}
  \left\langle
              \exp
                  \left\{
                         i g \int_{s_{1}}^{s_{2}}d\tau\,
                             {\dot x}(\tau )\cdot A[x(\tau )]
                  \right\}
  \right\rangle _{A}
$.
The contribution associated with the overlap of the soft boson
``clouds'' during the four-velocity transition process is encoded in
the factor involving the loop $C_{zz}$ (left panel of
Fig.~\ref{fig:softvertex}), the latter to be entered and exited
{\tenit smoothly} with four-velocities $u_{1}$ and $u_{2}$,
respectively.
Accordingly, 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, described by
infrapropagators,~\cite{KKS92} and a factor $G^{(1)}(z,z)$ which is
associated with a single loop that shrinks to zero (see for
details Ref.~8). %\cite{KKS95}
The analogous situation for the heavy meson transition in QCD is
shown in the right panel of Fig.~\ref{fig:softvertex}. Here the
rearrangement of the light degrees of freedom during a four-velocity
change gives rise to form-factor suppression, described by the
{\tenit universal} Isgur-Wise function.

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\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(22,-70){\psboxscaled{660}{loop.eps}}
\end{picture}
\begin{picture}(0,40)
  \put(220,-60){\psboxscaled{600}{mesondec.eps}}
\end{picture}
\vspace{1.7 true cm}
\caption[fig:cont]
{\eightrm  The left panel shows an infraparticle contour with an
           infinitesimal loop which serves to emulate the soft
           interactions during a heavy-meson transition. The latter
           is illustrated by the graph in the right panel.}
\label{fig:softvertex}
\end{figure}
%F I G U R E   1
%
%********************************************************************

A uniform parametrization of the loop integral
$
 \oint_{{C}_{zz}^{(1)}}^{}dx_{\mu}
 A_{\mu}[x(\tau )]
$
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 Witten's ribbon regularization,~\cite{Wit84} 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 (3) 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 pertaining
to 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
$.
The parameter $k$ may be viewed as relating long- and short-distance
regimes.
Performing renormalization, the result for the three-point function
reads
\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 (4) Renormalized three-point function with loop contribution
from which 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 (5) Removal of kinematics in the vertex function
The final result is the following {\tenit 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 (6) Vertex function in terms of the invariant velocity change
which can be analytically continued to Minkowski space~\cite{KKS95}
by replacing
$
  \theta
\to
  w
\equiv
      {\tilde u}_{\mu}^{1}\,{\tilde u}^{2\mu}
  = \frac{1}{\theta} \; ,
$
where ${\tilde u^{1}}$ and ${\tilde u^{2}}$ are the Minkowski
counterparts of $u^{1}$ and $u^{2}$, respectively.
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
=
 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.
In this way we arrive at
\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 (7) Final result for the vertex function (Isgur-Wise form factor)
The result of this first-principles computation is correctly
normalized $\left(\xi (w=1)=1\right)$, and yields for the rather
typical values
$k\approx 8$ and $\alpha _{s}\simeq 0.21$ the model form
\begin{equation}
  \xi (w)
=
  1\; - \;
  \exp
      \left(
            -\frac{0.582}{\sqrt{w^{2} - 1}}
      \right) \; .
\label{eq:model}
\end{equation}
%Eq (8) Model form of the infraparticle vertex function

This vertex function is shown in Fig.~\ref{fig:wisgur} in comparison
with other theoretical calculations together with the experimental
values reported by different collaborations
(see Ref.~8 %\cite{KKS95}
for details and references).
In the kinematic region accessible to semileptonic decays
($1\le w\le 1.6$), Eq.~(\ref{eq:model}) has physical characteristics
close to those expected for the ``true'' Isgur-Wise function, except
for the slope at zero recoil which turns out to violate the
Bjorken bound. This is due to the Abelian setup, which is too
simplifying a scheme to accomodate all nonperturbative phenomena of
QCD.

%********************************************************************
%*                         F I G U R E  2                           *
%********************************************************************
%
%\input psbox.tex
\begin{figure}%[h]
%\unitlength 1mm
\begin{picture}(0,40)
  \put(110,-130){\psboxscaled{500}{wisgurn.eps}}
\end{picture}
\vspace{4.5 true cm}
\caption[fig:data]
{\eightrm  Selected theoretical predictions in comparison with recent
           experimental data of different collaborations indicated in
           the figure. The curves correspond to our result (solid line),
           and to those obtained by Isgur~\cite{Isg91} (dotted line),
           Radyushkin~\cite{Rad91} (dashed-dotted line), and
           Neubert~\cite{Neu94} (dashed line).}
\label{fig:wisgur}
\end{figure}
%F I G U R E   2
%
%********************************************************************
\newpage

\section{\indent {\tenbf Conclusions}}
\label{sec:CONCL}
The investigations presented here indicate that our low-energy
effective approach, which is based on the quantum mechanical particle
path integral rather than Feynman graphs in field theory, is useful
for the nonperturbative calculation of quantities relating to soft
interactions.
The extension of the method to the ultraviolet regime, sketched
in Ref.~9 %\cite{Ste96}
in connection with fractal contours, seems also feasible.
Work in this direction is in progress.
\bigskip

\noindent{\tenbf References}
\begin{thebibliography}{99}
\bibitem{Neu94}M. Neubert,
               {\tenit Phys. Reports} {\tenbf 245}, 259 (1994).
%[1]
\bibitem{IW89}N. Isgur and M. Wise,
              {\tenit Phys. Lett.} {\tenbf B232}, 113 (1989);
              {\tenbf B237}, 527 (1990).
%[2]
\bibitem{KKS93}A.I. Karanikas, C.N. Ktorides, and N.G. Stefanis,
               {\tenit Phys. Lett.} {\tenbf B301}, 397 (1993).
%[3]
\bibitem{Sch63}B. Schroer,
               {\tenit Fortschr. Phys.} {\tenbf 11}, 1 (1963).
%[4]
\bibitem{KS89}A. Kernemann and N.G. Stefanis,
              {\tenit Phys. Rev.} {\tenbf D40}, 2103 (1989).
%[5]
\bibitem{KK92}A.I. Karanikas and C.N. Ktorides,
              {\tenit Phys. Lett.} {\tenbf B275}, 403 (1992);
              {\tenit Int. J. Mod. Phys.} {\tenbf A7}, 5563 (1992);
              {\tenit Phys. Rev.} {\tenbf D52}, 5883 (1995).
%[6]
\bibitem{SKK94}N.G. Stefanis, A.I. Karanikas, and C.N. Ktorides,
               in {\tenit Proceedings of the International Conference
               on Hadron Structure '94}, Ko\v sice, Slovakia, 1994,
               edited by J. Urb\'an and J. Vli\'akov\'a
               (Ko\v sice, Slovakia, 1994), p. 134.
%[7]
\bibitem{KKS95}A.I. Karanikas, C.N. Ktorides, and N.G. Stefanis,
               {\tenit Phys. Rev.} {\tenbf D52}, 5898 (1995).
%[8]
\bibitem{Ste96}N.G. Stefanis,
               in {\tenit International Conference on Problems of
               Quantum Field Theory},
               Alushta, Crimea, Ukraine, May 13-18, 1996.
               Bochum preprint RUB-TPII-08/96 (June 1996),
               [\hbox, 8 July, 1996].
%[9]
\bibitem{Wit84}E. Witten,
               {\tenit Commun. Math. Phys.} {\tenbf 92}, 451 (1984).
%[10]
\bibitem{Isg91}N. Isgur,
               {\tenit Phys. Rev.} {\tenbf D43}, 810 (1991).
%[11]
\bibitem{Rad91}A.V. Radyushkin,
               {\tenit Phys. Lett.} {\tenbf B271}, 218 (1991).
%[12]
\bibitem{KKS92}A.I. Karanikas, C.N. Ktorides, and N.G. Stefanis,
               {\tenit Phys. Lett.} {\tenbf B289}, 176 (1992).
%[13]
\end{thebibliography}
\end{document}

