%
% acknowledgments.tex
%
\begin{Large}
\begin{center}
%
{\bf ACKNOWLEDGMENTS}
%
\end{center}
\end{Large}

I am most grateful to Prof.~H.~G.~Dosch for the friendly supervision
of this thesis, for his strong support, and for the many interesting
and intensive discussions. I had many invaluable opportunities to
learn non-perturbative methods in QCD from one of the experts. I also
enjoyed very much his lectures on ``The Physics of Music.''

Next I want to thank Prof.~H.~J.~Pirner for the close and fruitful
collaboration, for his interest in my work, and for the very many
creative and instructive ideas. I also appreciated very much his
suggestions on the preliminary versions of this work. It has been a
pleasure to attend his interesting lectures and seminars.

I would like to thank Arif Shoshi, with whom I collaborated most
closely during large parts of this investigation, for countless
discussions and for the vivid exchange of ideas that helped enormously
to master the model in all its details.

I thank Prof.~B.~Povh for his friendly interest and readiness to
referee this thesis.

I want to express my sincere gratitude to Prof.~O.~Nachtmann for his
continuous willingness to clarify and help in subtle issues, for
inviting me to participate regularly in the pleasant Wednesday lunch
of his group, and for the letter of recommendation.

I am thankful to Prof.~J.~H\"ufner for many suggestions that have
helped to find the right perspective in many cases and for his
hospitality in the theoretical nuclear physics group. With his
invitation, I enjoyed very often the pleasant Friday afternoon tea.

I would like to thank Dr.~Carlo Ewerz for his proofreading, the many
suggestions, and for bringing birdtrack notation to my attention.
Dr.~Hilmar Forkel deserves a special thank for his magnificent lecture
on ``Instantons in QCD'' and many illuminating discussions. I am also
grateful to Dr.~Matthias Jamin for helping me to better understand
renormalization in QCD. Moreover, I thank Dr.~Peter John for advice in
computational issues, Dr.~Eduard Thommes and his crew for
administrative support, and Prof.~W.~Wetzel for providing an extremely
stable and reliable computer environment. For careful readings of the
manuskipt I would like to express my gratitude to Sonja Bartsch and
Felix Schwab.

For helpful introductions to the stochastic vacuum model (SVM), useful
computer code, and notes on analytic SVM computations, I thank
Dr.~Edgar Berger, Dr.~Gerhard Kulzinger, Dr.~Timo Paulus, and
Dr.~Steffen Weinstock. Volker Schatz is thanked for help in
mathematical problems.

I enjoyed many stimulating discussions related to this thesis for
which I thank Dr.~Nora Brambilla, Prof.~A.~Di Giacomo, Dr.~Alberto
Polleri, Prof.~E.~Meggiolaro, Prof.~A.~H.~Mueller, Dr.~J\"org
Raufeisen, Prof.~I.~Stamatescu, and Dr.~Antonio Vairo.

It is a great pleasure to work at the Institute for Theoretical
Physics in Heidelberg. I would like to thank the many friendly,
open-minded, and helpful members for making this institute a unique
place with an extraordinarily pleasant atmosphere. I enjoyed very much
to be part of the crew under the roof. I thank my colleagues Lala
Adueva (``party party''), Dr.~Tobias Baier, Juliane Behrend, Dr.~Eike
Bick (thanks for the nice gatherings and the excellent menues),
Dr.~Michael Doran (``daradada dadada daaaahhh''), Dr.~Markus
Eidem\"uller, Bj\"orn Feuerbacher, Dietrich Foethke, Gero von
Gersdorff, J\"org J\"ackel, Bj\"orn O.\ Lange, Christian M.\ M\"uller,
Markus M.\ M\"uller, Filipe Paccetti, Tassilo Ott, Tania Robens,
Gregor Sch\"afer, Xaver Schlagberger, Kai Schwenzer (sorry for the
``Badegumpen'' trip), Jan Schwindt and Claus Zahlten for many cheerful
conversations in and outside of physics. I am particularly grateful to
Joachim Holk for the delicious tee and the many kind Fanta Pina de
Coco offers during our night sessions. It was always pleasant not to
be alone with the cat at two o' clock in the morning. My sincere
thanks goes to Mrs.~G.~Rumpf for countless friendly conversations and
the valuable permanent catering service. Felix Schwab deserves a
special thank not only for the Bruce tickets but also for being the
best drummer in the institute.

My sincere thanks to the Graduiertenkolleg ``Physical Systems with
many Degrees of Freedom'' for supporting me since January 2000 and for
providing an ideal framework to organize our own workshops and
schools. In particular, I want to thank Prof.~F.~Wegner for his trust
in our organization of the autumn school ``Topology and Geometry in
Physics.'' I am extremely grateful to Prof.~J.-W.~van Holten,
Prof.~F.~Lenz, Prof.~T.~Sch\"ucker, Prof.~M.~Shifman, and
Prof.~J.~Zinn-Justin for making this autumn school a very special
event and for writing such highly pedagogical lecture notes for the
proceedings.

Now I come to the world outside of the physics. I have enjoyed very
much the sports program at the University of Heidelberg and the many
nice conversations with the friendly participants. I particularly
thank Matthias Wolf for the excellent TAEBO specials that helped a lot
in relaxing from physics. Moreover, I would like to thank Janine
Fritz, Angela Klein, Verena Schmidt-Steffens, Edgitha Stork, Zhe Xu
for their loyal friendship during all these years.

In a very special way I want to thank Natascha Kunert for her love,
all the wonderful moments, and her patience during the many weekends I
spent at the institute. I am also indebted to her family for the
continuous warm-hearted friendship and hospitality.

Finally, I am extremely thankful to my mother and my father for their
care and love. It is always a great pleasure to be at home.

%%% Local Variables: 
%%% mode: latex
%%% TeX-master: "master"
%%% End: 
% ComparisonData.tex
%
% ___ Comparison with Experimental Data ________________________________________
\chapter{Comparison with Experimental Data}
\label{Sec_Comparison_Data}
% ______________________________________________________________________________

In this chapter we present the phenomenological performance of our
model. We compute total, differential, and elastic cross sections,
structure functions, and diffractive slopes for hadron-hadron,
photon-proton, and photon-photon scattering, compare the results with
experimental data including cosmic ray data, and provide predictions
for future experiments. Having studied the saturation of the impact
parameter profiles, we show here how this manifestation of unitarity
translates into the quantities mentioned above and how it could become
observable.

Using the
$T$-matrix~(\ref{Eq_model_purely_imaginary_T_amplitude_final_result})
with the wave functions and parameters from
Secs.~\ref{Sec_Wave_Functions} and~\ref{Sec_Model_Parameters}, we
compute the {\em pomeron} contribution to $pp$, $p\pbar$,
$\pi^{\pm}p$, $K^{\pm}p$, $\gamma^{*} p$, and $\gamma \gamma$
reactions in terms of the universal dipole-dipole scattering amplitude
$S_{DD}$. This allows one to compare reactions induced by hadrons and
photons in a systematic way. In fact, it is our aim to provide a
unified description of all these reactions and to show in this way
that the pomeron contribution to the above reactions is universal and
can be traced back to the dipole-dipole scattering amplitude $S_{DD}$.

Our model describes pomeron ($C=+1$ gluon exchange) but neither
odderon ($C=-1$ gluon exchange) nor reggeon exchange (quark-antiquark
exchange) as discussed in Sec.~\ref{Sec_DD_Scattering_LLCM}. Only in the
computation of the hadronic total cross sections the reggeon
contribution is added~\cite{Donnachie:1992ny,Donnachie:2000kp}. This
improves the agreement with the data for $\sqrt{s} \ltsim 100\,\GeV$
and describes exactly the differences between $ab$ and $\bar{a}b$
reactions.

The fine tuning of the model and wave function parameters was
performed on the data shown below. The resulting parameter set given
in Secs.~\ref{Sec_Wave_Functions} and~\ref{Sec_Model_Parameters}
is used throughout this chapter.

% ______________________________________________________________________________
\section{Total Cross Sections}
\label{Sec_Total_Cross_Sections}
% _____________________________________________________________________________

The total cross section for the high-energy reaction $ab \to X$ is
related via the {\em optical theorem} to the imaginary part of the
forward elastic scattering amplitude and can also be expressed in
terms of the profile function~(\ref{Eq_profile_function_def})
%
\be
        \sigma^{tot}_{ab}(s) 
        \;=\; \inv{s}\,\im\,T(s, t=0) 
        \;=\; 2 \int \!d^2b_{\!\perp}\,J_{ab}(s,|\vec{b}_{\!\perp}|)
        \ ,  
\label{Eq_optical_theorem}
\ee
%
where $a$ and $b$ label the initial particles whose masses were
neglected as they are small in comparison to the c.m.\ energy
$\sqrt{s}$. 

We compute the pomeron contribution to the total cross section,
$\sigma^{tot, \Pomeron}_{ab}(s)$, from the
$T$-matrix~(\ref{Eq_model_purely_imaginary_T_amplitude_final_result}),
as explained above, and add only here a reggeon contribution of the
form~\cite{Donnachie:1992ny,Donnachie:2000kp}
%
\be
        \sigma^{tot, \Reggeon}_{ab}(s)
        = X_{ab}\, \left(  \frac{s}{1\,\GeV^2} \right)^{-0.4525} 
        \ ,
\label{Eq_DL_reggeon_contribution}
\ee
%
where $X_{ab}$ depends on the reaction considered: $X_{pp} =
56.08\,\mb$, $X_{p\pbar} = 98.39\,\mb$, $X_{\pi^+p} = 27.56\,\mb$,
$X_{\pi^-p} = 36.02\,\mb$, $X_{K^+p} = 8.15\,\mb$, $X_{K^-p} =
26.36\,\mb$, $X_{\gamma p} = 0.129\,\mb$, and $X_{\gamma \gamma} =
605\,\nb$. Accordingly, we obtain the total cross section
%
\be
        \sigma^{tot}_{ab}(s)
        = \sigma^{tot, \Pomeron}_{ab}(s) 
        + \sigma^{tot, \Reggeon}_{ab}(s)
\label{Eq_total_cross_section_final_result}
\ee
%
for $pp$, $p\pbar$, $\pi^{\pm}p$, $K^{\pm}p$, $\gamma p$, and $\gamma
\gamma$ scattering.

The good agreement of the computed total cross sections with the
experimental data is shown in Fig.~\ref{Fig_sigma_tot}.
% 
\begin{figure}[p]
\setlength{\unitlength}{1.cm}
\begin{center}
\epsfig{file=sigtot_pp_kp_pip.eps,width=14.5cm}
\end{center}
\caption{ \small 
  The total cross section $\sigma^{tot}$ as a function of the c.m.\ 
  energy $\sqrt{s}$ for $pp$, $p\pbar$, $\pi^{\pm}p$, $K^{\pm}p$,
  $\gamma p$, and $\gamma \gamma$ scattering.  The solid lines
  represent the model results for $pp$, $\pi^+p$, $K^+p$, $\gamma p$
  and $\gamma \gamma$ scattering and the dashed lines the ones for
  $p\pbar$, $\pi^-p$, and $K^-p$ scattering. The $pp$, $p\pbar$,
  $\pi^{\pm}p$, $K^{\pm}p$, $\gamma p$~\cite{Hagiwara:fs} and $\gamma
  \gamma$ data~\cite{Abbiendi:2000sz+X} taken at accelerators are
  indicated by the closed circles while the closed squares (Fly's eye
  data)~\cite{Baltrusaitis:1984ka+X} and the open circles (Akeno
  data)~\cite{Honda:1993kv+X} indicate cosmic ray data.  The arrows at
  the top point to the LHC energy, $\sqrt{s} = 14\,\TeV$, and to the
  onset of the black disc limit in $pp$ ($p\pbar$) reactions,
  $\sqrt{s} \approx 10^6\,\GeV$.}
\label{Fig_sigma_tot}
\end{figure}
%
Here the solid lines represent the theoretical results for $pp$,
$\pi^+p$, $K^+p$, $\gamma p$, and $\gamma \gamma$ scattering and the
dashed lines the ones for $p\pbar$, $\pi^-p$, and $K^-p$ scattering.
The $pp$, $p\pbar$, $\pi^{\pm}p$, $K^{\pm}p$, $\gamma
p$~\cite{Hagiwara:fs} and $\gamma \gamma$
data~\cite{Abbiendi:2000sz+X} taken at accelerators are indicated by
the closed circles while the closed squares (Fly's eye
data)~\cite{Baltrusaitis:1984ka+X} and the open circles (Akeno
data)~\cite{Honda:1993kv+X} indicate cosmic ray data. Only real
photons are considered which are, of course, transverse polarized.

The prediction for the total $pp$ cross section at LHC ($\sqrt{s} =
14\,\TeV$) is $\sigma^{tot}_{pp} = 114.2\,\mb$ in good agreement with
cosmic ray data. Compared with other works, our LHC prediction is
close to the one of Block et al.~\cite{Block:1999hu},
$\sigma^{tot}_{pp} = 108 \pm 3.4\,\mb$, but considerably larger than
the one of Donnachie and Landshoff~\cite{Donnachie:1992ny},
$\sigma^{tot}_{pp} = 101.5\,\mb$.

The differences between $ab$ and $\bar{a}b$ reactions for $\sqrt{s}
\ltsim 100\,\GeV$ result solely from the different reggeon
contributions which die out rapidly as the energy increases. The
pomeron contribution to $ab$ and $\bar{a}b$ reactions is, in
contrast, identical and increases as the energy increases. It thus
governs the total cross sections for $\sqrt{s} \gtsim 100\,\GeV$ where
the results for $ab$ and $\bar{a}b$ reactions coincide.

The differences between $pp$ ($p\pbar$), $\pi^{\pm}p$, and $K^{\pm}p$
scattering result from the different transverse extension parameters,
$S_p = 0.86\,\fm > S_{\pi} = 0.607\,\fm > S_{K} = 0.55\,\fm$, cf.\ 
Sec.~\ref{Sec_Wave_Functions}.  Since a smaller transverse extension
parameter favors smaller dipoles, the total cross section becomes
smaller and the short distance physics described by the perturbative
component becomes more important which leads to a stronger energy
growth due to $\epsilon^{\pert} = 0.73 > \epsilon^{\nprt} = 0.125$. In
fact, the ratios $\sigma^{tot}_{pp}/\sigma^{tot}_{\pi p}$ and
$\sigma^{tot}_{pp}/\sigma^{tot}_{Kp}$ converge slowly towards unity
with increasing energy as can already be seen in
Fig.~\ref{Fig_sigma_tot}.

For real photons, the transverse size is governed by the constituent
quark masses $m_f(Q^2=0)$, cf.\ Sec.~\ref{Sec_Wave_Functions},
where the light quarks have the strongest effect, i.e.\ 
$\sigma^{tot}_{\gamma p} \propto 1/m_{u,d}^2$ and
$\sigma^{tot}_{\gamma \gamma} \propto 1/m_{u,d}^4$. Furthermore, in
comparison with hadron-hadron scattering, there is the additional
suppression factor of $\alphaEM$ for $\gamma p$ and $\alphaEM^2$ for
$\gamma \gamma$ scattering coming from the photon-dipole transition.
In the $\gamma \gamma$ reaction, also the box diagram
contributes~\cite{Budnev:1975zs,Donnachie:2000kp} but is neglected
since its contribution to the total cross section is less than
1\%~\cite{Donnachie:2001wt}.

It is worthwhile mentioning that total cross sections for $pp$
($p\pbar$), $\pi^{\pm}p$, and $K^{\pm}p$ scattering do not depend on
the longitudinal quark momentum distribution in the hadrons since the
underlying dipole-dipole cross section is independent of the
longitudinal quark momentum fraction $z_i$ for $t = 0$. We have shown
this analytically on the two-gluon-exchange level
in~\cite{Shoshi:2002fq}.

Saturation effects as a manifestation of the $S$-matrix unitarity can
be seen in Fig.~\ref{Fig_sigma_tot}. Having investigated the profile
function for $pp$ ($p\pbar$) scattering, we know that this profile
function becomes higher and broader with increasing energy until it
saturates the black disc limit first for zero impact parameter
($|\vec{b}_{\!\perp}|=0$) at $\sqrt{s} \approx 10^6\,\GeV$.
Beyond this energy, the profile function cannot become higher but
expands towards larger values of $|\vec{b}_{\!\perp}|$. Consequently,
the total cross section~(\ref{Eq_optical_theorem}) increases no longer
due to the growing blackness at the center but only due to the
transverse expansion of the hadrons. This tames the growth of the
total cross section as can be seen for the total $pp$ cross section
beyond $\sqrt{s} \approx 10^6\,\GeV$ in Fig.~\ref{Fig_sigma_tot}.

At energies beyond the onset of the black disc limit at zero impact
parameter, the profile function can be approximated by
%
\be 
      J_{ab}^{\mathrm{approx}}(s,|\vec{b}_{\!\perp}|) =
      N_a\,N_b\,\Theta\left(R(s)-|\vec{b}_{\!\perp}|\right) \,
\label{Eq_J_ab_asymptotic_energies}
\ee
%
where $N_{a,b}$ denotes the normalization of the wave functions of the
scattered particles and $R(s)$ the full width at half maximum of the
exact profile function $J_{ab}(s,|\vec{b}_{\!\perp}|)$ that reflects
the effective radii of the interacting particles. Thus, the energy
dependence of the total cross section~(\ref{Eq_optical_theorem}) is
driven exclusively by the increase of the transverse extension of the
particles $R(s)$
%
\be
        \sigma^{tot}_{ab}(s) = 2 \pi N_a N_b R(s)^2
        \ ,
\label{Eq_sigma_tot_ab_asymptotic_energies}
\ee
%
which is known as {\em geometrical
  scaling}~\cite{Amaldi:1980kd,Castaldi:1983ft}. The growth of $R(s)$
can at most be logarithmic for $\sqrt{s} \to \infty$ because of the
Froissart bound~\cite{Froissart:1961ux}. In fact, a transition from a
power-like to an $\ln^2$-increase of the total $pp$ cross section
seems to set in at about $\sqrt{s} \approx 10^6\,\GeV$ as visible in
Fig.~\ref{Fig_sigma_tot}. Moreover, since the hadronic cross sections
join for $\sqrt{s} \to \infty$, $R(s)$ becomes independent of the
hadron-hadron reaction considered at asymptotic energies as long as
$N_{a,b}=1$. Also for photons of different virtuality $Q_1^2$ and
$Q_2^2$ one can check that the ratio of the total cross sections
$\sigma^{tot}_{\gamma^* p}(Q_1^2)/\sigma^{tot}_{\gamma^* p}(Q_2^2)$
converges to unity at asymptotic energies in agreement with the
conclusion in~\cite{Schildknecht:2001qe}.

% ______________________________________________________________________________
\section{The Proton Structure Function}
\label{Sec_Structure_Functions}
% ______________________________________________________________________________

The total cross section for the scattering of a transverse ($T$) and
longitudinally ($L$) polarized photon off the proton,
$\sigma_{\gamma^*_{T\!,L}p}^{tot}(x,Q^2)$, at photon virtuality $Q^2$
and c.m.\ energy\footnote{Here $\sqrt{s}$ refers to the c.m.\ energy
  in the $\gamma^* p$ system.} squared $s=Q^2/x$ is equivalent to the
{\em structure functions} of the proton
%
\be
        F_{T,L}(x,Q^2) 
        = \frac{Q^2}{4\pi^2\alphaEM} 
        \sigma_{\gamma^*_{T\!,L}p}^{tot}(x,Q^2)
\label{Eq_FTL}
\ee
%
and
%
\be
        F_2(x,Q^2) = F_{T}(x,Q^2) + F_{L}(x,Q^2)
        \ .
\label{Eq_F2}
\ee
%

Reactions induced by virtual photons are particularly interesting
because the transverse separation of the quark-antiquark pair that
emerges from the virtual photon decreases as the photon virtuality
increases (cf.\ Sec.~\ref{Sec_Wave_Functions})
%
\be
        |\vec{r}_\gamma| \approx \frac{2}{\sqrt{Q^2+4m_{u,d}^2}} 
        \ ,
\label{pts}
\ee
%
where $m_{u,d}$ is a mass of the order of the constituent $u$-quark
mass. With increasing virtuality, one probes therefore smaller
transverse distance scales of the proton.

In Fig.~\ref{Fig_sigma_tot_gp_vs_Q^2} the $Q^2$-dependence of the
total $\gamma^* p$ cross section
%
\be
        \sigma^{tot}_{\gamma^*p}(s,Q^2)
        = \sigma^{tot}_{\gamma_T^*p}(s,Q^2)
        + \sigma^{tot}_{\gamma_L^*p}(s,Q^2)
\label{Eq_sigma_tot_gp_=_sigma_T_+_sigma_L}
\ee
%
is presented, where the model results (solid lines) are compared with
the experimental data for c.m.\ energies from $\sqrt{s} = 20\,\GeV$ up
to $245\,\GeV$. Note the indicated scaling factors at different
$\sqrt{s}$ values. The low-energy data at $\sqrt{s} = 20\,\GeV$ are
from~\cite{Benvenuti:1989rh+X} while the data at higher energies have
been measured at HERA by the H1~\cite{Aid:1996au,Adloff:1997mf} and ZEUS
collaboration~\cite{Derrick:1996ef+X,Breitweg:1997hz}. At $Q^2 = 0.012\,\GeV^2$, also
the photoproduction ($Q^2=0$) data from~\cite{Caldwell:1978yb+X} are
displayed.
%
\begin{figure}[p]
  \centerline{\psfig{figure=sigtot_gp_gvirt.eps,
      width=13cm}}\bigskip 
\protect\caption{\small 
%
  The total $\gamma^* p$ cross section
  $\sigma^{tot}_{\gamma^*p}(s,Q^2)$ as a function of the photon
  virtuality $Q^2$ for c.m.\ energies from $\sqrt{s} = 20\,\GeV$ to
  $245\,\GeV$, where the model results (solid lines) and the
  experimental data at different $\sqrt{s}$ values are scaled with the
  indicated factors. The low energy data at $\sqrt{s} = 20\,\GeV$ are
  from~\cite{Benvenuti:1989rh+X}, the data at higher energies from the
  H1~\cite{Aid:1996au,Adloff:1997mf} and ZEUS
  collaboration~\cite{Derrick:1996ef+X,Breitweg:1997hz}. The
  photoproduction ($Q^2=0$) data from~\cite{Caldwell:1978yb+X} are
  displayed at $Q^2 = 0.012\,\GeV^2$.
%
}
\label{Fig_sigma_tot_gp_vs_Q^2}
\end{figure}
%

The model results are in reasonable agreement with the experimental
data in the window shown in Fig.~\ref{Fig_sigma_tot_gp_vs_Q^2}. The
total $\gamma^* p$ cross section levels off towards small values of
$Q^2$ as soon as the photon size $|\vec{r}_\gamma|$, i.e\ the
resolution scale, becomes comparable to the proton size. Our model
reproduces this behavior by using the perturbative photon wave
functions with $Q^2$-dependent quark masses, $m_f(Q^2)$, that
interpolate between the current (large $Q^2$) and the constituent
(small $Q^2$) quark masses as explained in detail in
Sec.~\ref{Sec_Wave_Functions}. The decrease of $\sigma^{tot}_{\gamma^*
  p}$ with increasing $Q^2$ results from decreasing dipole sizes and
the fact that small dipoles do not interact as strongly as large
dipoles.

The $x$-dependence of the computed proton structure function
$F_2(x,Q^2)$ is shown in Fig.~\ref{F2_p} for $Q^2 = 0.3,\,2.5,\,12,$
and $120\,\GeV^2$ in comparison to the data measured by the
H1~\cite{Abt:1993cb+X} and ZEUS~\cite{Derrick:1993ft+X} detector.
Within our model, the increase of $F_2(x,Q^2)$ towards small Bjorken
$x$ becomes stronger with increasing $Q^2$ in agreement with the trend
in the HERA data. This behavior results from the fast energy growth of
the perturbative component that becomes more important with increasing
$Q^2$ due to the smaller dipole sizes involved.
%
\begin{figure}[h]
\centerline{\psfig{figure=F2_p_rise.eps, width=11.cm}}
\caption{\small 
  The $x$-dependence of the computed proton structure function
  $F_2(x,Q^2)$ (solid line) for $Q^2 = 0.3,\,2.5,\,12,$ and
  $120\,\GeV^2$ in comparison to the data measured by the
  H1~\cite{Abt:1993cb+X} and ZEUS~\cite{Derrick:1993ft+X} detector,
  and the low energy data at $\sqrt{s} = 20\,\GeV$
  from~\cite{Benvenuti:1989rh+X}.}
\label{F2_p}
\end{figure}
%

As can be seen in Fig.~\ref{F2_p}, the data show a faster increase
with decreasing $x$ than the model outside the low-$Q^2$ region. This
results from the weak energy boost of the non-perturbative component
that dominates $F_2(x,Q^2)$ in our model. In fact, even for large
$Q^2$ the non-perturbative contribution overwhelms the perturbative
one, which explains also the overestimation of the data for $x \gtsim
10^{-3}$.
 
This problem is typical for the \SVM\ model applied to the scattering
of a small size dipole off a proton. In an earlier application by
R\"uter~\cite{Rueter:1998up}, an additional cut-off was introduced to
switch from the non-perturbative to the perturbative contribution as
soon as one of the dipoles is smaller than $r_{cut} = 0.16\,\fm$. This
yields a better agreement with the data also for large $Q^2$ but leads
to a discontinuous dipole-proton cross section. In the model of
Donnachie and Dosch~\cite{Donnachie:2001wt}, a similar \SVM-based
component is used also for dipoles smaller than $R_c = 0.22\,\fm$ with
a strong energy boost instead of a perturbative component.
Furthermore, their \SVM-based component is tamed for large $Q^2$ by an
additional $\alphaS(Q^2)$ factor.

We did not follow these lines in order to keep a continuous,
$Q^2$-independent dipole-proton cross section and, therefore, cannot
improve the agreement with the $F_2(x,Q^2)$ data without losing
quality in the description of hadronic observables. Since our
non-perturbative component relies on lattice QCD, we are more
confident in describing non-perturbative physics and thus put more
emphasis on the hadronic observables. Admittedly, our model misses
details of the proton structure that become visible with increasing
$Q^2$. However, most other existing models provide neither the profile
functions nor a simultaneous description of hadronic and
$\gamma^*$-induced processes.

% ______________________________________________________________________________
\section[The Slope $B$ of Elastic Forward Scattering]
{The Slope \boldmath$B$ of Elastic Forward Scattering}
\label{Sec_Slope_B}
% ______________________________________________________________________________

The {\em local slope} of elastic scattering $B(s,t)$ is defined as
%
\be
        B(s,t) := 
        \frac{d}{dt} \left( \ln \left[ \frac{d\sigma^{el}}{dt}(s,t) \right] \right)
\label{Eq_elastic_local_slope}
\ee
%
and characterizes the diffractive peak of the differential elastic
cross section $d\sigma^{el}/dt(s,t)$ discussed below. Here we
concentrate on the slope for elastic forward ($t=0$) scattering also
called {\em slope parameter}
%
\be
        B(s) 
        := B(s,t=0) 
        = \inv{2} 
        \frac{\int\!d^2b_{\!\perp}\,|\vec{b}_{\!\perp}|^2\,J(s,|\vec{b}_{\!\perp}|)}
        {\int\!d^2b_{\!\perp}\,J(s,|\vec{b}_{\!\perp}|)}
        = \inv{2} \langle b^2 \rangle \ ,
\label{Eq_elastic_forward_slope}
\ee
%
which measures the root mean squared interaction radius $\langle b^2
\rangle$ of the scattered particles, and does not depend on the
opacity.

We compute the slope parameter with the profile function from the
$T$-matrix~(\ref{Eq_model_purely_imaginary_T_amplitude_final_result})
and neglect the reggeon contributions, which are relevant only at
small c.m.\ energies, so that the same result is obtained for $ab$ and
$\bar{a}b$ scattering.

In Fig.~\ref{Fig_B_pp} the resulting slope parameter $B(s)$ is shown
as a function of $\sqrt{s}$ for $pp$ and $p\pbar$ scattering (solid
line) and compared with the $pp$ (open circles) and $p\pbar$ (closed
circles) data from~\cite{Amaldi:1971kt+X,Bozzo:1984ri,Amos:1989at}.
%
\begin{figure}[t] 
\centerline{\epsfig{figure=B_pp.eps,width=10.cm}}
\caption{\small
  The elastic slope parameter $B(s)$ as a function of the c.m.\ energy
  $\sqrt{s}$ for $pp$ and $p\pbar$ forward ($t=0$) scattering. The
  solid line represents the model result that is compared to the data
  for $pp$ (open circles) and $p\pbar$ (closed circles) reactions
  from~\cite{Amaldi:1971kt+X,Bozzo:1984ri,Amos:1989at}.}
\label{Fig_B_pp}
\end{figure}
%
As expected from the opacity independence of the slope parameter
(\ref{Eq_elastic_forward_slope}), saturation effects as seen in the
total cross sections do not occur. Indeed, one observes an approximate
$B(s) \propto R^2(s) \propto \ln^2(\sqrt{s}/\sqrt{s_0})$ growth for
$\sqrt{s} \gtsim 10^4\,\GeV$.  This behavior agrees, of course, with
the transverse expansion of $J_{pp}(s,|\vec{b}_{\!\perp}|)$ for
increasing $\sqrt{s}$ shown in Fig.~\ref{Fig_J_pp(b,s)}. Analogous
results are obtained also for $\pi p$ and $Kp$ scattering.

For the good agreement of our model with the data, the finite width of
the longitudinal quark momentum distribution in the hadrons, i.e.\ 
$\Delta z_p,\,\Delta z_{\pi},\,\mbox{and}\,\Delta z_{K}\neq 0$
in~(\ref{Eq_hadron_wave_function}), is important as the numerator in
(\ref{Eq_elastic_forward_slope}) depends on this width. In fact,
$B(s)$ comes out more than 10\% smaller with $\Delta z_p,\,\Delta
z_{\pi},\,\mbox{and}\,\Delta z_{K}= 0$. Furthermore, a strong growth
of the perturbative component, $\epsilon^{\pert} = 0.73$, is important
to achieve the increase of $B(s)$ for $\sqrt{s} \gtsim 500\,\GeV$
indicated by the data.

It must be emphasized that only the simultaneous fit of the total
cross section and the slope parameter provides the correct shape of
the profile function $J(s,|\vec{b}_{\!\perp}|)$. This shape leads then
automatically to a good description of the differential elastic cross
section $d\sigma^{el}/dt(s,t)$ as demonstrated below. Astonishingly,
only few phenomenological models provide a satisfactory description of
both quantities~\cite{Block:1999hu,Kopeliovich:2001pc}. In the
approach of~\cite{Berger:1999gu}, for example, the total cross section
is described correctly while the slope parameter exceeds the data by
more than 20\% already at $\sqrt{s} = 23.5\,\GeV$ and thus indicates
deficiencies in the form of $J(s,|\vec{b}_{\!\perp}|)$.

% ______________________________________________________________________________
\section{The Differential Elastic Cross Section}
\label{Sec_Diff_El_Cross_Section}
% ______________________________________________________________________________

The {\em differential elastic cross section} obtained from the squared
absolute value of the $T$-matrix element
%
\be
        \frac{d\sigma^{el}}{dt}(s,t) 
        = \inv{16 \pi s^2}|T(s,t)|^2
\label{Eq_dsigma_el_dt}
\ee
%
can be expressed for our purely imaginary
$T$-matrix~(\ref{Eq_model_purely_imaginary_T_amplitude_final_result})
in terms of the profile function
%
\be
        \frac{d\sigma^{el}}{dt}(s,t) 
        = \inv{4\pi} \left[ 
        \int \!\!d^2b_{\!\perp}\,
        e^{i {\vec q}_{\!\perp} {\vec b}_{\!\perp}}\,
        J(s,|\vec{b}_{\!\perp}|)
        \right ]^2
        \ .
\label{Eq_dsigma_el_dt_model}
\ee
%
and is, thus, very sensitive to the transverse extension {\em and}
opacity of the scattered particles.
Equation~(\ref{Eq_dsigma_el_dt_model}) shows the analogy to optical
diffraction, where $J(s,|\vec{b}_{\!\perp}|)$ describes the
distribution of an absorber that causes the diffraction pattern
observed for incident plane waves.

In Fig.~\ref{Fig_dsigma_el_dt_pp} the differential elastic cross
section computed for $pp$ and $p\pbar$ scattering (solid line) is
shown as a function of $|t|=\vec{q}^{\,2}_{\!\perp}$ at $\sqrt{s} =
23.5,\,30.7,\,44.7,\,63,\,546$, and $1800\,\GeV$ and compared with
experimental data (open circles), where the $pp$ data at $\sqrt{s} =
23.5,\,30.7,\,44.7,\,\mbox{and}\,63\,\GeV$ were measured at the CERN
ISR~\cite{Amaldi:1980kd}, the $p\pbar$ data at $\sqrt{s} = 546\,\GeV$
at the CERN $Sp{\pbar}S$~\cite{Bozzo:1984ri}, and the $p\pbar$ data at
$\sqrt{s} = 1.8\,\TeV$ at the Fermilab
Tevatron~\cite{Amos:1989at,Amos:1990jh}. The prediction of our model
for the $pp$ differential elastic cross section at the CERN LHC,
$\sqrt{s} = 14\,\TeV$, is given in Fig.~\ref{Fig_dsigma_el_dt_pp_LHC}.
%
\begin{figure}[p]
  \centerline{\psfig{figure=dsigdt_pp.eps,width=13.5cm}} 
\protect\caption{ \small 
%
  The differential elastic cross section for $pp$ and $p\pbar$
  scattering as a function of $|t|$. The result of our model,
  indicated by the solid line, is compared for $\sqrt{s} =
  23.5,\,30.7,\,44.7,\,\mbox{and}\,63\,\GeV$ with the CERN ISR $pp$
  data~\cite{Amaldi:1980kd}, for $\sqrt{s} = 546\,\GeV$ with the CERN
  $Sp{\pbar}S$ data~\cite{Bozzo:1984ri}, and for $\sqrt{s} =
  1.8\,\TeV$ with the Fermilab Tevatron $p\pbar$
  data~\cite{Amos:1989at,Amos:1990jh}, all indicated by the open
  circles with error bars.
%
}
\label{Fig_dsigma_el_dt_pp}
\end{figure}
%
\begin{figure}[tb]
  \centerline{\psfig{figure=dsigdt_pp_W14TeV.eps,width=9.cm}}
  \protect\caption{ \small 
%
    The prediction of our model for the $pp$ differential elastic
    cross section at LHC ($\sqrt{s} = 14\,\TeV$) as a
    function of momentum transfer $|t|$ up to $1\,\GeV^2$.
%
}
\label{Fig_dsigma_el_dt_pp_LHC}
\end{figure} 
%

For all energies, the model reproduces the experimentally observed
diffraction pattern, i.e\ the characteristic {\em diffraction peak} at
small $|t|$ and the {\em dip} structure at medium $|t|$. As the energy
increases, also the {\em shrinking of the diffraction peak} is
described which reflects the rise of the slope parameter $B(s,t=0)$
already discussed above. The shrinking of the diffraction peak comes
along with a dip structure that moves towards smaller values of
$|t|$ as the energy increases. This motion of the dip is also
described approximately.

The dip in the theoretical curves reflects a change of sign in the
$T$-matrix
element~(\ref{Eq_model_purely_imaginary_T_amplitude_final_result}). As
the latter is purely imaginary, it is not surprising that there are
deviations from the data in the dip region. Here  the real part is
expected to be important~\cite{Amos:1990jh} which is in the small
$|t|$ region negligible in comparison to the imaginary part.

The difference between the $pp$ and $p\pbar$ data, a deep dip for $pp$
but only a bump or shoulder for $p\pbar$ reactions, requires a $C = -
1$ contribution. Besides the reggeon contribution at small
energies,\footnote{Zooming in on the result for $\sqrt{s} =
  23.5\,\GeV$, one finds an underestimation of the data for all $|t|$
  before the dip, which is correct as it leaves room for the reggeon
  contribution being non-negligible at small energies.} one expects
here an additional perturbative $C=-1$ contribution such as
three-gluon exchange~\cite{Fukugita:1979fe,Donnachie:1984hf+X} or an
odderon~\cite{Lukaszuk:1973nt+X,Rueter:1999gj,Dosch:2002ai}. In fact,
allowing a finite size diquark in the (anti-)proton an odderon appears
that supports the dip in $pp$ but leads to the shoulder in $p\pbar$
reactions~\cite{Dosch:2002ai}.

For the differential elastic cross section at the LHC energy,
$\sqrt{s} = 14\,\TeV$, the above findings suggest an accurate
prediction in the small-$|t|$ region but a dip at a position smaller
than the predicted value at $|t| \approx 0.35\,\GeV^2$. Our confidence
in the validity of the model at small $|t|$ is supported additionally
by the total cross section that fixes $d\sigma^{el}/dt(s,t=0)$ and is
in agreement with the cosmic ray data shown in
Fig.~\ref{Fig_sigma_tot}. Concerning our prediction for the dip
position, it is close to the value $|t| \approx 0.41\,\GeV^2$
of~\cite{Block:1999hu} but significantly below the value $|t| \approx
0.55\,\GeV^2$ of~\cite{Berger:1999gu}. Beyond the dip position, the
height of the computed shoulder is always above the data and, thus,
very likely to exceed also the LHC data. In comparison with other
works, the height of our shoulder is similar to the one
of~\cite{Block:1999hu} but almost one order of magnitude above the one
of~\cite{Berger:1999gu}.

Considering Figs.~\ref{Fig_dsigma_el_dt_pp}
and~\ref{Fig_dsigma_el_dt_pp_LHC} more quantitatively in the
small-$|t|$ region, one can use the well known parametrization of the
differential elastic cross section in terms of the slope parameter
$B(s)$ and the {\em curvature} $C(s)$
%
\be
        d\sigma^{el}/dt(s,t) 
        = d\sigma^{el}/dt(s,t=0)\,\exp\left[B(s)t+C(s)t^2\right]  
        \ .
\label{Eq_dsigma_el_dt_exp_parameterization}
\ee
%
Using $B(s)$ from the preceding section and assuming for the moment
$C(s) = 0$, one achieves a good description at small momentum
transfers and energies, which is consistent with the approximate
Gaussian shape of $J_{pp}(s,|\vec{b}_{\!\perp}|)$ at small energies
shown in Fig.~\ref{Fig_J_pp(b,s)}. The dip, of course, is generated by
the deviation from the Gaussian shape at small impact parameters.
According to (\ref{Eq_dsigma_el_dt_exp_parameterization}), the
shrinking of the diffraction peak with increasing energy simply
reflects the increasing interaction radius described by $B(s)$.

For small energies $\sqrt{s}$, our model reproduces the experimentally
observed change in the slope at $|t| \approx
0.25\,\GeV^2$~\cite{Barbiellini:1972ua+X} that is characterized by a
positive curvature. For LHC, we find clearly a negative value for the
curvature in agreement with~\cite{Block:1999hu} but in contrast
to~\cite{Berger:1999gu}. The change of sign in the curvature reflects
the transition of $J(s,|\vec{b}_{\!\perp}|)$ from the approximate
Gaussian shape at low energies to the approximate step-function
shape~(\ref{Eq_J_ab_asymptotic_energies}) at high energies.

Important for the good agreement with the data is the longitudinal
quark momentum distribution in the proton. Besides the slope
parameter, which characterizes the diffraction peak, also the dip
position is very sensitive to the distribution width $\Delta z_p$,
i.e.\ with $\Delta z_p= 0$ the dip position appears at more than 10\%
lower values of $|t|$. In the earlier \SVM\ 
approach~\cite{Berger:1999gu}, the reproduction of the correct dip
position was possible without the $z$-dependence of the hadronic wave
functions but a deviation from the data in the low-$|t|$ region had to
be accepted. In this low-$|t|$ region, we achieved a definite
improvement with the new correlation
functions~(\ref{Eq_SVM_correlation_functions}) and the minimal
surfaces used in our model.
%
\begin{figure}[htb]
  \centerline{\psfig{figure=dsigdt_Kp_pip.eps,width=11.cm}} 
%
  \protect\caption{ \small 
%
    The differential elastic cross section $d\sigma^{el}/dt(s,t)$ as a
    function of $|t|$ for $\pi^{\pm}p$ and $K^{\pm}p$ reactions at a
    c.m.\ energy of $\sqrt{s} = 19.5\,\GeV$. The model results (solid
    line) are compared with the data (closed circles with error bars)
    from~\cite{Akerlof:1976gk+X}.}
\label{Fig_dsigdt_kp_pip}
\end{figure} 
%

The differential elastic cross section computed for $\pi^{\pm}p$ and
$K^{\pm}p$ reactions has the same behavior as the one for $pp$
($p\pbar$) reactions. The only difference comes from the different
$z$-distribution widths, $\Delta z_{\pi}$ and $\Delta z_{K}$, and the
smaller extension parameters, $S_{\pi}$ and $S_{K}$, which shift the
dip position to higher values of $|t|$. This is illustrated in
Fig.~\ref{Fig_dsigdt_kp_pip}, where the model results (solid line) for
the $\pi^{\pm}p$ and $K^{\pm}p$ differential elastic cross section are
shown at $\sqrt{s} = 19.5\,\GeV$ as a function of $|t|$ in comparison
to experimental data (closed circles) from~\cite{Akerlof:1976gk+X}.
The deviations from the data towards large $|t|$ leave room for
odderon and reggeon contributions. Indeed, with a finite diquark size
in the proton, an odderon occurs that improves the description of the
data at large values of $|t|$~\cite{Berger:PhDthesis:1999}.

% ______________________________________________________________________________
\section[The Elastic Cross Section $\sigma^{el}$, $\sigma^{el}/ \sigma^{tot}$ and $\sigma^{tot}/B$]{The Elastic Cross Section \boldmath$\sigma^{el}$, \boldmath$\sigma^{el}/ \sigma^{tot}$ and \boldmath$\sigma^{tot}/B$}
% ______________________________________________________________________________

The {\em elastic cross section} obtained by integrating the
differential elastic cross section
%
\be
        \sigma^{el}(s) 
        = \int_0^{-\infty}\!dt\,\frac{d\sigma^{el}}{dt}(s,t) 
        = \int_0^{-\infty}\!dt\,\inv{16 \pi s^2}|T(s,t)|^2
\label{Eq_total_elastic_cross_section}
\ee
%
reduces for our purely imaginary
$T$-matrix~(\ref{Eq_model_purely_imaginary_T_amplitude_final_result})
to
%
\be
        \sigma^{el}(s) 
        = \int \!\!d^2b_{\!\perp}\,|J(s,|\vec{b}_{\!\perp}|)|^2 
        \ .
\label{Eq_total_elastic_cross_section_J}
\ee
%
Due to the squaring, it exhibits the saturation of
$J(s,|\vec{b}_{\!\perp}|)$ at the black disc limit more clearly than
$\sigma^{tot}(s)$. Even more transparent is the saturation in the
following ratios given here for a purely imaginary $T$-matrix
%
\bea
        \frac{\sigma^{el}}{\sigma^{tot}}(s) 
        & = & 
        \frac
        {\int\!d^2b_{\!\perp}\,|J(s,|\vec{b}_{\!\perp}|)|^2}
        {2\int\!d^2b_{\!\perp}\,J(s,|\vec{b}_{\!\perp}|)}
        \ ,
\label{Eq_sigma_el/sigma_tot} \\
        \frac{\sigma^{tot}}{B}(s) 
        & = & 
        \frac
        {\left(2\int\!d^2b_{\!\perp}\,J(s,|\vec{b}_{\!\perp}|)\right)^2}
        {\int\!d^2b_{\!\perp}\,|\vec{b}_{\!\perp}|^2\,J(s,|\vec{b}_{\!\perp}|)} 
\label{Eq_sigma_tot/B}
        \ ,
\eea
%
which are directly sensitive to the opacity of the particles. This
sensitivity can be illustrated within the approximation 
%
\be
        T(s,t) = i\, s\, \sigma^{tot}(s)\, \exp[B(s) t/2]
\label{Eq_T_matrix_exp_parameterization}
\ee
%
that leads to the differential cross
section~(\ref{Eq_dsigma_el_dt_exp_parameterization}) with $C(s) = 0$,
i.e.\ an exponential decrease over $|t|$ with a slope $B(s)$. As the
purely imaginary $T$-matrix
element~(\ref{Eq_T_matrix_exp_parameterization}) is equivalent to
%
\be
        J(s,|\vec{b}_{\!\perp}|)
        =(\sigma^{tot}/4\pi B)\,\exp[-|\vec{b}_{\!\perp}|^2/2B] 
        =(4\sigma^{el}/\sigma^{tot})\,\exp[-|\vec{b}_{\!\perp}|^2/2B]
        \ ,
\label{Eq_J(b,s)_exp_parameterization}
\ee
%
one finds that the above ratios are a direct measure for the opacity
at zero impact parameter
%
\be
        J(s,|\vec{b}_{\!\perp}|=0) 
        = (\sigma^{tot}/4\pi B)
        = (4\sigma^{el}/\sigma^{tot}) 
        \ .
\label{Eq_J(b=0,s)_exp_parameterization}
\ee
%
For a general purely imaginary $T$-matrix, $T(s,t) =
i\,s\,\sigma^{tot}\,g(|t|)$ with an arbitrary real-valued function
$g(|t|)$, $J(s,|\vec{b}_{\!\perp}|=0)$ is given by
$\sigma^{el}/\sigma^{tot}$ times a pure number which depends on the
shape of $g(|t|)$.

We compute the elastic cross section $\sigma^{el}$ and the ratios
$\sigma^{el}/ \sigma^{tot}$ and $\sigma^{tot}/B$ in our model without
taking into account reggeons. In Fig.~\ref{Fig_sigtot_el_and_ratios}
the results for $pp$ and $p\pbar$ reactions (solid lines) are compared
with the experimental data (open and closed circles). The data for the
elastic cross section are taken from~\cite{Hagiwara:fs} and the data
for $\sigma^{tot}$ and $B$ from the references given in previous
sections.
%
\begin{figure}[p]
  \centerline{\psfig{figure=sigel_sigtot_B_pp.eps, width=10.cm}}
  \protect\caption{ \small The elastic cross section $\sigma^{el}$ and
    the ratios $\sigma^{el}/ \sigma^{tot}$ and $\sigma^{tot}/B$ as a
    function of the c.m.\ energy $\sqrt{s}$. The model results for
    $pp$ ($p\pbar$), $\pi p$, and $K p$ scattering are represented by
    the solid, dashed and dotted lines, respectively. The experimental
    data for the $pp$ and $p\pbar$ reactions are indicated by the open
    and closed circles, respectively. The data for the elastic cross
    section are taken from~\cite{Hagiwara:fs} and the data for
    $\sigma^{tot}$ and $B$ from the references given in previous
    sections.}
\label{Fig_sigtot_el_and_ratios}
\end{figure}
%
For $pp$ ($p\pbar$) scattering, we indicate explicitly the prediction
for LHC at $\sqrt{s}=14\,\TeV$ and the onset of the black disc limit
at $\sqrt{s} = 10^6\,\GeV$. The model results for $\pi p$ and $K p$
reactions are presented by the dashed and dotted line, respectively.
For the ratios, the asymptotic limits are indicated: Since the maximum
opacity or black disc limit governs the $\sqrt{s} \to \infty$
behavior, $\sigma^{el}/\sigma^{tot}$ ($\sigma^{tot}/B$) cannot exceed
$0.5$ ($8\pi$) in hadron-hadron scattering.

In the investigation of $pp$ ($p\pbar$) scattering, our theoretical
curves successfully confront the experimental data for the elastic
cross section and both ratios. At low energies, the data are
underestimated as reggeon contributions are not taken into account.
Again, the agreement is comparable to the one achieved
in~\cite{Block:1999hu} and better than in the approach
of~\cite{Berger:1999gu}, where $\sigma^{el}$ comes out too small due
to an underestimation of $d\sigma^{el}/dt$ in the low-$|t|$ region.

Concerning the energy dependence, $\sigma^{el}$ shows a similar
behavior as $\sigma^{tot}$ but with a more pronounced flattening
around $\sqrt{s} \gtsim 10^6\,\GeV$. This flattening is even stronger
for the ratios, drawn on a linear scale, and reflects very clearly the
onset of the black disc limit. As expected from the simple
approximation~(\ref{Eq_J(b=0,s)_exp_parameterization}),
$\sigma^{el}/\sigma^{tot}$ and $\sigma^{tot}/B$ show a similar
functional dependence on $\sqrt{s}$. At the highest energy shown,
$\sqrt{s} = 10^8\,\GeV$, both ratios are still below the indicated
asymptotic limits, which reflects that $J(s,|\vec{b}_{\!\perp}|)$
still deviates from the step-function
shape~(\ref{Eq_J_ab_asymptotic_energies}). The ratios
$\sigma^{el}/\sigma^{tot}$ and $\sigma^{tot}/B$ reach their upper
limits $0.5$ and $8\pi$, respectively, at asymptotic energies,
$\sqrt{s} \to \infty$, where the hadrons become infinitely large,
completely black discs.

Comparing the $pp$ ($p\pbar$) results with the ones for $\pi p$ and
$Kp$, one finds that the results for $\sigma^{tot}/B$ converge at high
energies as shown in Fig.~\ref{Fig_sigtot_el_and_ratios}. This follows
from the identical normalizations of the hadron wave functions that
lead to an identical black disc limit for hadron-hadron reactions.

%%% Local Variables: 
%%% mode: latex
%%% TeX-master: "master"
%%% End: 
% EuclideanScattering.tex
%
% ______________________________________________________________________________
\chapter{Euclidean Approach to High-Energy Scattering}
\label{Sec_DD_Scattering}
% ______________________________________________________________________________

In this chapter we present a Euclidean approach to high-energy
reactions of color dipoles in the eikonal approximation. We give a
short review of the functional integral approach to high-energy
scattering, which is the basis for the presented Euclidean approach
and for our investigations of hadronic high-energy reactions in the
following chapters. We generalize the analytic continuation introduced
by Meggiolaro~\cite{Meggiolaro:1996hf+X} from parton-parton scattering
to dipole-dipole scattering. This shows how one can access high-energy
reactions directly in lattice QCD. We apply this approach to compute
the scattering of dipoles in the fundamental and adjoint
representation of $SU(N_c)$ at high-energy in the Euclidean LLCM. The
result shows the consistency with the analytic continuation of the
gluon field strength correlator used in all earlier applications of
the SVM and LLCM to high-energy scattering. Finally, we comment on the
QCD van der Waals potential which appears in the limiting case of two
static color dipoles.

% ______________________________________________________________________________
\section[Functional Integral Approach to High-Energy Scattering]
{\!\!\!\!\!Functional Integral Approach to High-Energy\! Scattering}
\label{Sec_Functional_Integral_Approach}
% ______________________________________________________________________________

In {\em Minkowski space-time} high-energy reactions of color dipoles
in the eikonal approximation are considered -- as basis for
hadron-hadron, photon-hadron, and photon-photon reactions -- in the
functional integral approach to high-energy collisions developed
originally for parton-parton
scattering~\cite{Nachtmann:1991ua+X,Nachtmann:ed.kt} and then extended
to gauge-invariant dipole-dipole
scattering~\cite{Kramer:1990tr,Dosch:1994ym,Dosch:RioLecture}. The
corresponding $T$-matrix element for the elastic scattering of two
color dipoles at transverse momentum transfer ${\vec q}_{\!\perp}$ ($t
= -{\vec q}_{\!\perp}^{\,\,2}$) and c.m.\ energy squared~$s$ reads
%
\be
        T^M_{r_1 r_2}(s,t,z_1,\vec{r}_{1\perp},z_2,\vec{r}_{2\perp}) =
        2is \int \!\!d^2b_{\!\perp} 
        e^{i {\vec q}_{\!\perp} {\vec b}_{\!\perp}}
        \left[1-S^M_{r_1 r_2}(s,{\vec b}_{\!\perp},z_1,\vec{r}_{1\perp},z_2,\vec{r}_{2\perp})\right]
\label{Eq_T_DD_Minkowski}
\ee
%
with the $S$-matrix element ($M$ refers to Minkowski space-time)
%
\be
        S^M_{r_1 r_2}(s,{\vec b}_{\!\perp},z_1,\vec{r}_{1\perp},z_2,\vec{r}_{2\perp})
        = \lim_{T \rightarrow \infty}
        \frac{\langle W_{r_1}[C_1] W_{r_2}[C_2]\rangle_M}
        {\langle W_{r_1}[C_1]\rangle_M \langle W_{r_2}[C_2]\rangle_M}
        \ .
\label{Eq_S_DD_Minkowski}
\ee
%
The color dipoles are considered in the $SU(N_c)$ representation $r_i$
and have transverse size and orientation ${\vec r}_{i\perp}$. The
longitudinal momentum fraction carried by the quark of dipole $i$ is
$z_i$.  (Here and in the following we use several times the term quark
generically for color sources in arbitrary $SU(N_c)$ representations.)
The impact parameter between the dipoles is~\cite{Dosch:1997ss}
%
\be
        {\vec b}_{\!\perp} 
        \,=\, {\vec r}_{1q} + (1-z_1) {\vec r}_{1\perp} 
            - {\vec r}_{2q} - (1-z_2) {\vec r}_{2\perp} 
        \,=\, {\vec r}_{1\,cm} - {\vec r}_{2\,cm} 
        \ ,
\label{Eq_impact_vector}
\ee
%
where ${\vec r}_{iq}$ (${\vec r}_{i\qbar}$) is the transverse position
of the quark (antiquark), ${\vec r}_{i\perp} = {\vec r}_{i\qbar} -
{\vec r}_{iq}$, and ${\vec r}_{i\,cm} = z_i {\vec r}_{iq} +
(1-z_i){\vec r}_{i\qbar}$ is the center of light-cone momenta.
Figure~\ref{Fig_loop_loop_scattering_surfaces} illustrates the (a)
space-time and (b) transverse arrangement of the dipoles.
%$
\befig[p!]
  \begin{center}
        \epsfig{file=loop_loop_scattering_surfaces_ite.eps,width=10.cm}
  \end{center}
\caption{\small High-energy dipole-dipole scattering in the eikonal
  approximation represented by Wegner-Wilson loops in the fundamental
  representation of $SU(N_c)$: (a) Space-time and (b) transverse
  arrangement of the Wegner-Wilson loops. The shaded areas represent
  the strings extending from the quark to the antiquark path in each
  color dipole.  The thin tube allows us to compare the gluon field
  strengths in surface $S_1$ with the gluon field strengths in surface
  $S_2$. The impact parameter $\vec{b}_{\perp}$ connects the centers
  of light-cone momenta of the dipoles.}
\label{Fig_loop_loop_scattering_surfaces}
\efig
%
The dipole trajectories $C_i$ defining the {\WW} loops
in~(\ref{Eq_S_DD_Minkowski}) are described as straight lines.  This is
a good approximation as long as the kinematical assumption behind the
eikonal approximation, $s \gg -t$, holds that allows us to neglect the
change of the dipole velocities $v_i = p_i/m$ in the scattering
process, where $p_i$ is the momentum and $m$ the mass of the
considered dipole.  Moreover, the paths $C_i$ are considered
light-like\footnote{In fact, exactly light-like trajectories ($\gamma
  \to \infty$) are considered in most applications of the functional
  integral approach to high-energy
  collisions~\cite{Kramer:1990tr,Dosch:1994ym,Dosch:RioLecture,Rueter:1996yb,Dosch:1997ss,Dosch:1998nw,Rueter:1998qy,Kulzinger:1999hw,Rueter:1998up,D'Alesio:1999sf,Berger:1999gu,Donnachie:2000kp,Donnachie:2001wt,Dosch:2001jg,Shoshi:2002in,Shoshi:2002ri,Shoshi:2002fq,Shoshi:2002mt}.
  A detailed investigation of the more general case of finite rapidity
  $\gamma$ can be found in~\cite{Kulzinger:2002iu}.} in line with the
high-energy limit, $m^2 \ll s \to \infty$. For the {\em hyperbolic
  angle} or {\em rapidity gap} between the dipole trajectories $\gamma
= (v_1 \cdot v_2)$ -- which is the central quantity in the analytic
continuation discussed below and also defined through $s =
4m^2\cosh^2(\gamma/2)$ -- the high-energy limit implies
%
\be
        \lim_{m^2\ll s\to\infty} \gamma \approx \ln(s/m^2) \to \infty 
        \ .
\label{Eq_rapidity_light-like_loops}
\ee
%
The QCD VEVs $\langle\ldots\rangle_M$ in the $S$-matrix
element~(\ref{Eq_S_DD_Minkowski}) represent {\em Minkowskian}
functional integrals~\cite{Nachtmann:ed.kt} in which -- as in the
Euclidean case discussed above -- the functional integration over the
fermion fields has already been carried out.

The $S$-matrix element $S^M_{DD}:=S^M_{\fundamental\fundamental}$ for
the scattering of light-like dipoles in the fundamental
$SU(N_c\!=\!3)$ representation ($r_1\!=\!r_2\!=\!\Fundamental\!=\!3$)
is the key to our unified description of hadron-hadron, photon-hadron,
and photon-photon reactions in the following chapters. With color
dipoles given by the quark and antiquark in the meson or photon or in
a simplified picture by a quark and diquark in the baryon, we describe
hadrons and photons as quark-antiquark or quark-diquark systems, i.e.\ 
fundamental $SU(3)$ dipoles, with size and orientation determined by
appropriate light-cone wave
functions~\cite{Dosch:1994ym,Dosch:RioLecture}.  Accordingly, the
$T$-matrix element for the reaction $ab \to cd$ factorizes into the
universal $S$-matrix element $S^M_{DD}$ and reaction-specific
light-cone wave functions $\psi_{a,b}$ and $\psi_{c,d}$ that describe
the ${\vec r}_i$ and $z_i$ distribution of the color
dipoles~\cite{Dosch:1994ym,Dosch:RioLecture,Nachtmann:ed.kt}
%
\bea
        \!\!\!\!\!\!
        &&\hspace{-1cm}
        T_{ab \rightarrow cd}(s,t) =
        2is \int \!\!d^2b_{\!\perp} 
        e^{i {\vec q}_{\!\perp} {\vec b}_{\!\perp}}
        \int \!\!dz_1 d^2r_1 \!\int \!\!dz_2 d^2r_2      
        \hphantom{\hspace*{5.cm}}   
\label{Eq_model_T_amplitude}\\ 
        & \!\! \times \!\! & 
        \psi_c^*(z_1,\vec{r}_{1\perp})\,\psi_d^*(z_2,\vec{r}_{2\perp})
        \left[1-S^M_{DD}(s,{\vec b}_{\!\perp},z_1,\vec{r}_{1\perp},z_2,\vec{r}_{2\perp})\right]
        \psi_a(z_1,\vec{r}_{1\perp})\,\psi_b(z_2,\vec{r}_{2\perp}) 
        \ . 
\nonumber
\eea
%
Concentrating in this work on reactions with $a = c$ and $b = d$, the
squared wave functions
$|\psi_1(z_1,\vec{r}_{1\perp})|^2:=\psi_c^*(z_1,\vec{r}_{1\perp})\,\psi_a(z_1,\vec{r}_{1\perp})$
and
$|\psi_2(z_2,\vec{r}_{2\perp})|^2:=\psi_d^*(z_2,\vec{r}_{2\perp})\,\psi_b(z_2,\vec{r}_{2\perp})$
are needed. We use for hadrons the phenomenological Gaussian wave
function~\cite{Dosch:2001jg,Wirbel:1985ji} and for photons the
perturbatively derived wave function with running quark masses
$m_f(Q^2)$ to account for the non-perturbative region of low photon
virtuality $Q^2$~\cite{Dosch:1998nw}. In Sec.~\ref{Sec_Wave_Functions}
we specify and discuss these wave functions explicitly. The scattering
of dipoles with fixed size $|\vec{r}_i|$ and fixed longitudinal quark
momentum fraction $z_i$ averaged over all orientations,
%
\be
        |\psi_{\!\mbox{\tiny\it D}_i}(z_i,\vec{r}_i)|^2 =
        \inv{2\pi|\vec{r}_{\!\mbox{\tiny\it D}_i}|}\,\delta
        (|\vec{r}_i|-|\vec{r}_{\!\mbox{\tiny\it D}_i}|)\,\delta
        (z_i-z_{\!\mbox{\tiny\it D}_i})
        \ ,
\label{Eq_dip_wf}
\ee
%
is considered in Sec.~\ref{Sec_S-Matrix_Unitarity} to show that
$S$-matrix unitarity constraints are respected in our model. For the
analytic continuation of high-energy scattering to Euclidean
space-time, we now return to the scattering of dipoles with fixed size
{\em and} orientation $\vec{r}_i$ and fixed longitudinal quark
momentum fraction $z_i$.

% ______________________________________________________________________________
\section{Analytic Continuation of Dipole-Dipole Scattering}
\label{Sec_Analytic_Continuation_Dipoles}
% ______________________________________________________________________________

The Euclidean approach to the described elastic scattering of dipoles
in the eikonal approximation is based on {\em Meggiolaro's analytic
  continuation} of the high-energy parton-parton scattering
amplitude~\cite{Meggiolaro:1996hf+X}. Meggiolaro's analytic
continuation has been derived in the functional integral approach to
high-energy collisions~\cite{Nachtmann:1991ua+X,Nachtmann:ed.kt} in
which parton-parton scattering is described in terms of {\WW} lines:
The Minkowskian amplitude, $g^M(\gamma,T,t)$, given by the expectation
value of two {\WW} lines, forming an hyperbolic angle $\gamma$ in
Minkowski space-time, and the Euclidean ``amplitude,''
$g^E(\Theta,T,t)$, given by the expectation value of two {\WW} lines,
forming an angle $\Theta \in [0,\pi]$ in Euclidean space-time, are
connected by the following analytic continuation in the angular
variables and the temporal extension $T$, which is needed as an IR
regulator in the case of {\WW} lines,
%
\bea
        g^E(\Theta,T,t) & = & g^M(\gamma\to i\Theta,T\to -iT,t) 
        \ ,
\label{Eq_gE=gM}\\
        g^M(\gamma,T,t) & = & g^E(\Theta\to -i\gamma, T\to iT,t)
        \ .
\label{Eq_gM=gE}
\eea
%
Generalizing this relation to {\em gauge-invariant} dipole-dipole
scattering described in terms of {\WW}
loops~\cite{Dosch:1994ym,Dosch:RioLecture,Nachtmann:ed.kt}, the IR
divergence known from the case of {\WW} lines vanishes and no finite
IR regulator $T$ is necessary. Thus, the Minkowskian $S$-matrix
element~(\ref{Eq_S_DD_Minkowski}), given by the expectation values of
two {\WW} loops, forming an hyperbolic angle $\gamma$ in Minkowski
space-time, can be computed from the Euclidean ``$S$-matrix element''
%
\be
        S^E_{r_1 r_2}(\Theta,{\vec b}_{\!\perp},z_1,\vec{r}_{1\perp},z_2,\vec{r}_{2\perp})
        = \lim_{T \rightarrow \infty}
        \frac{\langle W_{r_1}[C_1] W_{r_2}[C_2]\rangle_E}
        {\langle W_{r_1}[C_1]\rangle_E \langle W_{r_2}[C_2]\rangle_E}
\label{Eq_S_DD_Euclidean}
\ee
%
given by the expectation values of two {\WW} loops, forming an angle
$\Theta \in [0,\pi]$ in Euclidean space-time, via an analytic
continuation in the angular variable
%
\be
        S^M_{r_1 r_2}(\gamma\approx\ln[s/m^2],{\vec b}_{\!\perp},z_1,\vec{r}_{1\perp},z_2,\vec{r}_{2\perp})
        = S^E_{r_1 r_2}(\Theta\to -i\gamma,{\vec b}_{\!\perp},z_1,\vec{r}_{1\perp},z_2,\vec{r}_{2\perp})
        \ ,
\label{Eq_SM=SE(theta->-igamma)}
\ee
%
where $E$ indicates Euclidean space-time and the QCD VEVs
$\langle\ldots\rangle_E$ represent Euclidean functional integrals that
are equivalent to the ones denoted by $\langle\ldots\rangle_G$ in the
preceding sections, i.e.\ in which the functional integration over the
fermion fields has already been carried out.

The angle $\Theta$ is best illustrated in the relation of the
Euclidean $S$-matrix element~(\ref{Eq_S_DD_Euclidean}) to the van der
Waals potential between two static dipoles $V_{r_1 r_2}(\Theta=0,
\vec{b}, z_1, \vec{r}_1, z_2, \vec{r}_2)$, discussed in
Sec.~\ref{Sec_VDW_Potential},
%
\be
        S^E_{r_1 r_2}(\Theta,\vec{b}_{\!\perp},z_1,\vec{r}_{1\perp},z_2,\vec{r}_{2\perp})
        = \lim_{T \rightarrow \infty}
        \exp\!\left[-\,T\,V_{r_1 r_2}(\Theta,\vec{b}_{\!\perp},z_1,\vec{r}_{1\perp},z_2,\vec{r}_{2\perp})\right]
        \ .
\label{Eq_S_DD_<->_V_DD}
\ee
%
Figure~\ref{Fig_tilted_loops} shows the loop-loop geometry necessary
to compute $S^E_{r_1 r_2}(\Theta\neq 0, \cdots)$ and how it is
obtained by generalizing the geometry relevant for the computation of
the potential between two static dipoles ($\Theta=0$): While the
potential between two static dipoles is computed from two loops along
parallel ``temporal'' unit vectors, $t_1 = t_2 = (0,0,0,1)$, the
Euclidean $S$-matrix element~(\ref{Eq_S_DD_Euclidean}) involves the
tilting of one of the two loops, e.g.\ the tilting of $t_1$ by the
angle $\Theta$ towards the $X_3$\,-\,axis, $t_1 =
(0,0,-\sin\Theta,\cos\Theta)$. The ``temporal'' unit vectors $t_i$ are
also discussed in Appendix~\ref{Sec_Parameterizations} together with
another illustration of the tilting angle $\Theta$.
%$
\begin{figure}[t!]
\centerline{\epsfig{figure=loop_loop_tilting_ite.eps,width=10.cm}}
\caption{\small 
  The loop-loop geometry necessary to compute $S^E_{r_1
    r_2}(\Theta\neq 0, \cdots)$ illustrated as a generalization of the
  geometry relevant for the computation of the van der Waals potential
  between two static dipoles ($\Theta=0$). While the potential between
  two static dipoles is computed from two loops along parallel
  ``temporal'' unit vectors, $t_1 = t_2 = (0,0,0,1)$, the Euclidean
  $S$-matrix element~(\ref{Eq_S_DD_Euclidean}) involves the tilting of
  one of the two loops, e.g.\ the tilting of $t_1$ by the angle
  $\Theta$ towards the $X_3$\,-\,axis, $t_1
  =(0,0,-\sin\Theta,\cos\Theta)$.}
\label{Fig_tilted_loops}
\end{figure}
%

Since the Euclidean $S$-matrix element~(\ref{Eq_S_DD_Euclidean})
involves only configurations of {\WW} loops in Euclidean space-time
and {\em Euclidean} functional integrals, it can be computed directly
on a Euclidean lattice. First attempts in this direction have been
carried out but only very few signals could be extracted, while most
of the data was dominated by noise~\cite{DiGiacomo:2002PC}. Once
precise results are available, the analytic
continuation~(\ref{Eq_SM=SE(theta->-igamma)}) will allow us to access
hadronic high-energy reactions directly in lattice QCD, i.e.\ within a
non-perturbative description of QCD from first principles. More
generally, the presented gauge-invariant analytic
continuation~(\ref{Eq_SM=SE(theta->-igamma)}) makes any approach
limited to a Euclidean formulation of the theory applicable for
investigations of high-energy reactions.  Indeed, Meggiorlaro's
analytic continuation has already been used to access high-energy
scattering from the supergravity side of the AdS/CFT
correspondence~\cite{Janik:2000zk+X}, which requires a positive
definite metric in the definition of the minimal
surface~\cite{Rho:1999jm}, and to examine the effect of instantons on
high-energy scattering~\cite{Shuryak:2000df+X}.

% ______________________________________________________________________________
\section[Dipole-Dipole Scattering in the Loop-Loop Correlation Model]
{Dipole-Dipole Scattering in the LLCM}
\label{Sec_DD_Scattering_LLCM}
% ______________________________________________________________________________

Let us now perform the analytic continuation explicitly in our
Euclidean loop-loop correlation model. For the scattering of two color
dipoles in the {\em fundamental representation} of $SU(N_c)$, the
Euclidean $S$-matrix element becomes with the
VEVs~(\ref{Eq_final_result_<W[C]>})
and~(\ref{Eq_final_Euclidean_result_<W[C1]W[C2]>_fundamental})
%
\bea
        S^E_{DD}(\Theta,\vec{b}_{\!\perp},z_1,\vec{r}_{1\perp},z_2,\vec{r}_{2\perp})
         && \!\!\!\!\!\!
        :=\,\,S^E_{\fundamental\fundamental}
        (\Theta,\vec{b}_{\!\perp},z_1,\vec{r}_{1\perp},z_2,\vec{r}_{2\perp})
\nonumber\\
        && \hspace{-4cm} = \lim_{T \rightarrow \infty}
        \left(
        \frac{N_c\!+\!1}{2N_c}\exp\!\left[-\frac{N_c\!-\!1}{2 N_c}\chi_{S_1 S_2}\right]
        + \frac{N_c\!-\!1}{2N_c}\exp\!\left[ \frac{N_c\!+\!1}{2 N_c}\chi_{S_1 S_2}\right]
        \right)
        \ ,
\label{Eq_S_DD_1}
\eea
%
where $\chi_{S_i S_j}$ -- defined in~(\ref{Eq_chi_Si_Sj}) --
decomposes into a perturbative ($\pert$) and non-per\-tur\-ba\-tive
($\nprt$) component according to our decomposition of the gluon field
strength correlator~(\ref{Eq_F_decomposition}),
%
\be
        \chi_{S_1 S_2} 
        \,\,=\,\, 
        \chi_{S_1 S_1}^{\pert} 
        \,+\, \chi_{S_1 S_2}^{\nprt}
        \,\,=\,\, 
        \chi_{S_1 S_2}^{\pert} 
        \,+\, \left(\chi_{S_1 S_2}^{\nprt\,\,nc} 
          \,+\, \chi_{S_1 S_2}^{\nprt\,\,c}\right)
        \ .
\label{Eq_chi_decomposition}        
\ee
%
In the limit $T_1=T_2=T\to\infty$ and for $\Theta \in [0,\pi]$, the
components read
%
\be
        \chi_{S_1 S_2}^{\pert} 
        =\cot\Theta\,\,\chi^{\pert}
        \,\, , \quad
        \chi_{S_1 S_2}^{\nprt\,\,nc} 
        =\cot\Theta\,\,\chi^{\nprt\,\,nc}
        \,\, , \quad
        \chi_{S_i S_j}^{\nprt\,\,c} 
        =\cot\Theta\,\,\chi^{\nprt\,\,c}
\label{Eq_S_DD_p_npc_npnc_E}
\ee
%
with
%
\bea
        \!\!\!\!\!\!\!\!\!\!\!\!\!
        \chi^{\pert} &\!\!=\!\!& 
        \left[ 
        g^2 D^{\prime\,(2)}_{\pert}
        \left(|\vec{r}_{1q}-\vec{r}_{2\qbar}|\right)
        +g^2 D^{\prime\,(2)}_{\pert}
        \left(|\vec{r}_{1\qbar}-\vec{r}_{2q}|\right)
        \right.
\nonumber \\
        \!\!\!\!\!\!\!\!\!\!\!\!\!
        &&
        \left.
        -\,g^2 D^{\prime\,(2)}_{\pert}
        \left(|\vec{r}_{1q}-\vec{r}_{2q}|\right)
        -g^2 D^{\prime\,(2)}_{\pert}
        \left(|\vec{r}_{1\qbar}-\vec{r}_{2\qbar}|\right)
        \right]
\label{Eq_S_DD_chi_p_M}\\ 
        \!\!\!\!\!\!\!\!\!
        \chi^{\nprt\,\,nc} &\!\!=\!\!& 
        \frac{\pi^2 G_2 (1-\kappa)}{3(N_c^2-1)} 
        \left[ 
        D^{\prime\,(2)}_1
        \left(|\vec{r}_{1q}-\vec{r}_{2\qbar}|\right)
        +D^{\prime\,(2)}_1
        \left(|\vec{r}_{1\qbar}-\vec{r}_{2q}|\right)
         \right.
\nonumber \\
        \!\!\!\!\!\!\!\!\!\!\!\!\!
        &&
        \hphantom{-\frac{\pi^2 G_2 (1-\kappa)}{3(N_c^2-1)}}
        \left.
        -\,D^{\prime\,(2)}_1
        \left(|\vec{r}_{1q}-\vec{r}_{2q}|\right) 
        -D^{\prime\,(2)}_1
        \left(|\vec{r}_{1\qbar}-\vec{r}_{2\qbar}|\right)
       \right]
\label{Eq_S_DD_chi_np_nc_M}\\
        \!\!\!\!\!\!\!\!\!\!\!\!\!
        \chi^{\nprt\,\,c} &\!\!=\!\!& 
        \frac{\pi^2 G_2 \kappa}{3(N_c^2-1)}\,
        \left(\vec{r}_1\cdot\vec{r}_2\right)
        \int_0^1 \! dv_1 \int_0^1 \! dv_2 \,\, 
        D^{(2)}\left(|\vec{r}_{1q}\! +\! v_1\vec{r}_{1\perp} 
        \!-\! \vec{r}_{2q}\! -\! v_2\vec{r}_{2\perp}|\right)
\label{Eq_S_DD_chi_np_c_M}
\eea
%
as derived explicitly in Appendix~\ref{Sec_Chi_Computation} with the
minimal surfaces illustrated in Fig.~\ref{Fig_tilted_loops}. In
Eq.~(\ref{Eq_S_DD_chi_p_M}) the shorthand notation $g^2
D^{\prime\,(2)}_{\pert}(|\vec{Z_\perp}|) =
g^2(|\vec{Z_\perp}|)\,D^{\prime\,(2)}_{\pert}(|\vec{Z_\perp}|)$ is
used with $g^2(|\vec{Z_\perp}|)$ again understood as the running
coupling~(\ref{Eq_g2(z_perp)}). The transverse Euclidean correlation
functions
%
\be
        D_x^{(2)}(\vec{Z}^2)      
        := \int \frac{d^4K}{(2\pi)^2}\,e^{iKZ}\,
        \tilde{D}_x(K^2)\,\delta(K_3)\,\delta(K_4)
\label{Eq_D(2)x}
\ee
%
are obtained from the (massive) gluon
propagator~(\ref{Eq_massive_gluon_propagator}) and the exponential
correlation function~(\ref{Eq_SVM_correlation_functions})
%
\bea
        \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
        D^{\prime\,(2)}_{\pert}(\vec{Z}_{\!\perp}^2)
        & \!\!=\!\! &
        \inv{2\pi} K_0\left(m_G |\vec{Z}_{\!\perp}|\right)
\label{Eq_D'(2)p(z,mg)}\\ 
        \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
        D^{\prime\,(2)}_1(\vec{Z}_{\!\perp}^2)
        & \!\!=\!\! &
        \pi a^4  \Big(
        3 \!+\! 3\frac{|\vec{Z}_{\!\perp}|}{a} \!+\! \frac{|\vec{Z}_{\!\perp}|^2}{a^2} 
        \Big)
        \exp\!\Big(\!-\frac{|\vec{Z}_{\!\perp}|}{a}\Big)
\label{Eq_D'(2)np_nc(z,a)}\\
        \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
        D^{(2)}(\vec{Z}_{\!\perp}^2)      
        & \!\!=\!\! &
        2 \pi a^2 
        \Big(1\!+\!\frac{|\vec{Z}_{\!\perp}|}{a}\Big) 
        \exp\!\Big(\!-\frac{|\vec{Z}_{\!\perp}|}{a}\Big)
\label{Eq_D(2)np_c(z,a)}
\eea
%
With the full $\Theta$-dependence exposed
in~(\ref{Eq_S_DD_p_npc_npnc_E}), the analytic
continuation~(\ref{Eq_SM=SE(theta->-igamma)}) reads
%
\be
        \chi_{S_1 S_2} = \cot\Theta\,\,\chi
        \quad\underrightarrow{\,\,\Theta\to -i\gamma\,\,\,\,}\quad
        \cot(-i\gamma)\,\chi
        \quad\underrightarrow{\,\,s\to\infty\,\,\,\,}\quad
        i\chi
\label{Eq_analytic_continuation_of_chi}
\ee
%
and leads to the desired Minkowskian $S$-matrix element for elastic
dipole-dipole scattering ($DD$) in the high-energy limit in which the
dipoles move on the light-cone
%
\bea
        \lim_{s \rightarrow \infty} 
        S_{DD}^{M}(s,{\vec b}_{\!\perp},z_1,\vec{r}_{1\perp},z_2,\vec{r}_{2\perp}) 
        &&\!\!\!\!\!\!:=\,\, 
        \lim_{s \rightarrow \infty} 
        S^M_{\fundamental\fundamental}(s,{\vec b}_{\!\perp},z_1,\vec{r}_{1\perp},z_2,\vec{r}_{2\perp})
\nonumber\\
        && \hspace{-4cm} = 
        S^E_{DD}(\cot\Theta \to i,{\vec b}_{\!\perp},z_1,\vec{r}_{1\perp},z_2,\vec{r}_{2\perp})
\nonumber\\
        && \hspace{-4cm} = 
        \lim_{T \rightarrow \infty}
        \left(
        \frac{N_c\!+\!1}{2N_c}\exp\!\left[-i\frac{N_c\!-\!1}{2N_c}\chi\right]
        + \frac{N_c\!-\!1}{2N_c}\exp\!\left[i\frac{N_c\!+\!1}{2N_c}\chi\right]
        \right)
\label{Eq_S_DD_1_M}
\eea
%
where $\chi =\chi^{\pert}+\chi^{\nprt\,\,nc}+\chi^{\nprt\,\,c}$
with~(\ref{Eq_S_DD_chi_p_M}), (\ref{Eq_S_DD_chi_np_nc_M}),
and~(\ref{Eq_S_DD_chi_np_c_M}).

It is striking that exactly the same result has been obtained
in~\cite{Shoshi:2002in}\footnote{To see this identity, recall that
  $\langle W[C]\rangle = 1$ for light-like loops and consider
  in~\cite{Shoshi:2002in} the result~(2.30) for the loop-loop
  correlation function (2.3) together with the $\chi$-function (2.40)
  and its components given in (2.49), (2.54), and (2.57) with the
  transverse Minkowskian correlation functions (2.50), (2.55), and
  (2.58).} with the alternative analytic continuation introduced for
applications of the SVM to high-energy
reactions~\cite{Kramer:1990tr,Dosch:1994ym,Dosch:RioLecture}. In this
complementary approach the gauge-invariant bilocal gluon field
strength correlator is analytically continued from Euclidean to
Minkowskian space-time by the substitution $\delta_{\mu\rho}
\rightarrow - g_{\mu\rho}$ and the analytic continuation of the
Euclidean correlation functions to real time $D^E_x(Z^2) \rightarrow
D^M_x(z^2)$. In the subsequent steps, one finds $\langle W[C]\rangle_M
= 1$ due to the light-likeness of the loops and that the longitudinal
correlations can be integrated out $\langle
W_{r_1}[C_1]W_{r_2}[C_2]\rangle_M = f(s,{\vec b}_{\!\perp},\cdots)$.
One is left with exactly the Euclidean correlations in transverse
space that have been obtained above. This confirms the analytic
continuation used in the earlier LLCM investigations in Minkowski
space-time~\cite{Shoshi:2002in,Shoshi:2002ri,Shoshi:2002fq,Shoshi:2002mt}
and in all earlier SVM applications to high-energy
scattering~\cite{Kramer:1990tr,Dosch:1994ym,Dosch:RioLecture,Rueter:1996yb,Dosch:1997ss,Dosch:1998nw,Rueter:1998qy,Kulzinger:1999hw,Rueter:1998up,D'Alesio:1999sf,Berger:1999gu,Kulzinger:2002iu}.

In the limit of small $\chi$-functions, $|\chi^{\pert}| \ll 1$ and
$|\chi^{\nprt}| \ll 1$, (\ref{Eq_S_DD_1_M}) reduces to
%
\be
        \lim_{s \rightarrow \infty} 
        S_{DD}^{M}(s,{\vec b}_{\!\perp},z_1,\vec{r}_{1\perp},z_2,\vec{r}_{2\perp})
        \approx 1 + \frac{N_c^2-1}{8 N_c^2}\,\chi^2 
        = 1 + \frac{C_2(\Fundamental)}{4 N_c}\,\chi^2 
        \ .
\label{Eq_S_DD_M_small_chi}
\ee
%
The perturbative correlations, $(\chi^{\pert})^2$, describe the
well-known {\em two-gluon exchange}
contribution~\cite{Low:1975sv+X,Gunion:iy} to dipole-dipole
scattering, which is, of course, an important successful cross-check
of the presented Euclidean approach to high-energy scattering. The
non-perturbative correlations, $(\chi^{\nprt})^2$, describe the
corresponding non-perturbative two-point interactions that contain
contributions of the confining QCD strings to dipole-dipole
scattering.  We have analyzed these string contributions
systematically as manifestations of confinement in high-energy
scattering reactions and have indeed found a new characteristic
structure (different from the perturbative dipole factors) in momentum
space~\cite{Shoshi:2002fq}. This analysis has also shown explicitly
that the non-perturbative contribution governs -- as expected -- the
region of low transverse momenta $|\vec{k}_{\!\perp}|$.  Here, we
focus on the structure in space-time representation and refer the
reader for complementary insights to our momentum-space
analysis~\cite{Shoshi:2002fq}.

As evident from the $v_1$ and $v_2$ integrations
in~(\ref{Eq_S_DD_chi_np_c_M}) and
Fig.~\ref{Fig_loop_loop_scattering_surfaces}b, there are contributions
from the transverse projections of the minimal surfaces
$(S_{1,2})_{\!\perp}$ connecting the quark and antiquark in each of
the two dipoles. These are the contributions that we interpret as
manifestations of the strings confining the quark and antiquark in
each dipole. We thus understand the confining component
$\chi_{c}^{\nprt}$ as a {\em string-string interaction}.
Interestingly, we have found in dipole-hadron and dipole-photon
interactions that the strings confining the quark-antiquark pair in
the dipole can represented as an integral over stringless dipoles with
a given dipole number density. As already mentioned, this {\em
  decomposition of the confining string into dipoles} even allows us
to compute unintegrated gluon distributions of hadrons and photons and
thus gives new insights into the microscopic structure of the
non-perturbative SVM~\cite{Shoshi:2002fq}.

Both non-perturbative components, $\chi^{\nprt}_c$ and
$\chi^{\nprt}_{nc}$, show {\em color transparency} for small dipoles,
i.e.\ a dipole-dipole cross section with
$\sigma_{DD}(\vec{r}_1,\vec{r}_2) \propto |\vec{r}_1|^2|\vec{r}_2|^2$
for $|\vec{r}_{1,2}| \to 0$, as known for the perturbative
case~\cite{Nikolaev:1991ja}. This can be seen by
squaring~(\ref{Eq_S_DD_chi_np_nc_M}) and~(\ref{Eq_S_DD_chi_np_c_M}) to
obtain the leading terms in the $T$-matrix element for small dipoles;
see~(\ref{Eq_S_DD_M_small_chi}).

Due to the truncation of the cumulant expansion in the Gaussian
approximation, a considerable dependence of $\chi_{c}^{\nprt}$ on the
specific surface choice is observed. In fact, a different and more
complicated result for $\chi_{c}^{\nprt}$ was obtained with the
pyramid mantle choice for the surfaces $S_{1,2}$ in earlier
applications of the \SVM\ to high-energy
scattering~\cite{Dosch:1994ym,Dosch:RioLecture,Rueter:1996yb,Dosch:1997ss,Dosch:1998nw,Rueter:1998qy,Kulzinger:1999hw,Rueter:1998up,D'Alesio:1999sf,Berger:1999gu,Dosch:2001jg}.
However, we use minimal surfaces in line with our model applications
in Euclidean space-time discussed in the previous chapter.  Moreover,
the simplicity of the minimal surfaces allows us to give an analytic
expression for the leading term of the non-perturbative dipole-dipole
cross section~\cite{Shoshi:2002fq}.  Phenomenologically, in comparison
with pyramid mantles, the description of the slope parameter $B(s)$,
the differential elastic cross section $d\sigma^{el}/dt(s,t)$, and the
elastic cross section $\sigma^{el}(s)$ can be improved with minimal
surfaces as shown in Chap.~\ref{Sec_Comparison_Data}. In contrast to
the confining component $\chi_{c}^{\nprt}$, the non-confining
components, $\chi_{nc}^{\nprt}$ and $\chi^{\pert}$, depend only on the
transverse position between the quark and antiquark of the two dipoles
and are therefore independent of the surface choice.

With the insights from the small-$\chi$ limit, one sees clearly that
the full $S$-matrix element~(\ref{Eq_S_DD_1_M}) describes {\em
  multiple gluonic interactions}.  Indeed, the higher order terms in
the expansion of the exponential functions in~(\ref{Eq_S_DD_1_M}) are
crucial to respect $S$-matrix unitarity constraints in impact
parameter space~\cite{Berger:1999gu,Shoshi:2002in} as shown explicitly
in Chap.~\ref{Sec_Impact_Parameter}.

Concerning the energy dependence, the $S$-matrix
element~(\ref{Eq_S_DD_1_M}) leads to energy-independent cross sections
in contradiction to the experimental observation. Although
disappointing from the phenomenological point of view, this is not
surprising since our approach does not describe explicit gluon
radiation needed for a non-trivial energy dependence. However, based
on the $S$-matrix element~(\ref{Eq_S_DD_1_M}), a phenomenological
energy dependence can be constructed -- see
Sec.~\ref{Sec_Energy_Dependence} -- that allows a unified description
of high-energy hadron-hadron, photon-hadron, and photon-photon
reactions and an investigation of saturation effects in hadronic cross
sections manifesting $S$-matrix
unitarity~\cite{Shoshi:2002in,Shoshi:2002ri,Shoshi:2002mt}.  This, of
course, can only be an intermediate step. For a more fundamental
understanding of hadronic high-energy reactions in our model, gluon
radiation and quantum evolution have to be implemented explicitly.

Note that $\chi = \chi_{c}^{\nprt} + \chi_{nc}^{\nprt} + \chi^{\pert}$
is a real-valued function. Since, in addition, the wave functions
$|\psi_i(z_i,\vec{r}_{i\perp})|^2$ used in this work -- see
Sec.~\ref{Sec_Wave_Functions} -- are invariant under the replacement
$(\vec{r}_{i\perp} \rightarrow -\vec{r}_{i\perp}, z_i \rightarrow
1-z_i)$, the $T$-matrix element~(\ref{Eq_model_T_amplitude}) with
$S^M_{DD}$ given in~(\ref{Eq_S_DD_1_M}) becomes purely imaginary and
reads for $N_c=3$
%
\bea
        \!\!\!\!\!\!\!\!\!\!\!\!\!
        T(s,t) 
        & = & 2is \int \!\!d^2b_{\!\perp} 
                e^{i {\vec q}_{\!\perp} {\vec b}_{\!\perp}}
                \int \!\!dz_1 d^2r_1 \!
                \int \!\!dz_2 d^2r_2 \,\,
                |\psi_1(z_1,\vec{r}_{1\perp})|^2   \,\,
                |\psi_2(z_2,\vec{r}_{2\perp})|^2       
\label{Eq_model_purely_imaginary_T_amplitude}\\    
        && \!\!\!\!\!\!\!\!\!\!
        \times 
        \left[1-\frac{2}{3} 
        \cos\!\left(\frac{1}{3}
        \chi({\vec b}_{\!\perp},z_1,\vec{r}_{1\perp},z_2,\vec{r}_{2\perp})\!\right)
        - \frac{1}{3}
        \cos\!\left(\frac{2}{3}
        \chi({\vec b}_{\!\perp},z_1,\vec{r}_{1\perp},z_2,\vec{r}_{2\perp})\!\right)
        \right].
\nonumber
\eea
%
The real part averages out in the integration over ${\vec r}_i$ and
$z_i$ because of
%
\be
        \chi(\vec{b}_{\!\perp},1-z_1,-\vec{r}_{1\perp},z_2,\vec{r}_{2\perp})
        = - \chi(\vec{b}_{\!\perp},z_1,\vec{r}_{1\perp},z_2,\vec{r}_{2\perp})
        \ ,
\label{Eq_odd_eikonal_function}
\ee
%
which can be seen directly
from~(\ref{Eq_S_DD_chi_p_M}),(\ref{Eq_S_DD_chi_np_nc_M}), and
(\ref{Eq_S_DD_chi_np_c_M}) as $(\vec{r}_{1\perp} \rightarrow
-\vec{r}_{1\perp}, z_1 \rightarrow 1-z_1)$ implies $\vec{r}_{1q}
\rightarrow \vec{r}_{1\qbar}$. In physical terms, $(\vec{r}_{i\perp}
\rightarrow -\vec{r}_{i\perp}, z_i \rightarrow 1-z_i)$ corresponds to
{\em charge conjugation}, i.e.\ the replacement of each parton with
its antiparton and the associated reversal of the loop direction.
Consequently, the
$T$-matrix~(\ref{Eq_model_purely_imaginary_T_amplitude}) describes
only charge conjugation $C = +1$ exchange. Since in our quenched
approximation purely gluonic interactions are modeled,
(\ref{Eq_model_purely_imaginary_T_amplitude}) describes only
pomeron\footnote{Odderon $C = -1$ exchange is excluded in our model.
  It would survive in the following cases: (a) Wave functions are used
  that are not invariant under the transformation $(\vec{r}_i
  \rightarrow -\vec{r}_i, z_i \rightarrow 1-z_i)$. (b) The proton is
  described as a system of three quarks with finite separations
  modeled by three loops with one common light-like line. (c) The
  Gaussian approximation that enforces the truncation of the cumulant
  expansion is relaxed and additional higher cumulants are taken into
  account.}  but not reggeon exchange.

Although the scattering of two color dipoles in the fundamental
representation of $SU(N_c)$ is the most relevant case, we can derive
immediately also the Minkowskian $S$-matrix element for the scattering
of a fundamental ($D$) and an adjoint dipole (``glueball''
$\glueball$) in the Euclidean LLCM.
Using~(\ref{Eq_final_Euclidean_result_<Wf[C1]Wa[C2]>}) and proceeding
otherwise as above, we find in the high-energy limit
%
\bea
        && \!\!\!\!\!\!\!\!
        \lim_{s \rightarrow \infty} 
        S^M_{D\,\glueball}(s, \vec{b}, z_1, \vec{r}_{1\perp}, z_2, \vec{r}_{2\perp}) 
        \,\,:=\,\, \lim_{s \rightarrow \infty} 
        S^M_{\fundamental\,\adjoint}(\Theta, \vec{b}, z_1, \vec{r}_{1\perp}, z_2, \vec{r}_{2\perp}) 
\label{Eq_S_fa_final_result}\\
        && \!\!\!\!\!\!\!\!
        = \lim_{T \rightarrow \infty}
        \Bigg(\,\inv{N_c^2\!-\!1}\,\exp\!\Big[i\,\frac{N_c}{2}\,\chi\Big]
        +\frac{N_c\!+\!2}{2(N_c\!+\!1)}\exp\!\Big[\!-\,i\,\inv{2}\,\chi\Big]
        +\frac{N_c\!-\!2}{2(N_c\!-\!1)}\exp\!\Big[i\,\inv{2}\,\chi\Big]
        \Bigg)
        \ .
\nonumber
\eea
%
where $\chi =\chi^{\pert}+\chi^{\nprt\,nc}+\chi^{\nprt\,c}$
with~(\ref{Eq_S_DD_chi_p_M}), (\ref{Eq_S_DD_chi_np_nc_M}),
and~(\ref{Eq_S_DD_chi_np_c_M}).

% ______________________________________________________________________________
\section{Comments on the QCD van der Waals Potential}
\label{Sec_VDW_Potential}
% ______________________________________________________________________________

Finally, we would like to comment on the {\em QCD van der Waals
  interaction} between two color dipoles, which is -- as mentioned
together with~(\ref{Eq_S_DD_<->_V_DD}) -- related to the Euclidean
$S$-matrix element in the limiting case of $\Theta=0$: The QCD van der
Waals potential between two static dipoles reads in terms of {\WW}
loops~\cite{Appelquist:1978rt,Bhanot:1979af}
%
\be
        V_{r_1 r_2}(\Theta=0, \vec{b}, z_1=1/2, \vec{r}_1, z_2=1/2, \vec{r}_2) =
        - \lim_{T \rightarrow \infty} \frac{1}{T} 
        \ln \frac{\langle W_{r_1}[C_1] W_{r_2}[C_2] \rangle}
        {\langle W_{r_1}[C_1] \rangle \langle W_{r_2}[C_2] \rangle}
        \ .
\label{Eq_V_DD}        
\ee
%
In this limit ($\Theta=0$) intermediate octet states and their limited
lifetime become important as is well known from perturbative
computations of the QCD van der Waals potential between two static
color dipoles~\cite{Appelquist:1978rt,Bhanot:1979af,Peskin:1979va+X}:
Working with static dipoles, i.e.\ infinitely heavy color sources,
there is an energy degeneracy between the intermediate octet states
and the initial (final) singlet states that leads for perturbative
two-gluon exchange to a linear divergence in $T$ as $T\to\infty$. This
IR divergence can be lifted by introducing manually an energy gap
between the singlet ground state and the excited octet state and thus
a limit on the lifetime of the intermediate octet
state~\cite{Appelquist:1978rt,Bhanot:1979af,Peskin:1979va+X}.

In the perturbative limit of $g^2\to 0$ and $T$ large but finite,
i.e.\ $\chi^{\pert} \ll 1$, the perturbative component of our model
describes the two-gluon exchange contribution to the van der Waals
potential which is plagued by the discussed IR divergence resulting
from the static limit. In the more general case of $g^2$ finite and
$T\to\infty$, which does not exclude non-perturbative physics, one
cannot use the small-$\chi$ limit and multiple gluonic interactions
become important. Here our perturbative component describes multiple
gluon exchanges that reduce to an effective one-gluon exchange
contribution to the van der Waals potential whose interaction range
($\propto 1/m_G$) contradicts the common expectations. Indeed, it is
also in contradiction to our results for the glueball mass
$M_{\glueball}$ which determines the interaction range ($\propto
1/M_{\glueball}$) between two color dipoles for large dipole
separations. As already mentioned in Sec.~\ref{Sec_QCD_Components}, we
find for the perturbative component, $M_{\glueball}^{\pert} = 2 m_G$,
i.e.\ half of the interaction range of one-gluon exchange, by
computing the exponential decay of the correlation of two small
quadratic loops $P^{\alpha \beta}_{r_i}$ for large Euclidean times
$\tau\to\infty$
%
\be
         M_{\glueball} :=
        - \lim_{\tau \rightarrow \infty} \frac{1}{\tau} 
        \ln \frac{\langle P_{r_1}^{\alpha \beta}(0) P_{r_2}^{\alpha \beta}(\tau)\rangle}
        {\langle P_{r_1}^{\alpha\beta}(0) \rangle 
         \langle P_{r_2}^{\alpha\beta}(\tau) \rangle}
        \ .
\label{Eq_Glueball_mass}
\ee
%
Note that we find for the non-perturbative component,
$M_{\glueball}^{\nprt} = 2/a$, which is smaller than
$M_{\glueball}^{\pert} = 2 m_G$ with the LLCM parameters and thus
governs the long range correlations in the LLCM.

Thus, for a meaningful investigation of the QCD van der Waals forces
within our model, one has to go beyond the static limit in order to
describe the limited lifetime of the intermediate octet states
appropriately. This we postpone for future work since the focus in
this work is on high-energy scattering where the gluons are always
exchanged within a short time interval due to the light-likeness of
the scattered particles and the finite correlation lengths.
Nevertheless, going beyond the static limit in the dipole-dipole
potential means going beyond the eikonal approximation in high-energy
scattering and it is, of course, of utmost importance to see how such
generalizations alter our results.

%%% Local Variables: 
%%% mode: latex
%%% TeX-master: "master"
%%% End: 
% HEScattering.tex
%
% ______________________________________________________________________________
\chapter{Hadronic Wave Functions and Universal Energy Dependence}
\label{Sec_High-Energy_Scattering}
% ______________________________________________________________________________

In this chapter hadron and photon wave functions are provided and a
universal energy dependence is constructed. Together with the
$T$-matrix element~(\ref{Eq_model_purely_imaginary_T_amplitude}),
these ingredients are crucial for our unified description of
hadron-hadron, photon-hadron, and photon-photon reactions in the
following chapters. The model parameters adjusted in fits to
experimental data are summarized at the end of this chapter.

% ______________________________________________________________________________
\section{Hadron and Photon Wave Functions}
\label{Sec_Wave_Functions}
% ______________________________________________________________________________

The light-cone wave functions $\psi_i(z_i,\vec{r}_i)$ provide the
distribution of transverse size and orientation ${\vec r}_{i}$ and
longitudinal quark momentum fraction $z_i$ to the light-like
Wegner-Wilson loops $W[C_i]$ that represent the scattering
color dipoles. In this way, they specify the projectiles as mesons,
baryons described as quark-diquark systems, or photons that fluctuate
into a quark-antiquark pair before the interaction.

% ______________________________________________________________________________
\subsection*{The Hadron Wave Function}
% ______________________________________________________________________________

In this work mesons and baryons are assumed to have a quark-antiquark
and quark-diquark valence structure, respectively. As quark-diquark
systems are equivalent to quark-antiquark systems~\cite{Dosch:1989hu},
this allows us to model not only mesons but also baryons as color
dipoles represented by Wegner-Wilson loops. To characterize mesons and
baryons, we use the phenomenological Gaussian Wirbel-Stech-Bauer
ansatz~\cite{Wirbel:1985ji}
%
\be
        \psi_h(z_i,\vec{r}_i) 
        = \sqrt{\frac{z_i(1-z_i)}{2 \pi S_h^2 N_h}}\, 
        e^{-(z_i-\inv{2})^2 / (4 \Delta z_h^2)}\,  
        e^{-|\vec{r}_i|^2 / (4 S_h^2)} 
        \ ,
\label{Eq_hadron_wave_function}
\ee
%
where the hadron wave function normalization to unity
%
\be
        \int \!\!dz_i d^2r_i \ |\psi_i(z_i,\vec{r}_i)|^2 = 1  
        \ ,
\label{Eq_hadron_wave_function_normalization}
\ee
%
requires the normalization constant
%
\be
        N_h = \int_0^1 dz_i \ z_i(1-z_i) \ e^{-(z_i-\inv{2})^2 / (2
        \Delta z_h^2)} 
        \ .
\label{Eq_N_h}
\ee
%
The different hadrons considered -- protons, pions, and kaons -- are
specified by $\Delta z_h$ and $S_h$ providing the width for the
distributions of the longitudinal momentum fraction carried by the
quark $z_i$ and transverse spatial extension $|\vec{r}_i|$,
respectively. In this work the extension parameter $S_h$ is a fit
parameter that should resemble approximately the electromagnetic
radius of the corresponding hadron~\cite{Dosch:2001jg}, while $\Delta
z_h = w/(\sqrt{2}\,m_h)$~\cite{Wirbel:1985ji} is fixed by the hadron
mass $m_h$ and the value $w = 0.35 - 0.5\,\GeV$ extracted from
experimental data. We find for (anti-)protons $\Delta z_p = 0.3$ and
$S_p = 0.86\,\fm$, for pions $\Delta z_{\pi} = 2$ and $S_{\pi} =
0.607\,\fm$, and for kaons $\Delta z_{K} = 0.57$ and $S_{K} =
0.55\,\fm$ which are the values used in the main text. For convenience
they are summarized in Table~\ref{Tab_Hadron_Parameters}.
%
\begin{table}
\caption{\small Hadron Parameters} 
\vspace{0.3cm}
\centering      
\begin{tabular}{|l|l|l|}\hline
Hadron        & $\Delta z_h$    & $S_h\;[\fm]$  \\ [1ex] \hline\hline       
$p, \bar{p}$    & $0.3$           & $0.86$  \\ \hline
$\pi^{\pm}$     & $2$             & $0.607$ \\ \hline
$K^{\pm}$      & $0.57$          & $0.55$ \\ \hline
\end{tabular}
\label{Tab_Hadron_Parameters}
\end{table}
%

Concerning the quark-diquark structure of the baryons, the more
conventional three-quark structure of a baryon would complicate the
model significantly but would lead to similar predictions once the
model parameters are readjusted~\cite{Dosch:1994ym}. In fact, there
are also physical arguments that favor the quark-diquark structure of
the baryon such as the $\delta I = 1/2$ enhancement in
semi-leptonic decays of baryons~\cite{Dosch:1989hu} and the strong
attraction in the scalar diquark channel in the instanton
vacuum~\cite{Schafer:1994ra}.

% ______________________________________________________________________________
\subsection*{The Photon Wave Function}   
% ______________________________________________________________________________

The photon wave function $\psi_{\gamma}(z_i,\vec{r}_i,Q^2)$ describes
the fluctuation of a photon with virtuality $Q^2$ into a
quark-antiquark pair with longitudinal quark momentum fraction $z_i$
and spatial transverse size and orientation $\vec{r}_i$. The
computation of the corresponding transition amplitude $\langle
q\qbar(z_i,\vec{r}_i)|\gamma^*(Q^2)\rangle$ can be performed
conveniently in light-cone perturbation theory~\cite{Bjorken:1971ah+X}
and leads to the following squared wave functions for transverse $(T)$
and longitudinally $(L)$ polarized photons~\cite{Nikolaev:1991ja}
%
\bea
\!\!\!\!\!\!\!\!\!\!\!\!\!|\psi_{\gamma_T^*}(z_i,\vec{r}_i,Q^2)|^2\! 
        &\!\!=\!\!&\!\frac{3\,\alphaEM}{2\,\pi^2} \sum_f e_f^2
                \left\{ 
                  \left[ z_i^2 + (1-z_i)^2\right]\,\epsilon_f^2\,K_1^2(\epsilon_f\,|\vec{r}_i|) 
                  + m_f^2\,K_0^2(\epsilon_f\,|\vec{r}_i|) 
                \right\}
        \label{Eq_photon_wave_function_T_squared} \\
\!\!\!\!\!\!\!\!\!\!\!\!\!|\psi_{\gamma_L^*}(z_i,\vec{r}_i,Q^2)|^2\! 
        &\!\!=\!\!&\!\frac{3\,\alphaEM}{2\,\pi^2} \sum_f e_f^2
                \left\{ 4\,Q^2\,z_i^2(1-z_i)^2\,K_0^2(\epsilon_f\,|\vec{r}_i|) \right\},
        \label{Eq_photon_wave_function_L_squared}
\eea
%
where $\alphaEM$ is the fine-structure constant, $e_f$ is the electric
charge of the quark with flavor $f$, and $K_0$ and $K_1$ are the modified
Bessel functions (McDonald functions). In the above expressions,
%
\be
        \epsilon_f^2 = z_i(1-z_i)\,Q^2 + m_f^2
\label{Eq_photon_extension_parameter}
\ee
%
controlls the transverse size(-distribution) of the emerging dipole,
$|\vec{r}_i| \propto 1/ \epsilon_f$, that depends on the quark flavor
through the current quark mass $m_f$.

For small $Q^2$, the perturbatively derived wave functions,
(\ref{Eq_photon_wave_function_T_squared}) and
(\ref{Eq_photon_wave_function_L_squared}), are not appropriate since
the resulting color dipoles of size $|\vec{r}_i| \propto 1/m_f \gg
1\,\fm$ should encounter non-perturbative effects such as confinement
and chiral symmetry breaking. To take these effects into account the
vector meson dominance (VMD) model~\cite{Bauer:1978iq} is usually
used. However, the transition from the ``partonic'' behavior at large
$Q^2$ to the ``hadronic'' one at small $Q^2$ can be modeled as well
by introducing $Q^2$-dependent quark masses, $m_f = m_f(Q^2)$, that
interpolate between the current quarks at large $Q^2$ and the
constituent quarks at small $Q^2$~\cite{Dosch:1998nw}. Following this
approach, we use~(\ref{Eq_photon_wave_function_T_squared})
and~(\ref{Eq_photon_wave_function_L_squared}) also in the low-$Q^2$
region but with the running quark masses
%
\bea
        m_{u,d}(Q^2) 
        &=& 0.178\,\GeV\,(1-\frac{Q^2}{Q^2_{u,d}})\,\Theta(Q^2_{u,d}-Q^2) 
        \ , 
        \label{Eq_m_ud_(Q^2)}\\
        m_s(Q^2) 
        &=& 0.121\,\GeV + 0.129\,\GeV\,(1-\frac{Q^2}{Q^2_s})\,\Theta(Q^2_s-Q^2) 
        \label{Eq_m_s_(Q^2)}
        \ ,
\eea
%
and the fixed charm quark mass
%
\be
        m_c = 1.25\,\GeV
        \ ,
\ee
%
where the parameters $Q^2_{u,d} = 1.05\,\GeV^2$ and $Q^2_s =
1.6\,\GeV^2$ are taken directly from~\cite{Dosch:1998nw} while we
reduced the values for the constituent quark masses $m_f(Q^2 = 0)$
of~\cite{Dosch:1998nw} by about 20\%. The smaller constituent quark
masses are necessary to reproduce the total cross sections for
$\gamma^* p$ and $\gamma^* \gamma^*$ reactions at low $Q^2$.  Similar
running quark masses are obtained in a QCD-motivated model of
spontaneous chiral symmetry breaking in the instanton
vacuum~\cite{Petrov:1998kf} that improve the description of $\gamma^*
p$ scattering at low $Q^2$~\cite{Martin:1999bh+X}.

% ______________________________________________________________________________
\section{Universal Energy Dependence}
\label{Sec_Energy_Dependence}
% ______________________________________________________________________________

Until now the $T$-matrix
element~(\ref{Eq_model_purely_imaginary_T_amplitude}) leads to energy
independent total cross sections in contradiction to the experimental
observation. In this section we introduce the energy dependence in a
phenomenological way inspired by other successful models.

Most models for high-energy scattering are constructed to describe
either hadron-hadron or photon-hadron reactions.  For example,
Kopeliovich et al.~\cite{Kopeliovich:2001pc} as well as Berger and
Nachtmann~\cite{Berger:1999gu} focus on hadron-hadron scattering. In
contrast, Golec-Biernat and W\"usthoff~\cite{Golec-Biernat:1999js+X}
and Forshaw, Kerley, and Shaw~\cite{Forshaw:1999uf} concentrate on
photon-proton reactions. A model that describes the energy dependence
in both hadron-hadron and photon-hadron reactions up to large photon
virtualities is the two-pomeron model of Donnachie and
Landshoff~\cite{Donnachie:1998gm+X}. Based on Regge
theory~\cite{Donnachie:en}, they find a soft pomeron trajectory with
intercept $ 1 + \epsilon_{\mathrm{soft}} \approx 1.08$ that governs the weak
energy dependence of hadron-hadron or $\gamma^{*} p$ reactions with
low $Q^2$~\cite{Donnachie:1992ny} and a hard pomeron trajectory with
intercept $1 + \epsilon_{\mathrm{hard}} \approx 1.4$ that governs the strong
energy dependence of $\gamma^*p$ reactions with high
$Q^2$~\cite{Donnachie:1998gm+X}. Similarly, we aim at a simultaneous
description of hadron-hadron, photon-proton, and photon-photon
reactions involving real and virtual photons as well.

In line with other two-component (soft $+$ hard)
models~\cite{Donnachie:1998gm+X,Forshaw:1999uf,Rueter:1998up,D'Alesio:1999sf,Donnachie:2001wt}
and the different hadronization mechanisms in soft and hard
collisions, our physical ansatz demands that the perturbative and
non-perturbative contributions do not interfere. Therefore, we modify
the cosine-summation in~(\ref{Eq_model_purely_imaginary_T_amplitude})
allowing only even numbers of soft and hard correlations, $\left
  (\chi^{\nprt} \right)^{2n} \left ( \chi^{\pert} \right)^{2m}$ with
$n,m \in I\!\!N$.  Interference terms with odd numbers of soft and
hard correlations are subtracted by the replacement
%
\be
        \cos\left[ c\chi \right] =  
        \cos\left[c\left( \chi^{\nprt} + \chi^{\pert} \right)\right] 
        \rightarrow 
        \cos\left[c\chi^{\nprt}\right]\cos\left[c\chi^{\pert}\right] 
        \ ,
\label{Eq_interference_term_subtraction}
\ee
%
where $c = 1/3$ or $2/3$. This prescription leads to the following
factorization of soft and hard physics in the $T$-matrix
element,
%
\bea
        && \!\!\!\!\!\!\!\!\!\!
        T(s,t) 
        = 2is \int \!\!d^2b_{\!\perp} 
        e^{i {\vec q}_{\!\perp} {\vec b}_{\!\perp}}
        \int \!\!dz_1 d^2r_1 \!\int \!\!dz_2 d^2r_2\,\,
        |\psi_1(z_1,\vec{r}_1)|^2 \,\, 
        |\psi_2(z_2,\vec{r}_2)|^2       
        \nonumber \\    
        &&\!\!\!\!\!\!\!\!\!\!\!\! 
        \times \left[ 1 - \frac{2}{3} 
        \cos\!\left(\!\frac{1}{3}\chi^{\nprt}\!\right)
        \cos\!\left(\!\frac{1}{3}\chi^{\pert}\!\right)         
        - \frac{1}{3}
        \cos\!\left(\!\frac{2}{3}\chi^{\nprt}\!\right)
        \cos\!\left(\!\frac{2}{3}\chi^{\pert}\!\right)
        \right] \ . 
%        \nonumber \\
\label{Eq_model_purely_imaginary_T_amplitude_almost_final_result}
\eea
%
In the limit of small $\chi$-functions, $\chi^{\nprt}
\ll 1$ and $\chi^{\pert} \ll 1$, one gets
%
\bea
        T(s,t) 
        & = & 2is \!\int \!\!d^2b_{\!\perp} 
        e^{i {\vec q}_{\!\perp} {\vec b}_{\!\perp}}
        \!\int \!\!dz_1 d^2r_1 \!\int \!\!dz_2 d^2r_2\,\,
        |\psi_1(z_1,\vec{r}_1)|^2 \,\,
        |\psi_2(z_2,\vec{r}_2)|^2       
        \nonumber \\  
        && \times \frac{1}{9}\left[
        \left(\chi^{\nprt}\right)^2
        +\left(\chi^{\pert}\right)^2
        \right]
\label{Eq_model_purely_imaginary_T_amplitude_small_chi_limit}
\eea
%
so that the $T$-matrix element becomes a sum of a perturbative and a
non-perturbative component. As already discussed in
Sec.~\ref{Sec_DD_Scattering_LLCM}, the perturbative component
$(\chi^{\pert})^2$ coincides with the well-known perturbative {\em
  two-gluon exchange}~\cite{Low:1975sv+X,Gunion:iy} and the
non-perturbative component $(\chi^{\nprt})^2$ represents the
corresponding non-perturbative gluonic interaction on the
``two-gluon-exchange'' level~\cite{Shoshi:2002fq}.

As the two-component structure of
(\ref{Eq_model_purely_imaginary_T_amplitude_small_chi_limit}) reminds
of the two-pomeron model of Donnachie and
Landshoff~\cite{Donnachie:1998gm+X}, we adopt the powerlike energy
increase and ascribe a weak energy dependence to the non-perturbative
component $\chi^{\nprt}$ and a strong one to the perturbative component
$\chi^{\pert}$
%
\bea
        \left(\chi^{\nprt}\right)^2 \quad & \rightarrow & \quad 
        \left(\chi^{\nprt}(s)\right)^2 := \left(\chi^{\nprt}\right)^2 
        \left(\frac{s}{s_0}
        \frac{\vec{r}_1^{\,2}\,\vec{r}_2^{\,2}}{R_0^4}\right)^{\epsilon^{\nprt}}
        \nonumber \\
        \left(\chi^{\pert}\right)^2 \quad & \rightarrow & \quad 
        \left(\chi^{\pert}(s)\right)^2 := \left(\chi^{\pert}\right)^2
        \left(\frac{s}{s_0} 
        \frac{\vec{r}_1^{\,2}\,\vec{r}_2^{\,2}}{R_0^4}\right)^{\epsilon^{\pert}}
\label{Eq_energy_dependence}
\eea
%
with the scaling factor $s_0 R_0^4$. The powerlike energy dependence
with the exponents $0\approx \epsilon^{\nprt} < \epsilon^{\pert} < 1$
guarantees Regge type behavior at moderately high energies, where the
small-$\chi$
limit~(\ref{Eq_model_purely_imaginary_T_amplitude_small_chi_limit}) is
appropriate. In~(\ref{Eq_energy_dependence}) the energy variable $s$
is scaled by the factor $\vec{r}_1^{\,2}\,\vec{r}_2^{\,2}$ that allows
to rewrite the energy dependence in photon-hadron scattering in terms
of the appropriate Bjorken scaling variable $x$
%
\be
        s\,\vec{r}_1^{\,2} \propto \frac{s}{Q^2} = \inv{x}
        \ ,
\label{Eq_x_Bj_<->_s}
\ee
%
where $|\vec{r}_1|$ is the transverse extension of the $q\qbar $
dipole in the photon. A similar factor has been used before in the
dipole model of Forshaw, Kerley, and Shaw~\cite{Forshaw:1999uf} and
also in the model of Donnachie and Dosch~\cite{Donnachie:2001wt} in
order to respect the scaling properties observed in the structure
function of the proton.\footnote{In the model of Donnachie and
  Dosch~\cite{Donnachie:2001wt}, $s\,|\vec{r}_1|\,|\vec{r}_2|$ is used
  as the energy variable if both dipoles are small, which is in
  accordance with the choice of the typical BFKL energy scale but
  leads to discontinuities in the dipole-dipole cross section. In
  order to avoid such discontinuities, we use the energy
  variable~(\ref{Eq_energy_dependence}) also for the scattering of two
  small dipoles.} In the dipole-proton cross section of Golec-Biernat
and W\"usthoff~\cite{Golec-Biernat:1999js+X}, Bjorken $x$ is used
directly as energy variable which is important for the success of the
model. In fact, also in our model, the
$\vec{r}_1^{\,2}\,\vec{r}_2^{\,2}$ factor improves the description of
$\gamma^{*} p$ reactions at large $Q^2$.

The powerlike Regge type energy dependence introduced
in~(\ref{Eq_energy_dependence}) is, of course, not mandatory but
allows successful fits and can also be derived in other theoretical
frameworks: A powerlike energy dependence is found for hard
photon-proton reactions from the BFKL equation~\cite{BFKL} and for
hadronic reactions by Kopeliovich et al.~\cite{Kopeliovich:2001pc}.
However, these approaches need unitarization since their powerlike
energy dependence will ultimately violate $S$-matrix unitarity at
asymptotic energies. In our model we use the following $T$-matrix
element for investigations in the remaining chapters
%
\bea
        &&\!\!\!\!\!\!\!\!\!\!
        T(s,t)  
        =  2is \int \!\!d^2b_{\!\perp} 
        e^{i {\vec q}_{\!\perp} {\vec b}_{\!\perp}}
        \int \!\!dz_1 d^2r_1 \!\int \!\!dz_2 d^2r_2\,\, 
        |\psi_1(z_1,\vec{r}_1)|^2 \,\, 
        |\psi_2(z_2,\vec{r}_2)|^2       
\label{Eq_model_purely_imaginary_T_amplitude_final_result}\\    
        &&\!\!\!\!\!\!\!\!\!\!\!\! 
        \times \left[1 - \frac{2}{3} 
        \cos\!\left(\!\frac{1}{3}\chi^{\nprt}(s)\!\right)
        \cos\!\left(\!\frac{1}{3}\chi^{\pert}(s)\!\right)         
        - \frac{1}{3}
        \cos\!\left(\!\frac{2}{3}\chi^{\nprt}(s)\!\right)
        \cos\!\left(\!\frac{2}{3}\chi^{\pert}(s)\!\right)
        \right] 
        \ , \nonumber
\eea
%
where the cosine functions ensure the unitarity condition in impact
parameter space as shown in Chap.~\ref{Sec_Impact_Parameter}. Indeed,
the multiple gluonic interactions associated with the higher order
terms in the expansion of the cosine functions are important for the
saturation effects observed within our model at ultra-high energies.

Having ascribed the energy dependence to the $\chi$-function, the
energy behavior of hadron-hadron, photon-hadron, and photon-photon
scattering results exclusively from the {\em universal} dipole-dipole
scattering kernel.

% ______________________________________________________________________________
\section{Model Parameters for High-Energy Scattering}
\label{Sec_Model_Parameters}
% ______________________________________________________________________________

Lattice QCD simulations provide important information and constraints
on the model parameters as discussed in Chap.~\ref{Sec_The_Model}. The
fine tuning of the parameters was, however, directly performed on
high-energy scattering data for hadron-hadron, photon-hadron, and
photon-photon reactions~\cite{Shoshi:2002in} where an error ($\chi^2$)
minimization was not fea\-si\-ble because of the non-trivial
multi-dimensional integrals in the $T$-matrix
element~(\ref{Eq_model_purely_imaginary_T_amplitude_final_result}).

The parameters $a$, $\kappa$, $G_2$, $m_G$, $M^2$, $s_0R^4_0$,
$\epsilon^{\nprt}$ and $\epsilon^{\pert}$ determine the dipole-dipole
scattering amplitude and are universal for all reactions considered.
In addition, there are reaction-dependent parameters in the wave
functions which are given in Sec.~\ref{Sec_Wave_Functions}.

The non-perturbative component involves the correlation length $a$,
the gluon condensate $G_2$, and the parameter $\kappa$ indicating the
non-Abelian character of the correlator. With the simple exponential
correlation function~(\ref{Eq_SVM_correlation_functions}), we obtain
the values given in~(\ref{Eq_MSV_scattering_fit_parameter_results})
that have already been applied in Chap.~\ref{Sec_Static_Sources}
%
\be
        a =  0.302\,\fm, \quad 
        \kappa = 0.74, \quad 
        G_2 = 0.074\,\GeV^4
        \ .
\nonumber
\ee
%

The perturbative component involves the gluon mass $m_G$ as IR
regulator (or inverse ``perturbative correlation length'') and the
parameter $M^2$ that freezes the running
coupling~(\ref{Eq_g2(z_perp)}) for large distance scales. The
following values are used in the following investigations
%
\be
        m_G =  m_{\rho} = 0.77\,\GeV 
        \quad \mbox{and }\quad 
        M^2 = 1.04\,\GeV^2
        \ .
\label{Eq_PGE_scattering_fit_parameter_results}
\ee
%

The energy dependence of the model is associated with the energy
exponents $\epsilon^{\nprt}$ and $\epsilon^{\pert}$, and the  scaling
parameter $s_0R^4_0$ 
%
\be
        \epsilon^{\nprt} = 0.125, \quad 
        \epsilon^{\pert} = 0.73, \quad \mbox{and} \quad 
        s_0 R_0^4 = (\,47\,\GeV\,\fm^2\,)^2
        \ .
\label{Eq_energy_dependence_scattering_fit_parameter_results}
\ee
%
In comparison to the energy exponents of Donnachie and
Landshoff~\cite{Donnachie:1992ny,Donnachie:1998gm+X}, $\epsilon_{\mathrm{soft}}
\approx 0.08$ and $\epsilon_{\mathrm{hard}} \approx 0.4$, our exponents are
significantly larger. However, the cosine functions in our $T$-matrix
element~(\ref{Eq_model_purely_imaginary_T_amplitude_final_result})
reduce the large exponents so that the energy dependence of the cross
sections agrees with the experimental data as illustrated in
Chap.~\ref{Sec_Comparison_Data}.

%%% Local Variables: 
%%% mode: latex
%%% TeX-master: "master"
%%% End: 
% Parameterizations.tex
%
% ___ Parameterizations ______________________________________
\chapter{Loop and Minimal Surface Parametrizations}
\label{Sec_Parameterizations}
% ______________________________________________________________________________

A rectangular {\em loop} $C_i$ with ``spatial'' extension $R_i$ and
``temporal'' extension $T_i$ placed in four-dimensional Euclidean
space --- as shown in Fig.~\ref{Fig_OneLoop_MinimalSurface} --- has
the following parameter representation
%
\be
        C_i 
        \,\,=\,\, 
        C_i^A \,\cup\, C_i^B \,\cup\, C_i^C \,\cup\, C_i^D
\label{Eq_Ci_parameterization}
\ee
%
with
%
\bea
        C_i^A \,\,=\,\,  
        \Big\{ 
        X_i^A(u_i) 
        & = & 
        X_{\!0\,i} - (1-z_i)\,r_i + u_i\,t_i,\quad\hphantom{v_i\,r_i}
        u_i \in [-T_i,T_i]
        \Big\} 
\label{Eq_Ci^A_parameterization}\\
        C_i^B \,\,=\,\, 
        \Big\{ 
        X_i^B(v_i) 
        & = & 
        X_{\!0\,i} - (1-z_i)\,r_i + v_i\,r_i + T_i\,t_i,\;\quad
        v_i \in [0,1]
        \Big\} 
\label{Eq_Ci^B_parameterization}\\
        C_i^C \,\,=\,\, 
        \Big\{ 
        X_i^C(u_i) 
        & = & 
        X_{\!0\,i} + z_i\,r_i + u_i\,t_i,\quad\hphantom{(1-z_i)\,r_i}
        u_i \in [T_i,-T_i]
        \Big\} 
\label{Eq_Ci^C_parameterization}\\
        C_i^D \,\,=\,\,
        \Big\{ 
        X_i^D(v_i) 
        & = & 
        X_{\!0\,i} - (1-z_i)\,r_i + v_i\,r_i + T_i\,t_i,\;\quad
        v_i \in [1,0]
        \Big\} 
\label{Eq_Ci^D_parameterization}
\eea
%
where
%
\be
        r_i
        := \left( \barray{c} 
        R_i\,\sin\theta_i\,\cos\phi_i \\ 
        R_i\,\sin\theta_i\,\sin\phi_i \\ 
        R_i\,\cos\theta_i\,\cos\Theta_i \\ 
        R_i\,\cos\theta_i\,\sin\Theta_i
        \earray \right)
        \quad \mbox{and} \quad
        t_i
        := \left( \!\! \barray{c} 
        0 \\ 
        0 \\
        -\sin\Theta_i \\
        \hphantom{-}\cos\Theta_i
        \earray \right)
        \ .
\label{Eq_ri_ti_four_vectors}
\ee
%
The ``center'' of the loop $C_i$ is given by $X_{\!0\,i}$. The
parameters $z_i$, $R_i$, $\theta_i$, $\phi_i$, and $\Theta_i$ are
defined in Fig.~\ref{Fig_OneLoop_MinimalSurface} that illustrates (a)
the spatial arrangement of a color dipole and (b) its world-line $C_i$
in Euclidean ``longitudinal'' space. The tilting angle $\Theta_i\neq
0$ is the central quantity in the analytic continuation presented in
Chap.~\ref{Sec_DD_Scattering}.  Moreover, $\Theta_1 =\pi/2$ together
with $\Theta_2 = 0$ allows us to compute conveniently the
chromo-magnetic field distributions in
Appendix~\ref{Sec_Chi_Computation}.

The {\em minimal surface} $S_i$ is the planar surface bounded by the
loop $C_i=\partial S_i$ given in~(\ref{Eq_Ci_parameterization}). It
can be parametrized as follows
%
\be
        S_i =  
        \Big\{ 
        X_i(u_i,v_i) 
        = X_{\!0\,i} - (1-z_i)\,r_i + v_i\,r_i + u_i\,t_i,
        \;u_i \in [-T_i,T_i], \;v_i \in [0,1] 
        \Big\}
\label{Eq_Si_parameterization}
\ee
%
with $r_i$ and $t_i$ given in~(\ref{Eq_ri_ti_four_vectors}). The
corresponding infinitesimal surface element reads
%
\be
        d\sigma_{\mu\nu}(X_i)
        = \Bigg( 
        \frac{\partial X_{\!i\mu}}{\partial u_i} 
        \frac{\partial X_{\!i\nu}}{\partial v_i}
        - \frac{\partial X_{\!i\mu}}{\partial v_i} 
        \frac{\partial X_{\!i\nu}}{\partial u_i} 
        \Bigg)\,du_i\,dv_i
        = \Bigg( 
        t_{i\mu} r_{i\nu} - r_{i\mu} t_{i\nu} 
        \Bigg)\,du_i\,dv_i
        \ .
\label{Eq_Si_surface_element}
\ee
%

\bigskip

%
\begin{figure}[h]
\centerline{\epsfig{figure=loop_parameterization_ite.eps,width=9.cm}}\hskip 0.5cm
\caption{\small
  (a) Spatial arrangement of a color dipole and (b) its world-line in
  Euclidean ``longitudinal'' space given by the rectangular {\em loop}
  $C_i$ that defines the {\em minimal surface} $S_i$ with $\partial
  S_i = C_i$. The minimal surface is represented by the shaded area.
  In our model, it is interpreted as the world-sheet of the QCD string
  that confines the quark and antiquark in the dipole.}
\label{Fig_OneLoop_MinimalSurface}
\end{figure}
%

%%% Local Variables: 
%%% mode: latex
%%% TeX-master: "master"
%%% End: 
% Saturation.tex
%
% ______________________________________________________________________________
\chapter{Impact Parameter Profiles and Gluon Saturation}
\label{Sec_Impact_Parameter}
% ______________________________________________________________________________

In this chapter $S$-matrix unitarity constraints are considered in our
model. On the basis of the impact parameter dependence of the
scattering amplitude, saturation effects can be exposed that manifest
the $S$-matrix unitarity. For each impact parameter the energy at
which the unitarity limit becomes important can be determined. The
results are used to discuss gluon saturation and to localize
saturation effects in experimental observables.

The impact parameter dependence of the scattering amplitude is given
by $\impactT(s,|\vec{b}_{\!\perp}|)$,
%
\be
        T(s,t=-{\vec q}_{\!\perp}^{\,\,2}) \;=\;
        4s\!\int \!\!d^2b_{\!\perp}\,
        e^{i {\vec q}_{\!\perp} {\vec b}_{\!\perp}}\,
        \impactT(s,|\vec{b}_{\!\perp}|)
        \ ,
\label{Eq_Fourier_transformed_T-matrix_element}
\ee
%
and in particular by the {\em profile function}
%
\be
        J(s,|\vec{b}_{\!\perp}|) 
        = 2\,\im\impactT(s,|\vec{b}_{\!\perp}|)
        \ ,
\label{Eq_profile_function_def}
\ee 
%
which describes the {\em blackness} or {\em opacity} of the
interacting particles as a function of the impact parameter $|{\vec
  b}_{\!\perp}|$ and the c.m.\ energy $\sqrt{s}$. In fact, the profile
function~(\ref{Eq_profile_function_def}) determines all observables if
the $T$-matrix is -- as in our model -- purely imaginary.

% ______________________________________________________________________________
\section[$S$-Matrix Unitarity Constraints]
{\boldmath$S$-Matrix Unitarity Constraints}
\label{Sec_S-Matrix_Unitarity}
% ______________________________________________________________________________

The $S$-matrix unitarity, $SS^{\dagger} = S^{\dagger}S = \Identity$,
leads directly to the {\em unitarity condition} in impact parameter
space\footnote{Integrating
  (\ref{Eq_unitarity_condition}) over the impact parameter space and
  multiplying by a factor of $4$ one obtains the relation
  $\sigma^{tot}(s) = \sigma^{el}(s) + \sigma^{inel}(s)$.}~\cite{Amaldi:1976gr,Castaldi:1983ft}
%
\be
%       \inv{2}\,J(s,\vec{b}_{\!\perp}) = 
        \im\impactT(s,|\vec{b}_{\!\perp}|)
        = |\impactT(s,|\vec{b}_{\!\perp}|)|^2 + G_{inel}(s,|\vec{b}_{\!\perp}|)
        \ ,
\label{Eq_unitarity_condition}
\ee 
%
where $G_{inel}(s,|\vec{b}_{\!\perp}|) \ge 0$ is the inelastic overlap
function~\cite{VanHove:1964rp}. This
unitarity condition imposes an absolute limit on the profile function
%
\be
        0 \;\;\leq\;\;
        2\,|\impactT(s,|\vec{b}_{\!\perp}|)|^2
        \;\;\leq\;\; 
        J(s,|\vec{b}_{\!\perp}|) 
        \;\;\leq\;\; 2
\label{Eq_absolute_unitarity_limit}
\ee 
%
and the inelastic overlap function, $G_{inel}(s,|\vec{b}_{\!\perp}|)
\le 1/4$.  At high energies, the elastic amplitude is expected to be
purely imaginary. Consequently, the solution
of~(\ref{Eq_unitarity_condition}) reads
%
\be
        J(s,|\vec{b}_{\!\perp}|) = 1 \pm \sqrt{1-4\,G_{inel}(s,|\vec{b}_{\!\perp}|)}
\label{Eq_solution_unitarity_condition}
\ee
%
and leads with the minus sign corresponding to the physical situation
to the {\em reduced unitarity bound}
%
\be
        0 \;\;\leq\;\;
        J(s,|\vec{b}_{\!\perp}|) 
        \;\;\leq\;\; 1
        \ .
\label{Eq_reduced_unitarity_bound}
\ee 
%
Reaching the {\em black disc limit} or {\em maximum opacity} at a
certain impact parameter $|\vec{b}_{\!\perp}|$,
$J(s,|\vec{b}_{\!\perp}|) = 1$, corresponds to maximal inelastic
absorption $G_{inel}(s,|\vec{b}_{\!\perp}|) = 1/4$ and equal elastic
and inelastic contributions to the total cross section at that impact
parameter.

In our model every reaction is reduced to dipole-dipole scattering
with well defined dipole sizes $|{\vec r}_i|$ and longitudinal quark
momentum fractions $z_i$ as discussed in
Sec.~\ref{Sec_Functional_Integral_Approach}. Thus, the most general
test of our model with respect to the unitarity constraints is
performed with the profile function
%
\be
        J_{DD}(s,|\vec{b}_{\!\perp}|,z_1,|\vec{r}_1|,z_2,|\vec{r}_2|)  
        = \int \frac{d\phi_1}{2\pi}  \int \frac{d\phi_2}{2\pi} 
        \left[1 - S_{DD}(s,\vec{b}_{\!\perp},z_1,{\vec r}_1,z_2,{\vec r}_2)\right]
        \ ,
\label{Eq_DD_profile_function}
\ee
%
where $\phi_i$ describes the dipole orientation, i.e.\ the angle
between ${\vec r}_i$ and $\vec{b}_{\!\perp}$, and $S_{DD}$ describes
{\em elastic dipole-dipole scattering}
%
\be
        S_{DD}
        = \frac{2}{3} 
        \cos\!\left(\frac{1}{3}\chi^{\nprt}(s)\right)
        \cos\!\left(\frac{1}{3}\chi^{\pert}(s)\right)         
        + \frac{1}{3}
        \cos\!\left( \frac{2}{3}\chi^{\nprt}(s)\right)
        \cos\!\left( \frac{2}{3}\chi^{\pert}(s)\right)
\label{Eq_S_DD_final_result}
\ee
%
with the purely real-valued eikonal functions $\chi^{\nprt}(s)$ and
$\chi^{\pert}(s)$ defined in~(\ref{Eq_energy_dependence}). Because of
$|S_{DD}| \leq 1$, a consequence of the cosine functions
in~(\ref{Eq_S_DD_final_result}) describing multiple gluonic
interactions, $J_{DD}$ respects the absolute
limit~(\ref{Eq_absolute_unitarity_limit}). Thus, the elastic
dipole-dipole scattering respects the unitarity
condition~(\ref{Eq_unitarity_condition}).  At high energies, the
arguments of the cosine functions in $S_{DD}$ become so large that
these cosines average to zero in the integration over the dipole
orientations. This leads to the black disc limit $J_{DD}^{\mathrm{max}} = 1$
reached at high energies first for small impact parameters.

% ______________________________________________________________________________
\section{The Profile Function for Proton-Proton Scattering}
\label{Sec_PP_Profile_Function}
% ______________________________________________________________________________

The profile function for proton-proton scattering
%
\be
        J_{pp}(s,|\vec{b}_{\!\perp}|)  = 
        \int \!\!dz_1 d^2r_1 \!\int \!\!dz_2 d^2r_2      
        |\psi_p(z_1,\vec{r}_1)|^2 |\psi_p(z_2,\vec{r}_2)|^2
        \left[1-S_{DD}(s,\vec{b}_{\!\perp},z_1,{\vec r}_1,z_2,{\vec r}_2)\right]
\label{Eq_model_pp_profile_function}
\ee
%
is obtained from~(\ref{Eq_DD_profile_function}) by weighting the
dipole sizes $|{\vec r}_i|$ and longitudinal quark momentum fractions
$z_i$ with the proton wave function $|\psi_p(z_i,\vec{r}_i)|^2$ from
Sec.~\ref{Sec_Wave_Functions}.

Using the model parameters from Sec.~\ref{Sec_Model_Parameters}, we
obtain the profile function $J_{pp}(s,|\vec{b}_{\!\perp}|)$ shown in
Fig.~\ref{Fig_J_pp(b,s)} for c.m.\ energies from $\sqrt{s} = 10\,\GeV$
to $10^8\,\GeV$.
%
\befig[t!]  
\centerline{\epsfig{figure=J_pp_W.eps,width=11.cm}}
\protect\caption{\small 
%
  The profile function for proton-proton scattering
  $J_{pp}(s,|\vec{b}_{\!\perp}|)$ as a function of the impact
  parameter $|\vec{b}_{\!\perp}|$ for c.m.\ energies from $\sqrt{s} =
  10\,\GeV$ to $10^8\,\GeV$. The unitarity
  limit~(\ref{Eq_absolute_unitarity_limit}) corresponds to
  $J_{pp}(s,|\vec{b}_{\!\perp}|) = 2$ and the black disc
  limit~(\ref{Eq_reduced_unitarity_bound}) to
  $J_{pp}(s,|\vec{b}_{\!\perp}|) = 1$.
%
}
\label{Fig_J_pp(b,s)}
\efig
%
Up to $\sqrt{s} \approx 100\,\GeV$, the profile has approximately a
Gaussian shape. Above $\sqrt{s}=1\,\TeV$, it develops into a broader
and higher profile until the black disc limit is reached for $\sqrt{s}
\approx 10^6\,\GeV$ and $|\vec{b}_{\!\perp}|=0$.  At this point, the
cosine functions in $S_{DD}$ average to zero
%
\be
        \int \!\!dz_1 d^2r_1 \!\int \!\!dz_2 d^2r_2  
        |\psi_p(z_1,{\vec r}_1)|^2|\psi_p(z_2,{\vec r}_2)|^2
        S_{DD}(\sqrt{s}\gtsim10^6\,\GeV,|\vec{b}_{\!\perp}|=0,\dots) 
        \approx 0
\label{Eq_pp_black_disc_limit_explained}
\ee
%
so that the proton wave function normalization determines the maximum
opacity
%
\be
        J_{pp}^{\mathrm{max}}
        =\int \!\!dz_1 d^2r_1 \!\int \!\!dz_2 d^2r_2\,        
        |\psi_p(z_1,{\vec r}_1)|^2\,|\psi_p(z_2,{\vec r}_2)|^2
        = 1
        \ .
\label{Eq_pp_black_disc_limit}
\ee
%
Once the maximum opacity is reached at a certain impact parameter, the
height of the profile function saturates at that $|\vec{b}_{\!\perp}|$
while the width of the profile function extends towards larger impact
parameters with increasing energy. Thus, multiple gluonic interactions
important to respect the $S$-matrix unitarity
constraint~(\ref{Eq_unitarity_condition}) lead to saturation for
$\sqrt{s} \gtsim 10^6\,\GeV$.

The above behavior of the profile function illustrates the evolution
of the proton with increasing c.m.\ energy. The proton is gray and of
small transverse size at small $\sqrt{s}$ but becomes blacker and more
transversally extended with increasing $\sqrt{s}$ until it reaches the
black disc limit in its center at $\sqrt{s} \approx 10^6\,\GeV$.
Beyond this energy, the proton cannot become blacker in its central
region but in its periphery with continuing transverse growth.
Furthermore, the proton boundary stays diffusive as claimed also
in~\cite{Frankfurt:2001nt+X}.

According to our model the black disc limit will not be reached at
LHC. Our prediction of $\sqrt{s} \approx 10^6\,\GeV = 10^3\,\TeV$ for
the onset of the black disc limit in proton-proton collisions is about
two orders of magnitude beyond the LHC energy $\sqrt{s} = 14\,\TeV$.
This is in contrast to, e.g.~\cite{Desgrolard:1999pr}, where the value
predicted for the onset of the black disc limit is $\sqrt{s} =
2\,\TeV$, i.e.\ small enough to be reached at LHC. However, note that
our profile function $J_{pp}(s,|\vec{b}_{\!\perp}|)$ yields good
agreement with experimental data for cross sections up to the highest
energies as shown in Chap.~\ref{Sec_Comparison_Data}.

For hadron-hadron reactions in general, the wave function
normalization of the hadrons determines the maximum opacity analogous
to~(\ref{Eq_pp_black_disc_limit}) and the transverse hadron size the
c.m.\ energy at which it is reached. Consequently, the maximum opacity
obtained for $\pi p$ and $K p$ scattering is identical to the one for
$pp$ scattering due to the
normalization~(\ref{Eq_hadron_wave_function_normalization}). The
smaller size of pions and kaons in comparison to protons, however,
demands slightly higher c.m.\ energies to reach this maximum opacity.
Such size effects become more pronounced in longitudinal photon-proton
scattering, where the size of the dipole emerging from the photon can
be controlled by the photon virtuality.

% ______________________________________________________________________________
\section{The Profile Function for Photon-Proton Scattering}
\label{Sec_GP_Profile_Function}
% ______________________________________________________________________________

The profile function for a longitudinal photon $\gamma_L^*$ scattering
off a proton $p$
%
\bea
        J_{\gamma_L^* p}(s,|\vec{b}_{\!\perp}|,Q^2)  
        & = & 
        \int \!\!dz_1 d^2r_1 \!\int \!\!dz_2 d^2r_2\,
        |\psi_{\gamma_L^*}(z_1,\vec{r}_1,Q^2)|^2 \,
        |\psi_p(z_2,\vec{r}_2)|^2
        \nonumber \\
        && 
        \times
        \left[1-S_{DD}(\vec{b}_{\!\perp},s,z_1,{\vec r}_1,z_2,{\vec r}_2)\right]
\label{Eq_model_gp_profile_function}
\eea
%
is calculated with the longitudinal photon wave function
$|\psi_{\gamma_L^*}(z_i,\vec{r}_i,Q^2)|^2$ given
in~(\ref{Eq_photon_wave_function_L_squared}). In this way, the profile
function~(\ref{Eq_model_gp_profile_function}) is ideally suited for
the investigation of dipole size effects since the photon virtuality
$Q^2$ determines the transverse size of the dipole into which the
photon fluctuates before it interacts with the proton.

Figure~\ref{Fig_J_gp_(b,s,Q^2)} shows the $|\vec{b}_{\!\perp}|$
dependence of the profile function
$J_{\gamma_L^*p}(s,|\vec{b}_{\!\perp}|,Q^2)$ divided by
$\alphaEM/{\pi}$ at a photon virtuality of $Q^2 = 1\,\GeV^2$ for c.m.\ 
energies $\sqrt{s}$ from $10\,\GeV$ to $10^9\,\GeV$, where $\alphaEM$
is the fine-structure constant.
%
\befig[t] \centerline{\epsfig{figure=J_gp_L.eps,width=11cm}}
\protect\caption{\small 
%
  The profile function for a longitudinal photon scattering off a
  proton $J_{\gamma_L^* p}(s,|\vec{b}_{\!\perp}|,Q^2)$ divided by
  $\alphaEM/{\pi}$ as a function of the impact parameter
  $|\vec{b}_{\!\perp}|$ at a photon virtuality of $Q^2 = 1\,\GeV^2$
  and c.m.\ energies from $\sqrt{s} = 10\,\GeV$ to $10^9\,\GeV$. The
  value of the black disc limit is
  $J_{\gamma_L^*p}^{\mathrm{max}}(Q^2=1\,\GeV^2) = 0.00164$\ .
%
}
\label{Fig_J_gp_(b,s,Q^2)}
\end{figure}
%
One clearly sees that the qualitative behavior of this rescaled
profile function is similar to the one for proton-proton scattering.
However, the black disc limit induced by the underlying dipole-dipole
scattering depends on the photon virtuality $Q^2$ and is given by the
normalization of the longitudinal photon wave function
%
\bea
        J_{\gamma_L^* p}^{\mathrm{max}}(Q^2)  
        = \int \!\!dz d^2r |\psi_{\gamma_L^*}(z,\vec{r},Q^2)|^2
\label{Eq_gp_black_disc_limit}
\eea
%
since the proton wave function is normalized to one.

The photon virtuality $Q^2$ does not only determine the absolute value
of the black disc limit but also the c.m.\ energy at which it is
reached. This is illustrated in Fig.~\ref{Fig_J_gp_(b=0,s,Q^2)}, where
the $\sqrt{s}$ dependence of $J_{\gamma_L^*
  p}(s,|\vec{b}_{\!\perp}|=0,Q^2)$ divided by $\alphaEM/{\pi}$ is
presented for $Q^2 = 1,\,10,\,\mbox{and}\,100\,\GeV^2$.
%
\befig[t]
\centerline{\epsfig{figure=Jb0_gp_L.eps,width=10.5cm}}
\caption{\small 
  The profile function for a longitudinal photon scattering off a
  proton $J_{\gamma_L^*p}(s,|\vec{b}_{\!\perp}|,Q^2)$ divided by
  $\alphaEM/{\pi}$ as a function of the c.m.\ energy $\sqrt{s}$ at
  zero impact parameter ($|\vec{b}_{\!\perp}|=0$) for photon
  virtualities of $Q^2 = 1,\,10,\,\mbox{and}\,100\,\GeV^2$.}
\label{Fig_J_gp_(b=0,s,Q^2)}
\end{figure}
%
With increasing resolution $Q^2$, i.e.\ decreasing dipole sizes,
$|\vec{r}_{\gamma_L^*}|^2 \propto 1/Q^2$, the absolute value of the
black disc limit grows and higher energies are needed to reach this
limit.\footnote{Note that the Bjorken $x$ at which the black disc
  limit is reached decreases with increasing photon virtuality $Q^2$.
  See also Fig.~\ref{Fig_xg(x,Q^2,b=0)_vs_x}.} The growth of the
absolute value of the black disc limit is simply due to the
normalization of the longitudinal photon wave function while the need
for higher energies to reach this limit is due to the decreasing
interaction strength with decreasing dipole size. The latter explains
also why the energies needed to reach the black disc limit in $\pi p$
and $K p$ scattering are higher than in $pp$ scattering. Comparing
$\gamma_L^*p$ scattering at $Q^2=1\,\GeV^2$ with $pp$ scattering
quantitatively, the black disc limit
$J_{\gamma_L^*p}^{\mathrm{max}}(Q^2=1\,\GeV^2) = 0.00164$ is about three orders
of magnitude smaller because of the photon wave function normalization
($\propto \alphaEM/{\pi}$). At $|\vec{b}_{\!\perp}|=0$ it is reached
at an energy of $\sqrt{s} \approx 10^8\,\GeV$, which is about two
orders of magnitude higher because of the smaller dipoles involved.

The way in which the profile function $J_{\gamma_L^*
  p}(s,|\vec{b}_{\!\perp}|,Q^2)$ approaches the black disc limit at
high energies depends on the shape of the proton and longitudinal
photon wave function at small dipole sizes $|\vec{r}_{1,2}|$. At high
energies, dipoles of typical sizes $0 \leq |\vec{r}_{1,2}| \leq
R_0\,(s_0/s)^{1/4}$ give the main contribution to
$S_{\gamma_L^*p} = 1 - J_{\gamma_L^*p}$ because
of~(\ref{Eq_energy_dependence}) and the fact that the contribution of
the large dipole sizes averages to zero upon integration over the
dipole orientations,
cf.~also~(\ref{Eq_pp_black_disc_limit_explained}). Since
$S_{\gamma_L^*p}$ is a measure of the proton transmittance, this means
that only small dipoles can penetrate the proton at high energies.
Increasing the energy further, even these small dipoles are absorbed
and the black disc limit is reached. However, the dependence of the
profile function on the short distance behavior of normalizable wave functions is
weak which can be understood as follows. Because of color
transparency, the eikonal functions $\chi^{\nprt}(s)$ and
$\chi^{\pert}(s)$ are small for small dipole sizes $0 \leq
|\vec{r}_{1,2}| \leq R_0\,(s_0/s)^{1/4}$ at large
$\sqrt{s}$. Consequently, $S_{DD} \approx 1$ and
%
\bea
        \!\!\!\!\!\!\!\!
        && 
        \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
        J_{\gamma_L^* p}(s,|\vec{b}_{\!\perp}|,Q^2)  
\nonumber \\
        \!\!\!\!\!\!\!\!
        &&
        \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
         \approx 
        J_{\gamma_L^* p}^{\mathrm{max}}(Q^2) - 4\pi^2\!\!
        \int\limits_0^1 \!\!dz_1 \!\!
        \int\limits_0^{r_c(s)}\!\!\!dr_1 r_1 
        |\psi_{\gamma_L^*}(z_1,r_1,Q^2)|^2 
        \int\limits_0^1 \!\!dz_2 \!\!
        \int\limits_0^{r_c(s)}\!\!\!dr_2 r_2 
        |\psi_p(z_2,r_2)|^2
\label{Eq_model_gp_profile_function_wavefunction_independence}
\eea
%
where $r_c(s) \approx R_0\,(s_0/s)^{1/4}$. Clearly, the
linear behavior from the phase space factors $r_{1,2}$ dominates over
the $r_{1,2}$-dependence of normalizable wave functions.\footnote{For our
  choice of the wave functions
  in~(\ref{Eq_model_gp_profile_function_wavefunction_independence}),
  one sees very explicitly that the specific Gaussian behavior of
  $|\psi_p(z_2,r_2)|^2$ and the logarithmic short distance behavior of
  $|\psi_{\gamma_L^*}(z_1,r_1,Q^2)|^2$ is dominated by the phase space
  factors $r_{1,2}$.} More generally, for any profile function
involving normalizable wave functions, the way in which the black disc
limit is approached depends only weakly on the short distance behavior
of the wave functions.

% ______________________________________________________________________________
\section{A Scenario for Gluon Saturation}
\label{Sec_Gluon_Saturation}
% ______________________________________________________________________________

In this section we discuss saturation of the {\em impact parameter
  dependent gluon distribution} of the proton
$xG(x,Q^2,|\vec{b}_{\!\perp}|)$.  Using a leading twist,
next-to-leading order QCD relation between $xG(x,Q^2)$ and the
longitudinal structure function $F_L(x,Q^2)$, we relate
$xG(x,Q^2,|\vec{b}_{\!\perp}|)$ to the profile function $J_{\gamma_L^*
  p}(s=Q^2/x,|\vec{b}_{\!\perp}|,Q^2)$ and find low-$x$ saturation of
$xG(x,Q^2,|\vec{b}_{\!\perp}|)$ as a manifestation of $S$-matrix
unitarity. The resulting $xG(x,Q^2,|\vec{b}_{\!\perp}|)$ is, of
course, only an estimate since our profile function contains also
higher twist contributions. Furthermore, in the considered low-$x$
region, the leading twist, next-to-leading order QCD formula may be
inadequate as higher twist contributions~\cite{Martin:1998kk+X} and
higher order QCD corrections~\cite{Gribov:1983tu,Mueller:1986wy} are
expected to become important. Nevertheless, still assuming a close
relation between $F_L(x,Q^2)$ and $xG(x,Q^2)$ at low $x$, we think
that our approach provides some insight into the gluon distribution as
a function of the impact parameter and into its saturation.

The {\em gluon distribution}\ of the proton $~xG(x,Q^2)~$ is defined
as follows: $xG(x,Q^2)dx$ gives the momentum fraction of the proton
which is carried by the gluons in the interval $[x, x+dx]$ as seen by
probes of virtuality $Q^2$. The {\em impact parameter dependent gluon
  distribution} $xG(x,Q^2,|\vec{b}_{\!\perp}|)$ is the gluon
distribution $xG(x,Q^2)$ at a given impact parameter
$|\vec{b}_{\!\perp}|$ so that
%
\be
        xG(x,Q^2) = \int
        \!\!d^2b_{\!\perp}\,xG(x,Q^2,|\vec{b}_{\!\perp}|) \ .
\label{Eq_def_xg(x,Q^2)}
\ee
%

In leading twist, next-to-leading order QCD, the gluon distribution
$xG(x,Q^2)$ is related to the structure functions $F_L(x,Q^2)$ and
$F_2(x,Q^2)$ of the proton~\cite{Martin:1988vw}
%
\be
        F_L(x, Q^2) 
        = \frac{\alphaS}{\pi}\!
        \left[
        \frac{4}{3}\int_x^1 \!
        \frac{dy}{y}\!\left(\frac{x}{y}\right)^{\!\!2} \!F_2(y,Q^2)
        + 2 \sum_f e_f^2\!\int_x^1 \!
        \frac{dy}{y}\!\left(\frac{x}{y}\right)^{\!\!2} \!\!
        \left(\!1-\frac{x}{y}\right) yG(y,Q^2)
        \right]
\label{Eq_FL_QCD_prediction}
\ee
%
where $\sum_f e_f^2$ is a flavor sum over the quark charges squared.
For four active flavors and $x \ltsim 10^{-3}$,
(\ref{Eq_FL_QCD_prediction}) can be approximated as
follows~\cite{Cooper-Sarkar:1988ds+X}
%
\be
        xG(x,Q^2) 
        \approx \frac{3}{5}\,5.8\, 
        \left[ 
        \frac{3\pi}{4\alphaS}\, F_L(0.417 x, Q^2)
        - \inv{1.97}\, F_2(0.75 x, Q^2) 
        \right] 
        \ .
\label{Eq_xg(x,Q^2)_approximation}
\ee
%
For typical $\Lambda_{QCD} = 100-300\,\MeV$ and $Q^2 = 50 -
100\,\GeV^2$, the coefficient of $F_L$ in
(\ref{Eq_xg(x,Q^2)_approximation}), $3\pi/(4\alphaS) = {\cal{O}}(10)$,
is large compared to the one of $F_2$. Taking into account also the
values of $F_2$ and $F_L$, the gluon distribution is mainly determined
by the longitudinal structure function for $x \ltsim 10^{-3}$ in this
$Q^2$ region. The longitudinal structure function can be expressed in
terms of the profile function for longitudinal photon-proton
scattering using the optical theorem (cf.~(\ref{Eq_optical_theorem}))
%
\be
        F_L(x,Q^2) 
        = \frac{Q^2}{4\,\pi^2\,\alphaEM}\,
        \sigma^{tot}_{\gamma^*_L p}(x,Q^2) 
        = \frac{Q^2}{4\,\pi^2\,\alphaEM}\, 
        2\!\int \!\!d^2b_{\!\perp}\,
        J_{\gamma_L^*p}(x,|\vec{b}_{\!\perp}|,Q^2) 
        \ ,
\label{fl}
\ee
%
where the $s$-dependence of the profile function is rewritten in terms
of the Bjorken scaling variable $x = Q^2/s$. Thus, neglecting the
$F_2$ term in~(\ref{Eq_xg(x,Q^2)_approximation}), the gluon
distribution reduces to
%
\be
        xG(x,Q^2) 
        \approx
        1.305\,\frac{Q^2}{\pi^2 \alphaS}\,\frac{\pi}{\alphaEM}
        \int \!\!d^2b_{\!\perp}\,
        J_{\gamma_L^*p}(0.417 x,|\vec{b}_{\!\perp}|,Q^2)
        \ . 
        % 1.305 = 9*5.8/40
\label{Eq_xg(x,Q^2)-J_gLp(x,b,Q^2)_connection}
\ee
%
Comparing (\ref{Eq_def_xg(x,Q^2)}) with
(\ref{Eq_xg(x,Q^2)-J_gLp(x,b,Q^2)_connection}), it seems natural to
relate the integrand of (\ref{Eq_xg(x,Q^2)-J_gLp(x,b,Q^2)_connection})
to the impact parameter dependent gluon distribution
%
\be
        xG(x,Q^2,|\vec{b}_{\!\perp}|) 
        \approx
        1.305\,\frac{Q^2}{\pi^2 \alphaS}\,\frac{\pi}{\alphaEM}\,
        J_{\gamma_L^*p}(0.417 x,|\vec{b}_{\!\perp}|,Q^2)
        \ .
\label{Eq_xg(x,Q^2,b)-J_gLp(x,b,Q^2)_relation}
\ee
%

The black disc limit of the profile function for longitudinal
photon-proton scattering~(\ref{Eq_gp_black_disc_limit}) imposes
accordingly an upper bound on $xG(x,Q^2,|\vec{b}_{\!\perp}|)$
%
\be
        xG(x,Q^2,|\vec{b}_{\!\perp}|)\ \leq \ 
        xG^{\mathrm{max}}(Q^2)
        \approx 
        1.305\,\frac{Q^2}{\pi^2 \alphaS}\,\frac{\pi}{\alphaEM}\,
        J_{\gamma^*_L p}^{\mathrm{max}}(Q^2)
        \ ,
\label{Eq_low_x_saturation} 
\ee
%
which is the low-$x$ saturation value of the gluon distribution
$xG(x,Q^2,|\vec{b}_{\!\perp}|)$ in our approach. With $\pi
J_{\gamma^*_L p}^{\mathrm{max}}(Q^2)/\alphaEM \approx 1$ as shown in
Fig.~\ref{Fig_J_gp_(b=0,s,Q^2)}, a compact approximation
of~(\ref{Eq_low_x_saturation}) is obtained
%
\be
        xG(x,Q^2,|\vec{b}_{\!\perp}|)\ \leq \ 
        xG^{\mathrm{max}}(Q^2)
        \approx 
        \frac{Q^2}{\pi^2 \alphaS}
        \ ,
\label{Eq_low_x_saturation_approximation} 
\ee
%
which is consistent with the results
in~\cite{Mueller:1986wy,Mueller:1999wm,Iancu:2001md} and indicates
strong color field strengths $G^a_{\mu \nu} \sim 1/ \sqrt{\alphaS}$ as
well.

According to our
relations~(\ref{Eq_xg(x,Q^2,b)-J_gLp(x,b,Q^2)_relation})
and~(\ref{Eq_low_x_saturation}), the {\em blackness} described by the
profile function is a measure for the gluon distribution and the {\em
  black disc limit} corresponds to the maximum gluon distribution
reached at the impact parameter under consideration. In accordance
with the behavior of the profile function $J_{\gamma_L^*p}$, see
Fig.~\ref{Fig_J_gp_(b,s,Q^2)}, the gluon distribution
$xG(x,Q^2,|\vec{b}_{\!\perp}|)$ decreases with increasing impact
parameter for given values of $x$ and $Q^2$. Consequently, the gluon
density has its maximum in the geometrical center of the proton, i.e.\ 
at zero impact parameter, and decreases towards the periphery.  With
decreasing $x$ at given $Q^2$, the gluon distribution
$xG(x,Q^2,|\vec{b}_{\!\perp}|)$ increases and extends towards larger
impact parameters just as the profile function $J_{\gamma_L^*p}$ for
increasing $s$.  The saturation of the gluon distribution
$xG(x,Q^2,|\vec{b}_{\!\perp}|)$ sets in first in the center of the
proton ($|\vec{b}_{\!\perp}|=0$) at very small Bjorken $x$.

In Fig.~\ref{Fig_xg(x,Q^2,b=0)_vs_x} the gluon distribution
$xG(x,Q^2,|\vec{b}_{\!\perp}|=0)$ is shown as a function of $x$ for
$Q^2 = 1,\,10,\,\mbox{and}\,100\,\GeV^2$, where
relation~(\ref{Eq_xg(x,Q^2,b)-J_gLp(x,b,Q^2)_relation}) has been used
also for low photon virtualities.
%
\begin{figure}[t]
  \centerline{\epsfig{figure=xG_x_q2_b0.eps,width=11cm}}
  \caption{\small 
    The gluon distribution of the proton at zero impact parameter
    $xG(x,Q^2,|\vec{b}_{\!\perp}|=0)$ as a function of $x$ for $Q^2 =
    1,\,10,\,\mbox{and}\,100\,\GeV^2$. The results are obtained within
    approximation~(\ref{Eq_xg(x,Q^2,b)-J_gLp(x,b,Q^2)_relation}).}
\label{Fig_xg(x,Q^2,b=0)_vs_x}
\end{figure}
%
Evidently, the gluon distribution $xG(x,Q^2,|\vec{b}_{\!\perp}|=0)$
saturates at very low values of $x \ltsim 10^{-10}$ for $Q^2 \gtsim
1\,\GeV^2$. The photon virtuality $Q^2$ determines the saturation
value~(\ref{Eq_low_x_saturation}) and the Bjorken $x$ at which it is
reached; cf.\ also Fig.~\ref{Fig_J_gp_(b,s,Q^2)}.  For larger $Q^2$,
the low-$x$ saturation value is larger and is reached at smaller
values of $x$, as claimed also in \cite{Gotsman:2001ku}.  Moreover, the
growth of $xG(x,Q^2,|\vec{b}_{\!\perp}|=0)$ with decreasing $x$ becomes
stronger with increasing $Q^2$. This results from the stronger energy
increase of the perturbative component, $\epsilon^{\pert} = 0.73$, that
becomes more important with decreasing dipole size.

According to our approach, the onset of the
$xG(x,Q^2,|\vec{b}_{\!\perp}|)$-saturation appears for $Q^2 \gtsim
1\,\GeV^2$ at $x \ltsim 10^{-10}$, which is far below the $x$-region
accessible at HERA ($x \gtsim 10^{-6}$). Even for THERA ($x\gtsim
10^{-7}$), gluon saturation is not predicted for $Q^2 \gtsim 1
\,\GeV^2$. However, since the HERA data can be described by models
with and without saturation embedded~\cite{Gotsman:2001ku}, the
present situation is not conclusive.\footnote{So far, the most
  striking hint for saturation in the present HERA data at $x\approx
  10^{-4}$ and $Q^2 < 2\,\GeV^2$ has been the turnover of
  $dF_2(x,Q^2)/d\ln(Q^2)$ towards small $x$ in the Caldwell
  plot~\cite{Abramowicz:1999ii}, which is still a controversial issue
  due to the correlation of $Q^2$ and $x$ values.}

Note that the $S$-matrix unitarity
condition~(\ref{Eq_unitarity_condition}) together
with~(\ref{Eq_xg(x,Q^2,b)-J_gLp(x,b,Q^2)_relation}) requires the
saturation of the impact parameter dependent gluon distribution
$xG(x,Q^2,|\vec{b}_{\!\perp}|)$ but not the saturation of the
integrated gluon distribution $xG(x,Q^2)$. Due to multiple gluonic
interactions in our model, this requirement is fulfilled, as can be
seen from Fig.~\ref{Fig_J_gp_(b,s,Q^2)} and
relation~(\ref{Eq_xg(x,Q^2,b)-J_gLp(x,b,Q^2)_relation}). Indeed,
approximating the gluon distribution $xG(x,Q^2,|\vec{b}_{\!\perp}|)$
in the saturation regime of very low $x$ by a step-function
%
\be
        xG(x,Q^2,|\vec{b}_{\!\perp}|) 
        \approx xG^{\mathrm{max}}(Q^2)\,
        \Theta(\,R(x,Q^2)-|\vec{b}_{\!\perp}|\,)
        \ ,
\label{Eq_J_gp_(x,b,Q^2)_Theta-approximation}
\ee
%
where $R(x,Q^2)$ denotes the full width at half maximum of the profile
function, one obtains with~(\ref{Eq_def_xg(x,Q^2)}),
(\ref{Eq_low_x_saturation}) and
(\ref{Eq_low_x_saturation_approximation}) the integrated gluon
distribution
%
\be
        xG(x,Q^2) 
        \;\approx\;
        1.305\,\frac{Q^2\,R^2(x,Q^2)}{\pi \alphaS}\,
        \frac{\pi}{\alphaEM}\,
        J_{\gamma^*_L p}^{\mathrm{max}}(Q^2)
        \;\approx\;
        \frac{Q^2\,R^2(x,Q^2)}{\pi\alphaS}
        \ ,
\label{Eq_xg(x,Q^2)_saturation_regime}
\ee
%
which does not saturate because of the increase of the effective
proton radius $R(x,Q^2)$ with decreasing $x$. Nevertheless, although
$xG(x,Q^2)$ does not saturate, the saturation of
$xG(x,Q^2,|\vec{b}_{\!\perp}|)$ leads to a slow-down in its growth
towards small $x$.\footnote{This is analogous to the rise of the total
  $pp$ cross section with growing c.m.\ energy that slows down as the
  corresponding profile function $J_{pp}(s,|\vec{b}_{\!\perp}|)$
  reaches its black disc limit as shown in
  Sec.~\ref{Sec_Total_Cross_Sections}.} Interestingly, our
result~(\ref{Eq_xg(x,Q^2)_saturation_regime}) coincides with the
result of Mueller and Qiu~\cite{Mueller:1986wy}.

Finally, it must be emphasized that the low-$x$ saturation of
$xG(x,Q^2,|\vec{b}_{\!\perp}|)$, required in our approach by the
$S$-matrix unitarity, is realized by {\em multiple gluonic
  interactions}. In other approaches that describe the evolution of
the gluon distribution with varying $x$ and $Q^2$, {\em gluon
  recombination} leads to gluon
saturation~\cite{Gribov:1983tu,Mueller:1986wy,McLerran:1994ni+X,Jalilian-Marian:1999dw+X,Iancu:2001hn+X},
which is reached when the probability of a gluon splitting up into two
is equal to the probability of two gluons fusing into one. A more
phenomenological understanding of saturation is attempted
in~\cite{Golec-Biernat:1999js+X,Capella:2001hq+X}.

%%% Local Variables: 
%%% mode: latex
%%% TeX-master: "master"
%%% End: 
% StaticSources.tex
%
% ______________________________________________________________________________
\chapter{Static Color Dipoles and Confining QCD Strings}
\label{Sec_Static_Sources}
% ______________________________________________________________________________

In this chapter we apply the loop-loop correlation model to compute
the QCD potential and the chromo-field distributions of static color
dipoles in the fundamental and adjoint representation of $SU(N_c)$.
Special emphasis is on Casimir scaling behavior and the interplay
between perturbative Coulomb behavior and non-perturbative formation
of the confining QCD string. Moreover, low-energy theorems are
discussed that relate the energy and action stored in the
chromo-fields to the static quark-antiquark potential.  These energy
and action sum rules allow us to show consistency of the model results
and to determine the values of $\beta$ and $\alphaS$ at the
renormalization scale at which the non-perturbative SVM component is
working.

% ______________________________________________________________________________
\section{The Static Color Dipole Potential}
\label{Sec_Static_Potential}
% ______________________________________________________________________________

The static color dipole -- two static color sources separated by a
distance $R$ in a net color singlet state -- is described by a {\WW}
loop $W_r[C]$ with a rectangular path $C$ of spatial extension $R$ and
temporal extension $T\to\infty$ where $r$ indicates the $SU(N_c)$
representation of the considered sources.  Figure~\ref{Fig_ONE_WWL}
%
\begin{figure}[b!]
  \centerline{\epsfig{figure=one_loop_potential_ite.eps,width=12.cm}}
\caption{\small 
  A static color dipole of size $R$ in the fundamental representation.
  The rectangular path $C$ of spatial extension $R$ and temporal
  extension $T$ indicates the world-line of the dipole described the
  {\WW} loop $W_{\fundamental}[C]$. The shaded area bounded by the
  loop $C=\partial S$ represents the minimal surface $S$ used to
  compute the static dipole potential.}
\label{Fig_ONE_WWL}
\end{figure}
%
illustrates a static color dipole in the fundamental representation
$r=\Fundamental$.  The potential of the static color dipole is
obtained from the VEV of the corresponding Wegner-Wilson
loop~\cite{Wilson:1974sk,Brown:1979ya}
%
\be
        V_r(R) 
        = - \lim_{T \to \infty} \inv{T} 
        \ln \langle W_r[C] \rangle_{\pot}
        \ ,
\label{Eq_static_potential}
\ee
%
where ``pot'' indicates the subtraction of the self-energy of the
color sources. The static quark-antiquark potential $V_{\fundamental}$
is obtained from a loop in the fundamental rep\-re\-sen\-ta\-tion
($r\!=\!\Fundamental$) and the potential of a static gluino pair
$V_{\adjoint}$ from a loop in the adjoint representation
($r\!=\!\Adjoint$).

With our result for $\langle W_r[C] \rangle$,
(\ref{Eq_final_result_<W[C]>}), obtained with the Gaussian
approximation in the gluon field strength, the static potential reads
%
\be
        V_r(R) = \frac{C_2(r)}{2}\,\lim_{T \to \infty} \inv{T}\,\chi_{SS\,\pot}
        \ , 
\label{Eq_Vr(R)_Gaussian_approximation}
\ee
%
with the self-energy subtracted, i.e.\ $\chi_{SS\,\pot} := \chi_{SS} -
\chi_{SS\,\self}$ (see Appendix~\ref{Sec_Chi_Computation}). According
to the structure of the gluon field strength correlator,
(\ref{Eq_Ansatz}) and~(\ref{Eq_F_decomposition}), there are
perturbative ($\pert$) and non-perturbative ($\nprt$) contributions to
the static potential
%
\be
        V_r(R) = \frac{C_2(r)}{2}\,\lim_{T \to \infty} \inv{T}\,
                 \left\{\chi_{SS\,\pot}^{\pert}
                   +\left(\chi_{SS\,\pot}^{\nprt\,\,nc} +
                     \chi_{SS\,\pot}^{\nprt\,\,c} \right) \right\}
        \ ,
\label{Eq_Vr(R)_P+NP}
\ee
%
where the explicit form of the $\chi$\,-\,functions is given
in~(\ref{Eq_chi_SS_NP_c_T->infty_V_E}),
(\ref{Eq_chi_SS_NP_nc_T->infty_pot_E}),
and~(\ref{Eq_chi_SS_P_T->infty_pot_E}).

The perturbative contribution to the static potential describes the
{\em color Yukawa potential} (which reduces to the {\em color Coulomb
  potential}~\cite{Kogut:1979wt} for $m_G=0$)
%
\be
        V_r^{\pert}(R) 
        = - C_2(r)\,\frac{g^2(R)}{4 \pi R} \exp[-m_G R] 
        \ .
\label{Eq_Vr(R)_color-Yukawa}
\ee
%
Here we have used the result for $\chi_{SS\,\pot}^{\pert}$ given
in~(\ref{Eq_chi_SS_P_T->infty_pot_E}) and the perturbative correlation
function
%
\be
        D^{\prime\,(3)}_{\pert}(\vec{Z}^2)
        := \int \frac{d^4K}{(2\pi)^3}\,e^{iKZ}\,
        \tilde{D}^{\prime\,(3)}_{\pert}(K^2)\,\delta(K_4)
         = -\,\frac{\exp[-\,m_G\,|\vec{Z}|]}{4\pi|\vec{Z}|}
\label{Eq_D'(3)p(z,mg)}
\ee
%
which is obtained from the massive gluon
propagator~(\ref{Eq_massive_gluon_propagator}). As shown below, the
perturbative contribution dominates the static potential for
small dipoles sizes $R$. 

The non-perturbative contributions to the static potential, the {\em
  non-confining} component ($nc$) and the {\em confining} component
($c$), read
%
\bea
        V_r^{\nprt\,\,nc}(R)
        & = & 
        C_2(r)\,\,
        \frac{\pi^2 G_2 (1-\kappa)}{3(N_c^2-1)}\,\,
        D_1^{\prime\,(3)}(R^2)
\label{Eq_Vr(R)_NP_nc}\\
        V_r^{\nprt\,\,c}(R) 
        & = & 
        C_2(r)\,\,
        \frac{\pi^2 G_2 \kappa}{3(N_c^2-1)}\,\,
        \int_0^R \!\! d\rho\,
        (R-\rho)\,
        D^{(3)}(\rho^2)
        \ ,
\label{Eq_Vr(R)_NP_c}
\eea
%
where we have used the results for $\chi_{SS\,\pot}^{\nprt\,\,nc}$ and
$\chi_{SS\,\pot}^{\nprt\,\,c}=\chi_{SS}^{\nprt\,\,c}$ given
respectively in~(\ref{Eq_chi_SS_NP_nc_T->infty_pot_E})
and~(\ref{Eq_chi_SS_NP_c_T->infty_V_E}) obtained with the minimal
surface, i.e.\ the planar surface bounded by the loop as indicated by
the shaded area in Fig.~\ref{Fig_ONE_WWL}. With the exponential
correlation function~(\ref{Eq_SVM_correlation_functions}), the
correlation functions in~(\ref{Eq_Vr(R)_NP_nc})
and~(\ref{Eq_Vr(R)_NP_c}) read
%
\bea
        D^{\prime\,(3)}_1(\vec{Z}^2)
        &\!\!:=\!\!&
        \int \frac{d^4K}{(2\pi)^3}\,e^{iKZ}\,
        \tilde{D}^{\prime\,(3)}_1(K^2)\,\delta(K_4)
        \,\,=\,\, -\,a\,|\vec{Z}|^2\,K_2[|\vec{Z}|/a]
        \ ,
\label{Eq_D'(3)np_nc(z,a)}\\
        D^{(3)}(\vec{Z}^2)      
        &\!\!:=\!\! & \int \frac{d^4K}{(2\pi)^3}\,e^{iKZ}\,\tilde{D}(K^2)\,\delta(K_4)
        \,\,=\,\, 2\,|\vec{Z}|\,K_1[|\vec{Z}|/a]
        \ .
\label{Eq_D(3)np_c(z,a)}
\eea
%
%
For large dipole sizes, $R \gtsim 0.5\ \fm$, the non-confining
contribution~(\ref{Eq_Vr(R)_NP_nc}) vanishes exponentially while the
confining contribution~(\ref{Eq_Vr(R)_NP_c}) -- as anticipated --
leads to {\em confinement}~\cite{Dosch:1987sk+X}, i.e.\ the confining
linear increase,
%
\be
        V_r^{\nprt\,\,c}(R)\Big|_{R\,\gtsim\,0.5\,\mbox{\scriptsize fm}} 
       = \sigma_r R  + \mbox{const.} \ .
\label{Eq_Vr(R)_NP_c_linear}
\ee
%
Thus, the QCD {\em string tension} is given by the confining SVM
component~\cite{Dosch:1987sk+X}: For a color dipole in the $SU(N_c)$
representation $r$, it reads
%
\be
        \sigma_r 
        = C_2(r)\,\,\frac{\pi^3 G_2 \kappa}{48} 
          \int_0^\infty dZ^2 D(Z^2) 
        = C_2(r)\,\,\frac{\pi^3 \kappa G_2 a^2}{24}
        \ ,
\label{Eq_string_tension}
\ee
%
where the exponential correlation
function~(\ref{Eq_SVM_correlation_functions}) is used in the final
step.  Since the string tension can be computed from first principles
within lattice QCD~\cite{Bali:2001gf},
relation~(\ref{Eq_string_tension}) puts an important constraint on the
three parameters of the non-perturbative QCD vacuum $a$, $G_2$, and
$\kappa$. With the values for $a$, $G_2$, and $\kappa$ given
in~(\ref{Eq_MSV_scattering_fit_parameter_results}), that are used
throughout this work, one obtains for the string tension of the
$SU(3)$ quark-antiquark potential ($r=3$) a reasonable value of
%
\be
        \sigma_3 
        = 0.22\,\GeV^2 \equiv 1.12 \,\GeV/\fm
        \ .
\label{Eq_sting_tension_from_exp_correlation}
\ee 
%

The static $SU(N_c = 3)$ quark-antiquark potential
$V_{\fundamental}(R) = V_3(R)$ is shown as a function of the
quark-antiquark separation $R$ in
Fig.~\ref{Fig_Static_Quark-Antiquark_Potential_Components},
%
\begin{figure}[t!]
  \centerline{\epsfig{figure=StaticPotential_SU3F_Components.eps,width=12.cm}}
\caption{\small 
  The static $SU(N_c = 3)$ quark-antiquark potential
  $V_{\fundamental}(R) = V_3(R)$ as a function of the quark-antiquark
  separation $R$. The solid, dotted, and dashed lines indicate the
  full static potential and its perturbative and non-perturbative
  contributions, respectively. For small quark-antiquark separations,
  $R \ltsim 0.5\,\fm$, the perturbative contribution dominates and
  gives rise to the well-known color Coulomb behavior at small
  distances. For medium and large quark-antiquark separations, $R
  \gtsim 0.5\,\fm$, the non-perturbative contribution dominates and
  leads to the confining linear rise of the static potential. As our
  model is working in the quenched approximation, string breaking
  cannot be described, which is expected to stop the linear increase
  for $R\,\gtsim\,1\,\fm$~\cite{Laermann:1998gm,Bali:2001gf}.}
\label{Fig_Static_Quark-Antiquark_Potential_Components}
\end{figure}
%
where the solid, dotted, and dashed lines indicate the full static
potential and its perturbative and non-perturbative contributions,
respectively.  For small quark-antiquark separations $R \ltsim
0.5\,\fm$, the perturbative contribution dominates giving rise to the
well-known color Coulomb behavior. For medium and large
quark-antiquark separations $R \gtsim 0.5\,\fm$, the non-perturbative
contribution dominates and leads to the confining linear rise of the
static potential. The transition from perturbative to string behavior
takes place at source separations of about $0.5\,\fm$ in agreement
with the recent results of L\"uscher and Weisz~\cite{Luscher:2002qv}.
This supports our value for the gluon mass $m_G=m_{\rho}=0.77\,\GeV$
which is only important around $R\approx 0.4\,\fm$, i.e.\ for the
interplay between perturbative and non-perturbative physics. For
$R\ltsim 0.3\,\fm$ and $R\gtsim 0.5\,\fm$, the effect of the gluon
mass, introduced as an IR regulator in our perturbative component, is
negligible. String breaking is expected to stop the linear increase
for $R\,\gtsim\,1\,\fm$ where lattice investigations show deviations
from the linear rise in full QCD~\cite{Laermann:1998gm,Bali:2001gf}.
As our model is working in the quenched approximation, string breaking
through dynamical quark-antiquark production is excluded.

As can be seen from~(\ref{Eq_Vr(R)_Gaussian_approximation}), the
static potential shows {\em Casimir scaling} which emerges in our
approach as a trivial consequence of the Gaussian approximation used
to truncate the cumulant
expansion~(\ref{Eq_matrix_cumulant_expansion}). Indeed, the Casimir
scaling hypothesis~\cite{Ambjorn:1984dp} has been verified to high
accuracy for $SU(3)$ on the lattice~\cite{Deldar:1999vi,Bali:2000un}
(see also Fig.~\ref{Fig_Static_Quark-Antiquark_Potential_F_vs_A}).
These lattice results have been interpreted as a strong hint towards
Gaussian dominance in the QCD vacuum and thus as evidence for a strong
suppression of higher cumulant
contributions~\cite{Shevchenko:2000du,Shevchenko:2001ij}. In contrast
to our model, the instanton model can neither describe Casimir
scaling~\cite{Shevchenko:2001ij} nor the linear rise of the confining
potential~\cite{Chen:1999ct}.

Figure~\ref{Fig_Static_Quark-Antiquark_Potential_F_vs_A}
%
\begin{figure}[t!]
\centerline{\epsfig{figure=StaticPotential_SU3_F_vs_A.eps,width=12.cm}}
\caption{\small 
  The static $SU(N_c = 3)$ potential of color dipoles in the
  fundamental representation $V_3(R)$ (solid line) and adjoint
  representation $V_8(R)$ (dashed line) as a function of the dipole
  size $R$ in comparison to $SU(3)$ lattice data for $\beta = 6.0$,
  6.2, and 6.4~\cite{Bali:2000un,Bali:2001gf}. The model results are
  in good agreement with the lattice data. This particularly holds for
  the obtained Casimir scaling behavior.}
\label{Fig_Static_Quark-Antiquark_Potential_F_vs_A}
\end{figure}
%
shows the static $SU(N_c = 3)$ potential for fundamental sources
$V_{\fundamental}(R) = V_3(R)$ (solid line) and adjoint sources
$V_{\adjoint}(R) = V_8(R)$ (dashed line) as a function of the dipole
size $R$ in comparison to $SU(3)$ lattice
data~\cite{Bali:2000un,Bali:2001gf}.  The model results are in good
agreement with the lattice data. In particular, the obtained Casimir
scaling behavior is strongly supported by $SU(3)$ lattice
data~\cite{Deldar:1999vi,Bali:2000un}.  This, however, points also to
a shortcoming of our model: From
Eq.~(\ref{Eq_Vr(R)_Gaussian_approximation}) and
Fig.~\ref{Fig_Static_Quark-Antiquark_Potential_F_vs_A} it is clear
that {\em string breaking} is neither described for fundamental nor
for adjoint dipoles in our model which indicates that not only
dynamical fermions (quenched approximation) are missing but also some
gluon dynamics.

% ______________________________________________________________________________
\section{Chromo-Field Distributions of Color Dipoles}
\label{Sec_Flux_Tube}
% ______________________________________________________________________________

As already explained in Sec.~\ref{Sec_Static_Potential}, the static
color dipole -- two static color sources separated by a distance $R$
in a net color singlet state -- is described by a {\WW} loop $W_r[C]$
with a rectangular path $C$ of spatial extension $R$ and temporal
extension $T\to\infty$ (cf.\ Fig.~\ref{Fig_ONE_WWL}) where $r$
indicates the $SU(N_c)$ representation of the considered sources.  A
second small quadratic loop or plaquette in the fundamental
representation placed at the space-time point $X$ with side length
$R_P\to 0$ and oriented along the $\alpha\beta$-axes
%
\be
        P_{\fundamental}^{\alpha \beta}(X) 
        = \tilde{\Tr}_{\fundamental}
        \exp\!\!\left[
        -i g \oint_{C_P}\!\!\!dZ_{\mu} \G_{\mu}^a(Z) t_{\fundamental}^a 
        \right] 
        = 1 
        - R_P^4\frac{g^2}{4N_c}\G_{\alpha\beta}^a(X)\G_{\alpha\beta}^a(X) 
        + \Order(R_P^6)
\label{Eq_plaquette}
\ee
%
is needed -- as a ``Hall probe'' -- to calculate the chromo-field
distributions at the space-time point $X$ caused by the static
sources~\cite{Fukugita:1983du,Flower:gs}
%
\bea
        \Delta G_{r\,\alpha \beta}^2(X) 
        & := &
        \Big\langle 
        \frac{g^2}{4\pi^2}\G_{\alpha\beta}^a(X)\G_{\alpha\beta}^a(X)
        \Big\rangle_{W_r[C]}
        - 
        \Big\langle 
        \frac{g^2}{4\pi^2}\G_{\alpha\beta}^a(X)\G_{\alpha\beta}^a(X)
        \Big\rangle_{\mbox{\scriptsize vac}}
\label{Eq_DeltaG2_definition}\\
        & = &
        -\,\lim_{R_P \to 0}\inv{R_P^4} \frac{N_c}{\pi^2} 
        \left[
        \frac
        {\langle W_r[C] P_{\fundamental}^{\alpha \beta}(X) \rangle}
        {\langle W_r[C] \rangle}
        - \langle P_{\fundamental}^{\alpha \beta}(X) \rangle
        \right]
\label{Eq_DeltaG2_formula}
\eea
%
with {\em no} summation over $\alpha$ and $\beta$
in~(\ref{Eq_plaquette}), (\ref{Eq_DeltaG2_definition}),
and~(\ref{Eq_DeltaG2_formula}). In
definition~(\ref{Eq_DeltaG2_definition})
$\langle\ldots\rangle_{W_r[C]}$ indicates the VEV in the presence of
the static color dipole while $\langle\ldots\rangle_{\mbox{\scriptsize
    vac}}$ indicates the VEV in the absence of any color sources.
Depending on the plaquette orientation indicated by $\alpha$ and
$\beta$, one obtains from~(\ref{Eq_DeltaG2_formula}) the squared
components of the chromo-electric and chromo-magnetic field at the
space-time point $X$
%
\be
        \Delta G_{r\,\alpha \beta}^2(X) 
        = \frac{g^2}{4\pi^2}
        \left( \barray{cccc}
        0       & B_z^2 & B_y^2 & E_x^2 \\
        B_z^2   & 0     & B_x^2 & E_y^2 \\
        B_y^2   & B_x^2 & 0     & E_z^2 \\
        E_x^2   & E_y^2 & E_z^2 & 0
        \earray \right)(X)
        \ , 
\label{Eq_chromo-electromagnetic_fields}        
\ee
%
i.e.\ space-time plaquettes ($\alpha\beta=i4$) measure chromo-electric
fields and space-space plaquettes ($\alpha\beta=ij$) chromo-magnetic
fields. As shown in Fig.~\ref{Fig_PW_arrangement}, we place the static
color sources on the $X_1$-axis at $(X_1 = \pm R/2,0,0,X_4)$ and use
the following notation plausible from symmetry arguments
%
\be
        E_{\parallel}^2 = E_x^2
%\,\,(14)
        \ ,\quad
        E_{\perp}^2 = E_y^2 = E_z^2
%\,\,(24)
        \ ,\quad
        B_{\parallel}^2 = B_x^2
%\,\,(23)
        \ ,\quad
        B_{\perp}^2 = B_y^2 = B_z^2
%\,\,(13)
        \ .
\label{Eq_E_B_para_perp}
\ee
%
Figure~\ref{Fig_PW_arrangement} illustrates also the plaquette
$P_{\fundamental}^{14}(X)$ at $X = (X_1, X_2,0,0)$ needed to compute
$E_{\parallel}^2(X)$. Due to symmetry arguments, the complete
information on the chromo-field distributions is obtained from
plaquettes in ``transverse'' space $\mbox{$X = (X_1, X_2,0,0)$}$ with
four different orientations, $\alpha\beta = 14,\,24,\,13,\,23$,
cf.~(\ref{Eq_E_B_para_perp}).
%
\begin{figure}[t]
\centerline{\epsfig{figure=plaquette_loop_surfaces_ite.eps,width=10.cm}}
\caption{\small
  The plaquette-loop geometry needed to compute the squared
  chromo-electric field $E_{\parallel}^2(X)$ generated by a static
  color dipole in the fundamental $SU(N_c)$ representation
  ($r=\Fundamental$).  The rectangular path $C$ indicates the
  world-line of the static dipole described the {\WW} loop
  $W_{\fundamental}[C]$. The square with side length $R_P$ illustrates
  the plaquette $P_{\fundamental}^{14}(X)$.  The shaded areas
  represent the minimal surfaces used in our computation of the
  chromo-field distributions.  The thin tube allows us to compare the
  gluon field strengths in surface $S_P$ with the gluon field
  strengths in surface $S_W$.}
\label{Fig_PW_arrangement}
\end{figure}
%

The {\em energy} and {\em action density distributions} around a
static color dipole in the $SU(N_c)$ representation $r$ are given by
the squared chromo-field distributions
%
\bea
        \varepsilon_r(X) 
        & = & 
        \inv{2}\left(-\vec{E}^2(X)+\vec{B}^2(X)\right) 
\label{Eq_energy_density}\\
        \actiondensity_r(X)
        & = &
        -\inv{2}\left(\vec{E}^2(X)+\vec{B}^2(X)\right)
\label{Eq_action_density}
\eea
%
with signs according to Euclidean space-time conventions. Low-energy
theorems that relate the energy and action stored in the chromo-fields
of the static color dipole to the corresponding ground state energy
are discussed in the next section.

For the chromo-field distributions of a static color dipole in the
{\em fundamental} representation of $SU(N_c)$, i.e.\ a static
quark-antiquark pair, we obtain with our results for the VEV of one
loop~(\ref{Eq_final_result_<W[C]>}) and the correlation of two loops
in the fundamental
representation~(\ref{Eq_final_Euclidean_result_<W[C1]W[C2]>_fundamental})
%
\bea
        &&\Delta G_{\fundamental\,\alpha\beta}^2(X) =
        -\lim_{R_P \to 0}\inv{R_P^4}\frac{N_c}{\pi^2} 
        \exp\left[-\frac{C_2(\!\Fundamental\!)}{2}\,\chi_{S_P S_P}\right]
\label{Eq_chromo_fields_F_1}\\
        &&\hspace*{2cm}\times 
        \Bigg(\frac{N_c+1}{2N_c}\exp\!\left[-\frac{N_c-1}{2N_c}\chi_{S_P S_W}\right]
        + \frac{N_c-1}{2N_c}\exp\!\left[ \frac{N_c+1}{2N_c}\chi_{S_P S_W}\right] - 1\Bigg)
\nonumber
\eea
%
where $\chi_{S_i S_j}$ is defined in~(\ref{Eq_chi_Si_Sj}). The
subscripts $P$ and $W$ indicate surface integrations to be performed
over the surfaces spanned by the plaquette and the Wegner-Wilson-loop,
respectively. Choosing the surfaces -- as illustrated by the shaded
areas in Fig.~\ref{Fig_PW_arrangement} -- to be the minimal surfaces
connected by an infinitesimal thin tube (which gives no contribution
to the integrals) it is clear that $\chi_{S_P S_P} \propto R_P^4$ and
$\chi_{S_P S_W} \propto R_P^2$. Being interested in the chromo-fields
at the space-time point $X$, the extension of the quadratic plaquette
is taken to be infinitesimally small, $R_P \rightarrow 0$, so that one
can expand the exponential functions and keep only the term of lowest
order in $R_P$
%
\be
        \Delta G_{\fundamental\,\alpha\beta}^2(X) = 
        -\,C_2(\!\Fundamental\!)\,\lim_{R_P \to 0}\inv{R_P^4}\,\inv{4\pi^2}\,\chi_{S_P S_W}^2
        \ .
\label{Eq_chromo_fields_F_final_result}
\ee
%
This result -- obtained with the matrix cumulant expansion in a very
straightforward way -- agrees exactly with the result derived
in~\cite{Rueter:1994cn} with the expansion method. Indeed, the
expansion method agrees for small $\chi$-functions with the matrix
cumulant expansion (Berger-Nachtmann approach) used in this work but
breaks down for large $\chi$-functions, where the matrix cumulant
expansion is still applicable.

The chromo-field distributions of a static color dipole in the {\em
  adjoint} representation of $SU(N_c)$, i.e.\ a static gluino pair,
are computed analogously. Using our
result~(\ref{Eq_final_Euclidean_result_<Wf[C1]Wa[C2]>}) for the
correlation of one loop in the fundamental representation (plaquette)
with one loop in the adjoint representation (static sources), one
obtains
%
\bea
        \!\!\!\!&&\!\!\!\!\!\!\!\!
        \Delta G_{\adjoint\,\alpha\beta}^2(X) =
        -\lim_{R_P \to 0}\inv{R_P^4}\frac{N_c}{\pi^2} 
        \exp\left[-\frac{C_2(\!\Fundamental\!)}{2}\chi_{S_P S_P}\right]\,
        \Bigg(\!\inv{N_c^2\!-\!1}\,\exp\!\Big[\frac{N_c}{2}\,\chi_{S_P S_W}\Big]
\nonumber\\
        && \hskip 1.5cm
%\!\!\!\!\!\!\!\!
        +\,\frac{N_c\!+\!2}{2(N_c\!+\!1)}\exp\!\Big[\!-\inv{2}\,\chi_{S_P S_W}\Big]
        +\frac{N_c\!-\!2}{2(N_c\!-\!1)}\exp\!\Big[\inv{2}\,\chi_{S_P S_W}\Big]
        -1\!\Bigg)
\label{Eq_chromo_fields_A_1}
\eea
%
which reduces -- as explained for sources in the fundamental
representation -- to
%
\be
        \Delta G_{\adjoint\,\alpha\beta}^2(X) = 
        -\,C_2(\!\Adjoint\!)\,\lim_{R_P \to 0}\inv{R_P^4}\,\inv{4\pi^2}\,\chi_{S_P S_W}^2
        \ .
\label{Eq_chromo_fields_A_final_result}
\ee
%
Thus, the squared chromo-electric fields of an adjoint dipole differ
from those of a fundamental dipole only in the eigenvalue of the
corresponding quadratic Casimir operator $C_2(r)$. In fact, {\em
  Casimir scaling} of the chromo-field distributions holds for dipoles
in any representation $r$ of $SU(N_c)$ in our model. As can be seen
with the low-energy theorems discussed below, this is in line with the
Casimir scaling of the static dipole potential found in the previous
section. Besides lattice investigations that show Casimir scaling of
the static dipole potential to high accuracy in
$SU(3)$~\cite{Deldar:1999vi,Bali:2000un}, Casimir scaling of the
chromo-field distributions has been considered on the lattice as well
but only for $SU(2)$~\cite{Trottier:1995fx}. Here only slight
deviations from the Casimir scaling hypothesis have been found that
were interpreted as hints towards adjoint quark screening.

In our model the shape of the field distributions around the color
dipole is identical for all $SU(N_c)$ representations $r$ and given by
$\chi_{S_P S_W}^2$. This again illustrates the shortcoming of our
model discussed in the previous section. Working in the quenched
approximation, one expects a difference between fundamental and
adjoint dipoles: {\em string breaking} cannot occur in fundamental
dipoles as dynamical quark-antiquark production is excluded but should
be present for adjoint dipoles because of gluonic vacuum polarization.
Comparing~(\ref{Eq_chromo_fields_F_final_result})
with~(\ref{Eq_chromo_fields_A_final_result}) it is clear that this
difference is not described in our model. In fact, as shown in
Sec.~\ref{Sec_Static_Potential}, string breaking is neither described
for fundamental nor for adjoint dipoles. Interestingly, even on the
lattice there has been no striking evidence for adjoint quark
screening in quenched QCD~\cite{Kallio:2000jc}. It is even conjectured
that the {\WW} loop operator is not suited to studies of string
breaking~\cite{Gusken:1997sa+X}.

In the LLCM there are perturbative ($\pert$) and non-perturbative
($\nprt$) contributions to the chromo-electric fields according to the
structure of the gluon field strength correlator, (\ref{Eq_Ansatz})
and~(\ref{Eq_F_decomposition}),
%
\bea
        \Delta G_{r\,\alpha\beta}^2(X) 
        & = & 
        -C_2(r)\,\lim_{R_P \to 0}\inv{R_P^4}\,\inv{\pi^2}\,
\label{Eq_chromo_fields_F_no_interference}\\
        && \times\left\{
          \left(\chi_{S_P S_W}^{\pert}(X)\right)_{\alpha\beta}^2 
        + \left[
          \left(\chi_{S_P S_W}^{\nprt\,nc}(X)\right)_{\alpha\beta}
          +  \left(\chi_{S_P S_W}^{\nprt\,c}(X)\right)_{\alpha\beta}
        \right]^2
      \right\}
\nonumber
\eea
%
where we have demanded the non-interference of perturbative and
non-perturbative correlations in line with the Minkowskian
applications of our
model~\cite{Shoshi:2002in,Shoshi:2002ri,Shoshi:2002fq,Shoshi:2002mt}.
In the following we give only the final results of the
$\chi$\,-\,functions for the minimal surfaces shown in
Fig.~\ref{Fig_PW_arrangement}.  Details on their derivation can be
found in Appendix~\ref{Sec_Chi_Computation}.

The {\em perturbative contribution} ($P$) described by massive gluon
exchange leads, of course, to the well-known {\em color Yukawa field}
that reduces to the {\em color Coulomb field} for $m_g=0$. It
contributes only to the chromo-electric fields, $E_{\parallel}^2 =
E_x^2$ ($\alpha\beta=14$) and $E_{\perp}^2 = E_y^2 = E_z^2$
($\alpha\beta=24$), and reads explicitly for $X = (X_1, X_2, 0, 0)$
%
\bea
\!\!\!\!\!\!\!\!\!\!\!\!
        \left(\chi_{S_P S_W}^{\pert}(X)\right)_{14}
        &\!\!=\!\!& -\,\frac{R_P^2}{2}\!\int_{-\infty}^{\infty}\!\!\!d\tau
        \left\{
        (X_1 - R/2)\,
        g^2(Z_{1A}^2)\,D_\pert(Z_{1A}^2)
        \right.
\nonumber\\
        && \hphantom{-\,\frac{R_P^2}{2}\!\int_{-\infty}^{\infty}\!\!\!d\tau\Big(}
        \left.       
        - \,(X_1 + R/2)\,
        g^2(Z_{1C}^2)\,D_\pert(Z_{1C}^2)
        \right\}
\label{Eq_Chi_PW_p_14}\\
\!\!\!\!\!\!\!\!\!\!\!\!
        \left(\chi_{S_P S_W}^{\pert}(X)\right)_{24}
        &\!\!=\!\!& -\,\frac{R_P^2}{2}\!\int_{-\infty}^{\infty}\!\!\!d\tau\,X_2
        \left\{
        g^2(Z_{1A}^2)\,D_\pert(Z_{1A}^2)
        - g^2(Z_{1C}^2)\,D_\pert(Z_{1C}^2)
        \right\}
\label{Eq_Chi_PW_p_24}
\eea
%
with the perturbative correlation function~(\ref{Eq_Dp(z,mg)}), the
running coupling~(\ref{Eq_g2(z_perp)}), and
%
\be
        Z_{1A}^2 = \left(X_1\!-\!\frac{R}{2}\right)^2+X_2^2+\tau^2
        \quad \mbox{and} \quad
        Z_{1C}^2 = \left(X_1\!+\!\frac{R}{2}\right)^2+X_2^2+\tau^2
        \ .
\label{Eq_|Z1A|_|Z1C|}
\ee
%

The {\em non-confining non-perturbative contribution} ($\nprt\,nc$) has
the same structure as the perturbative contribution -- as expected
from the identical tensor structure -- but differs, of course, in the
prefactors and the correlation function, $D_1 \neq D_p$. Its
contributions to the chromo-electric fields $E_{\parallel}^2 = E_x^2$
($\alpha\beta=14$) and $E_{\perp}^2 = E_y^2 = E_z^2$
($\alpha\beta=24$) read for $X = (X_1, X_2, 0, 0)$
%
\bea
        \left(\chi_{S_P S_W}^{\nprt\,\,nc}(X)\right)_{14}
        &=&
        -\,\frac{R_P^2 \pi^2 G_2 (1\!-\!\kappa)}{6\,(N_c^2\!-\!1)}
        \!\int_{-\infty}^{\infty}\!\!\!d\tau
        \Big\{
        (X_1 - R/2)\,D_1(Z_{1A}^2)
\nonumber\\
        && 
        \hphantom{-\,\frac{R_P^2 \pi^2 G_2}{6\,(N_c^2\!-\!1)}}
        - \,(X_1 + R/2)\,D_1(Z_{1C}^2)
        \Big\}
\label{Eq_Chi_PW_np_nc_14}\\
        \left(\chi_{S_P S_W}^{\nprt\,\,nc}(X)\right)_{24}
        &=&
        -\,\frac{R_P^2 \pi^2 G_2 (1\!-\!\kappa)}{6\,(N_c^2\!-\!1)}
        \!\int_{-\infty}^{\infty}\!\!\!d\tau\,X_2
        \Big\{D_1(Z_{1A}^2)-D_1(Z_{1C}^2)\Big\}
\label{Eq_Chi_PW_np_nc_24}
\eea
%
with the exponential correlation
function~(\ref{Eq_SVM_correlation_functions}) and $Z_{1A}^2$ and
$Z_{1C}^2$ as given in~(\ref{Eq_|Z1A|_|Z1C|}).

The {\em confining non-perturbative contribution} ($\nprt\,c$) has a
different structure that leads to confinement and flux-tube formation.
It gives only contributions to the chromo-electric field
$E_{\parallel}^2 = E_x^2$ ($\alpha\beta=14$) that read for $X = (X_1,
X_2, 0, 0)$
%
\bea
        \left(\chi_{S_P S_W}^{\nprt\,\,c}(X)\right)_{14}
        & = & 
        R_P^2 R 
        \frac{\pi^2 G_2 \kappa}{3\,(N_c^2\!-\!1)}
        \int_{0}^{1} d\rho\,
        D^{(3)}(\vec{Z_{\perp}}^2)
        \ ,
\label{Eq_Chi_PW_np_c_14}
\eea
%
with the correlation function given in~(\ref{Eq_D(3)np_c(z,a)}) as
derived from the exponential correlation
function~(\ref{Eq_SVM_correlation_functions}), and
%
\be
        \vec{Z}_{\perp}^2 = [X_1+(1/2-\rho)R]^2+X_2^2
        \ .
\label{Eq_|Z|}
\ee
%

In our model there are no contributions to the {\em chromo-magnetic
  fields}, i.e.\ the static color charges do not affect the magnetic
background field
%
\be
        B_{\parallel}^2 = B_x^2 = 0
        \quad \mbox{and} \quad
        B_{\perp}^2 = B_y^2 = B_z^2 = 0
        \ ,
\label{Eq_B^2=0}
\ee
%
which can be seen from the corresponding plaquette-loop geometries as
pointed out in Appendix~\ref{Sec_Chi_Computation}. Thus, the energy
and action densities are identical in our approach and completely
determined by the squared chromo-electric fields
%
\be
        \varepsilon_r(X) 
        \,\,=\,\,\actiondensity_r(X)
        \,\,=\,\, -\inv{2}\,\vec{E}^2(X)
        \ .
\label{Eq_energy=action_density}
\ee
%
This picture is in agreement with other effective theories of
confinement such as the `t~Hooft-Mandelstam
picture~\cite{Mandelstam:1976pi+X} or dual QCD~\cite{Baker:bc} and,
indeed, a relation between the dual Abelian Higgs model and the SVM
has been established~\cite{Baker:1998jw}. In contrast, lattice
investigations work at scales at which the chromo-electric and
chromo-magnetic fields are of similar
magnitude~\cite{Bali:1994de,Green:1996be}. Using low-energy theorems,
we will see in the next section, that the vanishing of the
chromo-magnetic fields determines the value of the $\beta$-function at
the renormalization scale at which the non-perturbative component of
our model is working.

%
\begin{figure}[h]
\centerline{
\epsfig{figure=EnergyDensity_XY_Profile_SU3F_R01fmBW_WL.eps,width=7.7cm}
\hspace*{-0.8cm}
%\hfill
\epsfig{figure=EnergyDensity_XY_Profile_SU3F_R1fmBW_WL.eps,width=7.7cm}}
\centerline{
\epsfig{figure=EnergyDensity_XY_Profile_SU3F_R05fmBW_WL.eps,width=7.7cm}
\hspace*{-0.8cm}
%\hfill
\epsfig{figure=EnergyDensity_XY_Profile_SU3F_R15fmBW_WL.eps,width=7.7cm}}
\caption{\small
  Energy density distributions $g^2\varepsilon_3(X_1,X_2\!=\!X_3)$
  generated by a color dipole in the fundamental $SU(3)$
  representation ($r\!=\!3$) for quark-antiquark separations of $R =
  0.1,\,0.5,\,1$ and $1.5\,\fm$. Flux-tube formation leads to the
  confining QCD string with increasing dipole size $R$.}
\label{Fig_3D_profiles}
\end{figure}
%
In Fig.~\ref{Fig_3D_profiles} the energy density distributions
$g^2\varepsilon_3(X_1,X_2\!=\!X_3)$ generated by a color dipole in the
fundamental $SU(3)$ representation ($r\!=\!3$) are shown for
quark-antiquark separations of $R = 0.1,\,0.5,\,1$ and $1.5\,\fm$.
With increasing dipole size $R$, one sees explicitly the formation of
the flux tube which represents the confining QCD string.

The {\em longitudinal} and {\em transverse energy density profiles}
generated by a color dipole in the fundamental representation ($r=3$)
of $SU(N_c=3)$ are shown for quark-antiquark separations (dipole
sizes) of $R = 0.1,\,0.5,\,1$ and $1.5\,\fm$ in
Figs.~\ref{Fig_L_profiles} and~\ref{Fig_T_profiles}. The perturbative
and non-perturbative contributions are given in the dotted and dashed
lines, respectively, and the sum of both in the solid lines. The open
and filled circles indicate the quark and antiquark positions. As can
be seen from~(\ref{Eq_DeltaG2_formula})
and~(\ref{Eq_chromo-electromagnetic_fields}), we cannot compute the
energy density separately but only the product $g^2\varepsilon_r(X)$.
Nevertheless, a comparison of the total energy stored in
chromo-electric fields to the ground state energy of the color dipole
via low-energy theorems yields $g^2 = 10.2$ $(\equiv \alphaS=0.81)$
for the non-perturbative SVM component as shown in the next section.
%
\begin{figure}[p]
\centerline{\epsfig{figure=XProfilesWSources.eps,width=10cm}}
\caption{\small
  Longitudinal energy density profiles
  $g^2\varepsilon_3(X_1,X_2\!=\!X_3\!=\!0)$ generated by a color
  dipole in the fundamental $SU(3)$ representation ($r\!=\!3$) for
  quark-antiquark separations of $R = 0.1,\,0.5,\,1$ and $1.5\,\fm$.
  The dotted and dashed lines give the perturbative and
  non-perturbative contributions, respectively, and the solid lines
  the sum of both.  The open and filled circles indicate the quark and
  antiquark positions. For small dipoles, $R=0.1\,\fm$, perturbative
  physics dominates and non-perturbative correlations are negligible.
  For large dipoles, $R\gtsim 1\,\fm$, the formation the confining
  string (flux tube) can be seen which dominates the chromo-electric
  fields between the color sources.}
\label{Fig_L_profiles}
\end{figure}
%
%
\begin{figure}[p]
\centerline{\epsfig{figure=YProfilesWSources.eps,width=10cm}}
\caption{\small
  Transverse energy density profiles
  $g^2\varepsilon_3(X_2,X_1\!=\!X_3\!=\!0)$ generated by a color
  dipole in the fundamental $SU(3)$ representation ($r\!=\!3$) for
  quark-antiquark separations of $R = 0.1,\,0.5,\,1$ and $1.5\,\fm$.
  The dotted and dashed lines give the perturbative and
  non-perturbative contributions, respectively, and the solid lines
  the sum of both.  The filled circles indicate the positions of the
  color sources. For small dipoles, $R=0.1\,\fm$, perturbative physics
  dominates and non-perturbative correlations are negligible. For
  large dipoles, $R\gtsim 1\,\fm$, the formation the confining string
  (flux tube) can be seen which dominates the chromo-electric fields
  between the color sources.}
\label{Fig_T_profiles}
\end{figure}
%

In Figs.~\ref{Fig_L_profiles} and~\ref{Fig_T_profiles} the formation
of the confining string (flux tube) with increasing source separations
$R$ can again be seen explicitly: For small dipoles, $R=0.1\,\fm$,
perturbative physics dominates and non-perturbative correlations are
negligible. For large dipoles, $R\gtsim 1\,\fm$, the non-perturbative
correlations lead to formation of a narrow flux tube which dominates
the chromo-electric fields between the color sources.

Figure~\ref{Fig_R_ms_and_g2epsilon(0)} 
%
\begin{figure}[t]
\centerline{\epsfig{figure=RDependencies_SU3F.eps,width=12cm}}
\caption{\small
  Root mean squared radius $R_{ms}$ of the flux tube and energy
  density in the center of a fundamental $SU(3)$ dipole
  $g^2\varepsilon_3(X=0)$ as a function of the dipole size $R$.
  Perturbative and non-perturbative contributions are given
  respectively in the dotted and dashed lines and the sum of both in
  the solid lines. For large $R$, both the width and height of the
  flux tube in the central region are governed completely by
  non-perturbative physics and saturate respectively at
  $R_{ms}^{R\to\infty}\approx 0.55\,\fm$ and
  $\varepsilon_3^{R\to\infty}(X=0)\approx 1\,\GeV/\fm^3$. The latter
  value is extracted with the result $g^2 = 10.2$ deduced from
  low-energy theorems in the next section.}
\label{Fig_R_ms_and_g2epsilon(0)}
\end{figure}
%
shows the evolution of the transverse width (upper plot) and height
(lower plot) of the flux tube in the central region of the {\WW} loop
as a function of the dipole size $R$ where perturbative and
non-perturbative contributions are given in the dotted and dashed
lines, respectively, and the sum of both in the solid lines. The width
of the flux tube is best described by the root mean squared ($ms$)
radius
%
\be
        R_{ms}
        = \sqrt{\frac{\int dX_{\perp}\,X_{\perp}^3\,g^2\varepsilon_r(X_1=0,X_{\perp})}
        {\int dX_{\perp}\,X_{\perp}\,g^2\varepsilon_r(X_1=0,X_{\perp})}}
        \ ,
\label{Eq_R_ms}
\ee
%
which is universal for dipoles in all $SU(N_c)$ representations $r$ as
the Casimir factors divide out. The height of the flux tube is given
by the energy density in the center of the considered dipole,
$g^2\varepsilon_r(X=0)$. For large source separations, $R \gtsim
1\,\fm$, both the width and height of the flux tube in the central
region of the {\WW} loop are governed completely by non-perturbative
physics and saturate for a fundamental $SU(3)$ dipole
($r=\Fundamental=3$) at reasonable values of
%
\be
        R_{ms}^{R\to\infty}\approx 0.55\,\fm
        \quad \mbox{and} \quad 
        \varepsilon_3^{R\to\infty}(X=0)\approx 1\,\GeV/\fm^3
        \quad \mbox{with} \quad g^2 = 10.2
        \ .
\label{Eq_R_ms_and_g2epsilon(0)_saturation_values}
\ee
%

Note that the qualitative features of the non-perturbative SVM
component do not depend on the specific choice for the parameters,
surfaces, and correlation functions and have already been discussed
with the pyramid mantle choice of the surface and different
correlation functions in the first investigation of flux-tube
formation in the SVM~\cite{Rueter:1994cn}. The quantitative results,
however, are sensitive to the parameter values, the surface choice,
and the correlation functions and are presented above with the LLCM
parameters, the minimal surfaces, and the exponential correlation
function~\cite{Shoshi:2002rd}.

% ______________________________________________________________________________
\section{Low-Energy Theorems}
\label{Sec_Low_Energy_Theorems}
% ______________________________________________________________________________

Many low-energy theorems have been derived in continuum theory by
Novikov, Shifman, Vainshtein, and Zakharov~\cite{Novikov:xi+X} and in
lattice gauge theory by Michael~\cite{Michael:1986yi}. Here  we
consider the energy and action sum rules -- known in lattice QCD as
{\em Michael sum rules} -- that relate the energy and action stored
in the chromo-fields of a static color dipole to the corresponding
ground state energy~\cite{Wilson:1974sk,Brown:1979ya}
%
\be
        E_r(R) = 
        - \lim_{T \to \infty} \inv{T} 
        \ln \langle W_r[C] \rangle
        \ .
\label{Eq_Er(R)_def}        
\ee
%
In their original form~\cite{Michael:1986yi}, however, the Michael sum
rules are incomplete~\cite{Dosch:1995fz,Rothe:1995hu+X}. In
particular, significant contributions to the energy sum rule from the
trace anomaly of the energy-momentum tensor have been
found~\cite{Rothe:1995hu+X} that modify the naively expected relation
in line with the importance of the trace anomaly found for hadron
masses~\cite{Ji:1995sv}. Taking all these contributions into account,
the {\em energy} and {\em action sum rule} read
respectively~\cite{Rothe:1995hu+X,Michael:1995pv,Green:1996be}
%
\bea
        && 
        E_r(R) 
        = \int d^3X\,\varepsilon_r(X)
        - \inv{2}\frac{\beta(g)}{g}
        \int d^3X\,\actiondensity_r(X)
\label{Eq_energy_sum_rule}\\
        &&
        E_r(R) + R\,\frac{\partial E_r(R)}{\partial R}
        = - \frac{2\beta(g)}{g}
        \int d^3X\,\actiondensity_r(X)
\label{Eq_action_sum_rule}
\eea
%
where $\beta(g)=\mu \partial g/\partial\mu$ with the renormalization
scale $\mu$. Inserting~(\ref{Eq_action_sum_rule})
into~(\ref{Eq_energy_sum_rule}), we find the following relation
between the total energy stored in the chromo-fields
$E_r^{\mbox{\scriptsize tot}}(R)$ and the ground state energy $E_r(R)$
%
\be
        E_r^{\mbox{\scriptsize tot}}(R) 
        := \int d^3X\,\varepsilon_r(X)
        = \inv{4}\left(3\,E_r(R) - R\frac{\partial E_r(R)}{\partial R} \right)
        \ .
\label{Eq_Etot-Er(R)_relation}
\ee
%
The difference from the naive expectation that the full ground state
energy of the static color sources is stored in the chromo-fields is
due to the trace anomaly contribution~\cite{Rothe:1995hu+X} described
by the second term on the right-hand side (rhs)
of~(\ref{Eq_energy_sum_rule}).

With the low energy theorems~(\ref{Eq_action_sum_rule})
and~(\ref{Eq_Etot-Er(R)_relation}) the ratio of the integrated squared
chromo-magnetic to the integrated squared chromo-electric field
distributions can be derived
%
\be
        Q(R) := \frac{\int d^3X \vec{B}^2(X)}{\int d^3X \vec{E}^2(X)}
        = \frac{\left(2+6\,\beta(g)/g\right)\,E_r(R) 
        + \left(1-\beta(g)/g\right)\,R\,\frac{\partial E_r(R)}{\partial R}}
        {\left(2-6\,\beta(g)/g\right)\,E_r(R) 
        + \left(1+\beta(g)/g\right)\,R\,\frac{\partial E_r(R)}{\partial R}}
      \ ,
\label{Eq_Q_ratio_general}
\ee
%
which becomes for $E_r(R)=\sigma_r R+E_{\self}$ after subtraction of
the self-energy contributions, i.e.\ the linear potential
$V_r(R)=\sigma_r R$ with string tension $\sigma_r$ in the considered
representation $r$,
%
\be
        Q(R) \Big|_{V_r(R)=\sigma_r R}
        = \frac{2+\beta(g)/g}{2-\beta(g)/g}
      \ .
\label{Eq_Q_ratio_linear_potential}
\ee
%

In our model there are no contributions to the chromo-magnetic
fields~(\ref{Eq_B^2=0}) so that -- as already discussed in the
previous section -- the energy and action densities are identical and
completely determined by the squared chromo-electric
fields~(\ref{Eq_energy=action_density}). Since the non-perturbative
SVM component of our model describes the confining linear potential
for large source separations $R$, this allows us to determine
from~(\ref{Eq_Q_ratio_linear_potential}) immediately the value of the
$\beta$\,-\,function at the scale $\mu_{\nprt}$ at which the
non-perturbative component is working
%
\be
        \frac{\beta(g)}{g}\Big|_{\mu =\mu_{\nprt}} = -2
        \ .
\label{Eq_beta/g=-2}
\ee
%

Concentrating on the confining non-perturbative component ($\nprt c$)
we now use (\ref{Eq_Etot-Er(R)_relation}) to determine the value of
$\alphaS = g^2/(4\pi)$ at which the non-perturbative SVM component is
working. The rhs of~(\ref{Eq_Etot-Er(R)_relation}) is obtained
directly from the confining contribution to the static potential
$E_r^{\nprt c}(R)=V_r^{\nprt c}(R)$ given in~(\ref{Eq_Vr(R)_NP_c}).
The lhs of~(\ref{Eq_Etot-Er(R)_relation}), however, involves a
division by the {\em a priori} unknown value of $g^2$ after
integrating $g^2\varepsilon_r(X)$ for the chromo-electric field of the
confining non-perturbative component~(\ref{Eq_Chi_PW_np_c_14}). As
discussed in the previous section, we cannot compute the energy
density separately but only the product $g^2\varepsilon_r(X)$.
Adjusting the value of $g^2$ such that~(\ref{Eq_Etot-Er(R)_relation})
is exactly fulfilled for source separations of $R=1.5\,\fm$, we find
that the non-perturbative component is working at the scale
$\mu_{\nprt}$ at which
%
\be
        g^2(\mu_{\nprt}) = 10.2
        \quad \equiv \quad
        \alphaS(\mu_{\nprt}) = 0.81
        \ .
\label{Eq_alphaS=0.81}
\ee
%
As already mentioned in Sec.~\ref{Sec_QCD_Components}, we use this
value as a practical asymptotic limit for the simple one-loop
coupling~(\ref{Eq_g2(z_perp)}) used in our perturbative component.
Note that earlier SVM investigations along these lines have found a
smaller value of $\mbox{$\alphaS(\mu_{\nprt}) = 0.57$}$ with the
pyramid mantle choice for the
surface~\cite{Rueter:1994cn,Dosch:1995fz} but were incomplete since
only the contribution from the traceless part of the energy-momentum
tensor has been considered in the energy sum rule.

In Fig.~\ref{Fig_NP_c_consistency}
%
\begin{figure}[t]
\centerline{\epsfig{figure=LET_c_SU3_F_vs_A_1d5fm.eps,width=12cm}}
\caption{\small
  The total energy stored in the chromo-field distributions around a
  static color dipole of size $R$ in the fundamental ($r=3$) and
  adjoint ($r=8$) representation of $SU(3)$ from the confining
  non-perturbative SVM component, $E_{3,8}^{\mbox{\scriptsize
      tot}\,\nprt_c}(R)$, for $\alphaS = 0.81$ (solid lines) compared
  with the relation to the corresponding ground state energy (dashed
  lines) given by the low-energy
  theorem~(\ref{Eq_Etot-Er(R)_relation}). Good consistency is found
  even down to very small values of $R$.}
\label{Fig_NP_c_consistency}
\end{figure}
%
we show the total energy stored in the chromo-field distributions
around a static color dipole in the fundamental ($r=3$) and adjoint
($r=8$) representation of $SU(3)$ from the confining non-perturbative
SVM component, $E_{3,8}^{\mbox{\scriptsize tot}\,\nprt_c}(R)$, for
$\alphaS = 0.81$ (solid lines) as a function of the dipole size $R$.
Comparing this total energy, which appears on the lhs
of~(\ref{Eq_Etot-Er(R)_relation}), with the corresponding rhs
of~(\ref{Eq_Etot-Er(R)_relation}) (dashed lines), we find good
consistency even down to very small values of $R$. This is a
nontrivial and important result as it confirms the consistency of our
loop-loop correlation result -- needed to compute the chromo-electric
field -- with the result obtained for the VEV of one loop -- needed
to compute the static potential $V_r^{\nprt_c}(R)$. Moreover, it shows
that the minimal surfaces ensure the consistency of our
non-perturbative component. The good consistency found for the pyramid
mantle choice of the surface relies on the naively expected energy sum
rule~\cite{Rueter:1994cn,Dosch:1995fz} in which the contribution from
the traceless part of the energy-momentum tensor is not taken into
account.

%%% Local Variables: 
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\begin{titlepage}
%
\pagestyle{empty}
%
% ___ Official First Page _____________________________________________
%
%\vspace*{2cm}
%
\begin{center}
  \Huge \bf \sc
  From Static Potentials\\
  to\\
  High-Energy Scattering%[3mm]
\end{center}
%
\vfill
%\vspace*{2.cm}
\begin{center}
  {%\sc
  \Large Dissertation\\}
  \vspace{0.3cm}
  {\large
  submitted to the\\
  Combined Faculties for the Natural Sciences and for Mathematics\\
  of the Ruperto--Carola University of Heidelberg, Germany\\
  for the degree of\\ 
  Doctor of Natural Sciences}
\end{center}
\vfill
\begin{center}
\begin{large}
  presented by\\
  \vspace{0.5cm}
  {\LARGE\sc Frank Daniel Steffen}\\
  \vspace{0.5cm}
    born in Wattenscheid\\
\end{large}
\end{center}
%
\vfill
%
\begin{center}
\begin{large}
\begin{tabular}{ll}
Referees: & Prof.~Dr.~Hans G\"unter Dosch\\
& Prof.~Dr.~Bogdan~Povh\\
\end{tabular}

Oral examination: \, February 14, 2003 \, \hphantom{.}

\end{large}
\end{center}
%
\newpage
%
% ___ Official Third Page _____________________________________________
%
\enlargethispage{2cm}
\vspace*{-2cm}
%$\mbox{ }$
%\vspace*{-1.25cm}
%
\begin{center}
  {\bf \large Von Statischen Potenzialen zur Hochenergiestreuung}\\[.2cm]
  {\bf Zusammenfassung}
\end{center}
%
%\parindent0cm
{\small
%
  Wir entwickeln ein Loop-Loop Korrelations Modell zur einheitlichen
  Beschreibung von sta\-tischen Farbdipol Potenzialen, einschliessenden
  QCD Strings, und hadronischen Hochenergiereaktionen mit besonderer
  Ber\"ucksichtigung von S\"attigungseffekten, die die $S$-Matrix
  Unitarit\"at bei extrem hohen Energien manifestieren.
%
  Das Modell verbindet st\"orungs\-theoretischen Gluonaustausch mit
  dem nicht-st\"orungstheoretischen Modell des Stochastischen Vakuums,
  das den Einschluss von Farbladungen durch Flussschlauchbildung der
  Farbfelder beschreibt.
%
  Wir berechnen Farbfeldverteilungen statischer Farbdipole in
  verschiedenen $SU(N_c)$ Darstellungen und finden Casimir
  Skalierungsverhalten in \"Ubereinstimmung mit aktuellen Gitter-QCD
  Ergebnissen.
%
  Wir untersuchen die im einschliessenden String gespeicherte Energie
  und zeigen mit Niederenergie\-theoremen die Konsistenz mit dem
  statischen Quark-Antiquark Potenzial.
%
  Wir ver\-allgemeinern Meggiolaros Analytische Fortsetzung von
  Parton-Parton auf Dipol-Dipol Streuung und er\-halten einen
  Euklidischen Zugang zur Hochenergiestreuung, der prinzipiell
  erlaubt, Streu\-matrix\-elemente in Gitter-QCD zu berechnen.
%
  Mit dem Euklidischen Loop-Loop Korrelations Modell berechnen wir in
  diesen Zugang Dipol-Dipol Streuung bei hohen Energien.
%
  Das Ergebnis bildet zusammen mit einer universellen
  Energieabh\"angigkeit und reaktionsspezifischen Wellenfunktionen die
  Grundlage f\"ur eine einheitliche Beschreibung von pp, $\pi$p, Kp,
  $\gamma^*$p und $\gamma\gamma$ Reaktionen in guter \"Ubereinstimmung
  mit experimentellen Daten f\"ur Wirkungsquerschnitte,
  Steigungsparameter und Strukturfunktionen.
%
  Die erhaltenen Stossparameterprofile f\"ur pp und $\gamma^*_{L}$p
  Reaktionen und die stossparameterabh\"angige Gluonverteilung des
  Protons $xG(x,Q^2,|\vec{b}_{\!\perp}|)$ zeigen S\"attigung bei
  extrem hohen Energien in \"Ubereinstimmung mit Unitarit\"atsgrenzen.
%
}
%
%\vspace{1cm}
%
\begin{center}
  {\bf \large From Static Potentials to High-Energy Scattering}\\[.2cm]
  {\bf  Abstract}
\end{center}
%\parindent0cm
{\small
%
  We develop a loop-loop correlation model for a unified description
  of static color dipole potentials, confining QCD strings, and
  hadronic high-energy reactions with special emphasis on saturation
  effects manifesting $S$-matrix unitarity at ultra-high energies.
%
  The model combines perturbative gluon exchange with the
  non-perturbative stochastic vacuum model which describes color
  confinement via flux-tube formation of color fields.
%
  We compute the chromo-field distributions of static color dipoles in
  various $SU(N_c)$ representations and find Casimir scaling in
  agreement with recent lattice QCD results.
%
  We investigate the energy stored in the confining string and use
  low-energy theorems to show consistency with the static
  quark-antiquark potential.
%
  We generalize Meggiolaro's analytic continuation from parton-parton
  to dipole-dipole scattering and obtain a Euclidean approach to
  high-energy scattering that allows us in principle to calculate
  $S$-matrix elements in lattice QCD.
%
  In this approach we compute high-energy dipole-dipole scattering
  with the Euclidean loop-loop correlation model.
%
  Together with a universal energy dependence and reaction-specific
  wave functions, the result forms the basis for a unified description
  of pp, $\pi$p, Kp, $\gamma^*$p, and $\gamma\gamma$ reactions in good
  agreement with experimental data for cross sections, slope
  parameters, and structure functions.
%
  The obtained impact parameter profiles for pp and $\gamma^*_{L}$p
  reactions and the impact parameter dependent gluon distribution of
  the proton $xG(x,Q^2,|\vec{b}_{\!\perp}|)$ show saturation at
  ultra-high energies in accordance with unitarity constraints.
%
} \vfill
%
\cleardoublepage
%
% ________________________________________________________________________
%
%
\end{titlepage}
%

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%\includeonly{Title,Introduction,TheModel,StaticPotential,CEMFields,LETheorems,DDScattering,Conclusion,Parameterizations,ChiComputation,Bibliography}
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%\includeonly{Title,Introduction,TheModel,StaticSources}
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\documentclass[11pt,a4paper,twoside,final]{report}
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%\usepackage{cmbright}  %a beautiful font
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% ___ pagestyle __________________________________________________
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% ___ aliases _______________________________________________________
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\input{aliases.tex} 
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% ___ Begin the actual document. ____________________________________
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\begin{document}
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% ___ Official Title Pages __________________________________________
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\pagenumbering{roman}
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\input{Title}
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% ___ Table of Contents _____________________________________________
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\tableofcontents
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%\addtocontents{toc}{\protect\enlargethispage{1.cm}}
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%\listoffigures
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% ___ The Chapters __________________________________________________
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% \newpage
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\setcounter{page}{1}
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\pagenumbering{arabic}
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% The lines below are necessary in order to enumerate the equations
% according to the sections where they are.
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\@addtoreset{equation}{chapter}
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%__________________________________________________________________
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\input{Introduction}
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\cleardoublepage
\input{TheModel}
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\cleardoublepage
\input{StaticSources}
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\cleardoublepage
\input{EuclideanScattering}
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\cleardoublepage
\input{HEScattering}
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\cleardoublepage
\input{Saturation}
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\cleardoublepage
\input{ComparisonData}
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\cleardoublepage
\input{Conclusion}
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% ___ Appendix ______________________________________________________
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\begin{appendix}
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\cleardoublepage
\input{Parameterizations}  
\cleardoublepage
\input{ChiComputation}  
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\end{appendix}
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% ___ Bibliography __________________________________________________
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%\newpage
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\cleardoublepage
\begin{thebibliography}{99}
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\input{Bibliography}
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\end{thebibliography}  
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% ___ Acknowledgements ______________________________________________
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\pagestyle{empty}
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\cleardoublepage
\input{Acknowledgments.tex}
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\end{document}

