% Euclidean.../AppendixParameterizations.tex
%
% ___ Parameterizations ______________________________________
\section{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

\smallskip

%
\be
        C_i 
        \,\,=\,\, 
        C_i^A \,\cup\, C_i^B \,\cup\, C_i^C \,\cup\, C_i^D
\label{Eq_Ci_parameterization}
\ee
%

\smallskip

with

\smallskip

%
\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
%

\smallskip

where

\smallskip

%
\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
%

\smallskip

\noindent
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.
%
\begin{figure}[p]
\centerline{\epsfig{figure=loop_parameterization_ite.eps,width=10.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}
%
The tilting angle $\Theta_i\neq 0$ is the central quantity in the
analytic continuation presented in Sec.~\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}.

\newpage

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
%


%%% Local Variables: 
%%% mode: latex
%%% TeX-master: "master"
%%% End: 
% SectionCEMFields.tex
%
% ______________________________________________________________________________
\section{Chromo-Field Distributions of Color-Dipoles}
\label{Sec_Flux_Tube}
% ______________________________________________________________________________

In this section we compute the chromo-electric fields generated by a
static color-dipole in the fundamental and adjoint representation of
$SU(N_c)$. We find formation of a color-flux tube that confines the
two color-sources in the dipole. This confining string is analysed
quantitatively. Its mean squared radius is calculated and transverse
and longitudinal energy density profiles are provided. The interplay
between perturbative and non-perturbative contributions to the
chromo-field distributions is investigated and exact Casimir scaling
is found for both contributions.

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 to compare the
  field strengths in surface $S_P$ with the 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 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 --- illustrated
explicitly in Appendix~\ref{Sec_Static_Potential} --- that emerges
trivially in our approach as a consequence of the Gaussian
approximation used to truncate the cumulant
expansion~(\ref{Eq_matrix_cumulant_expansion}). Indeed, Casimir
scaling hypothesis~\cite{Ambjorn:1984dp} has been verified to high
accuracy for $SU(3)$ on the lattice~\cite{Deldar:1999vi,Bali:2000un}
which has 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}.  Also the
Casimir scaling of the chromo-field distributions has been considered
on the lattice for $SU(2)$ where only slight deviations have been
found that were interpreted as hints towards adjoint quark
screening~\cite{Trottier:1995fx}.

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 illustrates clearly a shortcoming of our
model. 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
Appendix~\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 previous applications of
our model~\cite{Shoshi:2002in,Shoshi:2002ri,Shoshi:2002fq}. 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 color-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 color-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
%
\be
        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)}
\ee
%
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.

The {\em longitudinal} and {\em transverse energy density profiles}
generated by color-dipoles 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 color-dipoles
  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 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
  $g^2\varepsilon_3(X_2,X_1\!=\!X_3\!=\!0)$ of color-dipoles 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 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 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 for the first
time with the LLCM parameters, the minimal surfaces, and the
exponential correlation function.

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

In this section we present a Euclidean approach to high-energy
reactions of color-dipoles in the eikonal approximation. After a short
review of the functional integral approach to high-energy
dipole-dipole scattering in Minkowski space-time, 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.

In {\em Minkowski space-time}, high-energy reactions of color-dipoles
in the eikonal approximation have been 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_model_T_amplitude}
\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 an arbitary $SU(N_c)$
representation.)  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 field
  strengths in surface $S_1$ with the 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 $C_i$ 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+X,Dosch:1997ss,Berger:1999gu,Dosch:2001jg,Kulzinger:2002iu,Shoshi:2002in,Shoshi:2002ri,Shoshi:2002fq}.
  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 VEV's $\langle\ldots\rangle_M$ 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 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, 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 VEV's
$\langle\ldots\rangle_E$ 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 at the end of
this section,
%
\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}.

Let us now perform the analytic continuation explicitly in our
Euclidean 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 VEV's~(\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-perturbative
($\nprt$) 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} and in all
earlier SVM applications to high-energy
scattering~\cite{Kramer:1990tr,Dosch:1994ym,Dosch:RioLecture,Rueter:1996yb+X,Dosch:1997ss,Berger:1999gu,Donnachie:2000kp+X,Dosch:2001jg,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 string to dipole-dipole scattering.
We have analysed these string contributions systematically as
manifestations of confinement in high-energy scattering reactions in
our previous work~\cite{Shoshi:2002fq}.

From the small-$\chi$ limit, one sees that the full $S$-matrix
element~(\ref{Eq_S_DD_1_M}) describes multiple gluonic interactions.
Indeed, the higher order terms in the expansion of the exponential
functions ensure the fundamental $S$-matrix unitarity condition in
impact parameter space as shown discussed
in~\cite{Berger:1999gu,Shoshi:2002in}.

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 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 the $S$-matrix
unitarity~\cite{Shoshi:2002in,Shoshi:2002ri}. 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.

Although the scattering of two color-dipoles in the fundamental
representation of $SU(N_c)$ is, of course, 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, z_2, \vec{r}_2) 
        \,\,:=\,\, \lim_{s \rightarrow \infty} 
        S^M_{\fundamental\,\adjoint}(\Theta, \vec{b}, z_1, \vec{r}_1, z_2, \vec{r}_2) 
\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}).

Finally, we would like to comment on the {\em van der Waals
  interaction} of two color-dipoles, which is --- as already mentioned
--- related to the Euclidean $S$-matrix element in the limiting case
of $\Theta=0$ as can be seen from~(\ref{Eq_S_DD_<->_V_DD}): The QCD
van der Waals potential between two static dipoles can be expressed 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 this IR divergence due to the static
limit. In the more general case of $g^2$ finite and $T\to\infty$,
which is applicable also for the non-perturbative component of our
model, 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 correlations 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: 
% SectionLETheorems.tex
%
% ______________________________________________________________________________
\section{Low-Energy Theorems}
\label{Sec_Low_Energy_Theorems}
% ______________________________________________________________________________

In this section we use low-energy theorems to test the consistency of
the non-perturbative SVM component and to determine the value of the
$\beta$\,-\,function and $\alphaS = g^2/(4\pi)$ at the renormalization
scale at which this component is working. The considered energy and
action sum rules allow us to confirm the consistency of our loop-loop
correlation result with the result obtained for the VEV of one loop.

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}) in
Appendix~\ref{Sec_Static_Potential}. 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: 
%%% mode: latex
%%% TeX-master: "master"
%%% End: 
% SectionStaticPotential.tex
%
% ______________________________________________________________________________
\section{The Static Color-Dipole Potential}
\label{Sec_Static_Potential}
% ______________________________________________________________________________

In this Appendix the QCD potential of static color-dipoles in the
fundamental and adjoint representation of $SU(N_c)$ is computed in our
model.  Color-Coulomb behavior is found for small dipole sizes and the
confining linear rise for large dipole sizes. Casimir scaling is
obtained in agreement with lattice QCD investigations.

As already explained in Sec.~\ref{Sec_Flux_Tube}, 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 $V_r =
V_{\fundamental}$ is obtained from a loop in the fundamental
representation and the potential of a static gluino pair $V_r =
V_{\adjoint}$ from a loop in the adjoint representation.

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
Eqs.~(\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} 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 function in~(\ref{Eq_Vr(R)_NP_nc}) reads
%
\be
        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)}
\ee
%
and the correlation function in~(\ref{Eq_Vr(R)_NP_c}) is given
in~(\ref{Eq_D(3)np_c(z,a)}).  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} and reads for a color-dipole in the
representation $r$ of $SU(N_c)$
%
\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 fundamental 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 the 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}). This is in line with
the obtained Casimir scaling of the chromo-electric fields discussed
in Sec.~\ref{Sec_Flux_Tube}. 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 holds in particular
  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}.  A shortcoming of our model
discussed already in Sec.~\ref{Sec_Flux_Tube} reappears in the static
dipole potential: From Eq.~(\ref{Eq_Vr(R)_Gaussian_approximation}) and
Fig.~\ref{Fig_Static_Quark-Antiquark_Potential_F_vs_A} it is clear
that string breaking is neither described for fundamental nor for
adjoint dipoles in our model which indicates again that not only
dynamical fermions (quenched approximation) are missing but also some
gluon dynamics.


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% ___ Title, authors and addresses _____________________________________________
%
\title{ \vspace*{-2.cm} {\normalsize\rightline{HD-THEP-02-22}}
  {\normalsize\rightline}  \vspace*{0.cm} 
{\Large
\bf  
\boldmath 
Confining QCD Strings, Casimir Scaling,
and a Euclidean Approach to High-Energy Scattering
}}
%
%\footnote{Supported by the European Contract
%    No.~FMRX-CT96-0008 and the Graduiertenkolleg ``Physikalische Systeme
%    mit vielen Freiheitsgraden'', Universit\"at Heidelberg}
%
\author{}
\date{} \maketitle
%
\vspace*{-2.5cm}
%
\begin{center}
%
\renewcommand{\thefootnote}{\alph{footnote}}
%
{\large 
A.~I.~Shoshi$^{1,}$\footnote{shoshi@tphys.uni-heidelberg.de},
F.~D.~Steffen$^{1,}$\footnote{Frank.D.Steffen@thphys.uni-heidelberg.de},
H.~G.~Dosch$^{1,}$\footnote{H.G.Dosch@thphys.uni-heidelberg.de}, and
H.~J.~Pirner$^{1,2,}$\footnote{pir@tphys.uni-heidelberg.de}}

{\it $^1$Institut f\"ur Theoretische Physik, Universit\"at Heidelberg,\\
Philosophenweg 16 {\sl \&}\,19, D-69120 Heidelberg, Germany}

{\it $^2$Max-Planck-Institut f\"ur Kernphysik, Postfach 103980, \\
D-69029 Heidelberg, Germany}
%
\end{center}
%
% ___ Text of abstract _________________________________________________________
%
\begin{abstract}
  
  We compute the chromo-field distributions of static color-dipoles in
  the fundamental and adjoint representation of $SU(N_c)$ in the
  loop-loop correlation model and find Casimir scaling in agreement
  with recent lattice results. Our model combines perturbative gluon
  exchange with the non-perturbative stochastic vacuum model which
  leads to confinement of the color-charges in the dipole via a string
  of color-fields. We compute 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 gauge-invariant
  dipole-dipole scattering and obtain a Euclidean approach to
  high-energy scattering that allows us in principle to calculate
  $S$-matrix elements directly in lattice simulations of QCD. We apply
  this approach and compute the $S$-matrix element for high-energy
  dipole-dipole scattering with the presented Euclidean loop-loop
  correlation model. The result confirms the analytic continuation of
  the gluon field strength correlator used in all earlier applications
  of the stochastic vacuum model to high-energy scattering.

\vspace{1.cm}

% keywords here
\noindent
{\it Keywords}: 
Casimir Scaling, 
Confining String,
Flux Tube, 
High-Energy Scattering,
Low-Energy Theorems,
Static Potential,
Stochastic Vacuum Model

\medskip

% PACS codes here
\noindent
{\it PACS numbers}: 
%
11.15.-q,  % Gauge field theories 
%
11.15.Kc,  % Classical and semiclassical techniques
%
12.38.-t,  % Quantum chromodynamics
%
12.38.Aw,  % General properties of QCD (dynamics, confinement, etc.)  
%
%
%
\end{abstract}

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\input{AppendixChiComputation.tex}  
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