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\title{ \bf Electrostatics of Pagels-Tomboulis effective model}
\author{by \\ \\{\bf H. Arod\'z, M. \'Slusarczyk and A. Wereszczy\'nski}
\\
\\Marian Smoluchowski Institute of Physics,
\\ Jagellonian University, Reymonta 4, Krak\'ow, Poland}

\date{ }

\maketitle

\begin{abstract}
Long time ago Pagels and Tomboulis have proposed a model for the
nonperturbative gluodynamics which in the Abelian sector can be reduced to
a strongly nonlinear electrodynamics. In the present paper we investigate
Abelian, static solutions with external charges in that model. Nonzero
total charge implies that the corresponding field has infinite energy due
to slow fall off at large distances. For a pair of opposite charges the
energy is finite -- it grows like 
$R^{\alpha}$, $0<\alpha <1$ with the distance $R$ between the charges.

\end{abstract}

\vspace*{2cm}
\noindent
TPJU-3/2001
\pagebreak

\section{\bf{Introduction}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Nontrivial features of QCD such as the asymptotic freedom or the quark
confinement are due to its gluonic part.
Unfortunately, in spite of numerous efforts dynamics of non-Abelian gauge
fields in a nonperturbative regime
is not fully understood yet. Popular method of investigating it is
the lattice approximation. It is the most successful
one up to now,  yet it has well-known drawbacks.
There have been several attempts to apply the approach
which has turned out to be very fruitful in condensed matter physics, that
is to construct Ginzburg-Landau type effective models which would
capture the main features of the nonperturbative gluonic dynamics
already on the classical level. In the condensed matter physics, precisely
the Ginzburg-Landau models are the main tool for investigating
such nonperturbative phenomena as, for example, vortices in superfluids and
in superconductors, or hedgehogs in nematic liquid crystals.

In the literature there are several proposals for effective models for the
nonperturbative gluonic dynamics. The most popular ones are the dual
superconductor model [1,2], the colour dielectric model [3,4], and the
stochastic vacuum model [5,6]. Each of them has its own problems.
It seems that the satisfactory
effective model for the nonperturbative gluonic sector has not
been found yet, but this does not mean that such model necessarily will be
very different from the ones already considered in the literature.

The effective Ginzburg-Landau model is expected to give a good description
of physics already on the classical level (a mean field approximation).
When constructing such a model, the first step consists in choosing dynamical
variables, that is the field content of the model. In the case of
gluodynamics almost all models known to us
contain Yang-Mills fields or at least
a $U(1)$ field if one assumes a kind of Abelian projection, and sometimes
other fields. Then, it is clear that the effective model
for the nonperturbative gluonic dynamics has to be drastically different
from typical field theoretical models. First, it should not
allow for existence of massless, freely propagating vector particles (gluons).
Second, it should predict absence of coloured states, and a confining force
between (anti)quarks in colour neutral states.
Third, it should predict existence of massive glueballs which could
propagate freely.  Finally, the model should explicitly contain
a mass scale -- for the massive glueballs as well as in order to reproduce
the trace anomaly of energy-momentum tensor. It is a well-known
fact that classical Yang-Mills theory with the standard Lagrangian obeys
none of these requirements.

In the present paper we investigate a model which has been proposed
long time ago by Pagels and Tomboulis [7]. In a sense it can be regarded
as a version of the colour dielectric model without a dilaton type scalar
field - the pertinent Lagrangian is a function of Yang-Mills field strength
only. The model is too simple in order to provide a
very good approximation to the real-life nonperturbative gluodynamics in QCD,
but it looks rather promising
as far as the four requirements specified above are concerned. Our general
goal is to study such highly nonstandard field theoretical models in order
to clarify their physical contents.

Pagels and Tomboulis calculated electric field of a single point source
in their model. They found that in comparison with the standard
Coulomb field it is less singular close to the charge and
much stronger far away from it. The corresponding energy is infinite due
to the large distance behaviour rather than the singularity at
the origin. This result implies a confinement of charges.
However, one should also check that a pair of charges $Q$, $-Q$
has finite energy, otherwise also the quark-antiquark pairs would disappear
from the physical spectrum and the model would be wrong.
Because of nonlinearity of Gauss law in the model,
one can not compute the electric field of the
pair by a simple superposition.
In the present paper,  using a combination of
analytic and numerical methods, we check that
the energy of the pair $Q$, $-Q$ is finite if a parameter $\delta > 1/4$.
We also calculate distribution of electric field around the charges.
It turns out to be very similar to a flux-tube, that is it is
localised around the straight line connecting the charges, but an effective
string tension depends on the distance $R$ between the charges.
The total energy behaves like
$R^{\alpha}$, where $\alpha=(4 \delta-1)/(4\delta+1)$ and $\delta$ is a
parameter. Thus, the charges are confined if $\delta >1/4$ because then
$ 0 < \alpha <1$. The effective string tension behaves like $R^{\alpha -1}$.
For $\delta=1/4$ we obtain the logarithmic behaviour of the energy.

Interestingly enough, for $\delta = 3/4$ we obtain the $\sqrt{R}$ behaviour
of the energy, which is in agreement with a phenomenological potential found
in fits to spectra of heavy quarkonia \cite{8, 9}.

Our paper is organised as follows. In Section $2$ we recall the model and we
derive field equations. We also point out that in that model classical
vacuum is strongly degenerate. In Section $3$ we calculate the energy of
the system of two static, opposite charges.
Several remarks about the model are collected in Section $4$.


\section{\bf{The Pagels-Tomboulis model}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let us denote by ${\cal F}_2$ the standard Yang-Mills invariant\footnote{We
choose the metric in
Minkowski space-time with the signature $(+1,-1,-1,-1)$. }
\begin{equation}
{\cal F}_2 =
\frac{1}{2} F^{a}_{\mu\nu} F^{a \mu \nu} = B^{ai}B^{ai} - E^{ai}E^{ai},
\label{F}
\end{equation}
where the field strength $F^{a}_{\mu\nu}$ has the usual form
\begin{equation}
F^{a}_{\mu\nu} = \partial_{\mu} A^{a}_{\nu} - \partial_{\nu}
A^{a}_{\mu} -g f_{abc}A^{b}_{\mu}A^{c}_{\nu}
\label{Fa}
\end{equation}
and $E^{ai} = F^{a}_{0i}$, and $ B^{ai} = -
\frac{1}{2}\varepsilon_{ikl}F^{a}_{kl}$ are the
non-Abelian electric field and magnetic induction field,  correspondingly.
We will consider only $SU(2)$ gauge group, hence $a,b,c=1,2,3$.
The model we would like to investigate has Lagrangian $\cal L$ of the form
${\cal L} = \ell ({\cal F}_2)$, where $\ell$ is a nontrivial function
which we shall choose shortly. In a connection with QCD
models of this kind have been proposed long time ago [7], [10], [11].
The Abelian Born-Infeld electrodynamics is
even older [12], but its aim is to modify only physics of strong fields,
while in the QCD case the main problem is with weak fields.
The same remark applies to the quite popular recently non-Abelian
Born-Infeld actions [13].

Pagels and Tomboulis [7] considered Lagrangians of the form
\begin{equation}
L_{eff} (A_{\mu}) = \frac{1}{2} \frac{F_2}{\bar{g}^{2}(t)}
\label{Leff}
\end{equation}
in Euclidean space-time. In formula (\ref{Leff}) $F_2 =
F^{a}_{\mu\nu}F^{a}_{\mu\nu}/2$ is non-negative due to the Euclidean metric,
$t = \ln \frac{F_2}{\mu^{4}}$,
and $\bar{g}(t)$ is the effective coupling constant determined
from the equation
\begin{equation}
t = \int_{g_0}^{\bar{g}(t)} \frac{dg}{\beta(g)},
\label{t}
\end{equation}
where $\beta(g)$ is the Gell-Mann, Low function.
The Lagrangian (\ref{Leff}) has been proposed in [7] as an
Ansatz consistent with renormalization
group invariance of the effective action $\int d^{4}x L_{eff}$, and also with the trace anomaly.
The perturbative $1$-loop result for the $\beta(g)$ function:
\begin{equation}
\beta(g) \cong - b_{0} g^{3}, \;\; b_{0} > 0,
\label{beta_g}
\end{equation}
where $b_{0}$ is a positive constant, gives
\begin{equation}
\frac{1}{\bar{g}^{2}(t)} = \frac{1}{g_0^{2}} + 2 b_{0}t,
\label{1_gb}
\end{equation}
and in consequence
\begin{equation}
L_{eff} = \frac{1}{2} \frac{F_2}{g_0^{2}} + b_{0}F_2
\ln \frac{F_2}{\mu^{4}}.
\label{Leff_2}
\end{equation}
Here $1/g^{2}$ is the integration constant, equal to the
value of the coupling constant at the subtraction point $\mu$. Another
case, also considered in [7], corresponds to
\begin{equation}
\beta(g) = - \delta g,
\label{beta_g_2}
\end{equation}
where $\delta > 0$ is a constant. Now Eq. (\ref{t}) gives
\begin{equation}
\frac{1}{\bar{g}^{2}(t)} = \frac{1}{g_0^{2}} \left( \frac{F_2}{\mu^{4}}
\right)^{2 \delta}
\end{equation}
and
\begin{equation}
L_{eff} = \frac{1}{2 g_0^{2}} F_2 \left( \frac{F_2}{\mu^{4}}
\right)^{2 \delta}.
\label{Leff_3}
\end{equation}

Notice that
\begin{equation}
\left( \frac{F_2}{\mu^{4}} \right)^{2 \delta} =
\exp \left[ 2 \delta \ln \left( \frac{F_2}{\mu^{4}} \right) \right]
= 1 + 2 \delta \ln  \frac{F_2}{\mu^{4}} +\ldots ,
\label{Fmu}
\end{equation}
hence Lagrangian (\ref{Leff_3}) with $\delta = b_{0} g^{2}$ reduces to
(\ref{Leff_2}) when $2 \delta \ln \frac{F_2}{\mu^{4}} \ll 1$, that is when
$F_2$ is close to the subtraction point $\mu$. One may regard Lagrangian
(\ref{Leff_3}) as a resumation of powers of $\ln(F_2/\mu^{4})$ in
the effective action functional $\Gamma (A_{\mu})$ for the
Yang-Mills field. From this viewpoint, $\delta$ can be associated with the
anomalous dimension of the $F_2$ operator.

The corresponding effective models in Minkowski
space-time are obtained by replacing the Euclidean $F_2$ by ${\cal F}_2$.
In order to avoid problems with the sign of ${\cal F}_2$ and
complex-valued Lagrangians we
write $\ln \frac{F_2}{\mu^{4}}$ in
(\ref{Leff_2}) as $\frac{1}{2} \ln \left( \frac{F_2}{\mu^{4}} \right)^{2}$,
and $ \left( \frac{F_2}{\mu^{4}} \right)^{2 \delta}$ as
$\left( \frac{F_2^{2}}{\mu^{8}} \right)^{\delta}$ in (\ref{Leff_3}).


The model corresponding to $ L_{eff}$ (\ref{Leff_2}) was
considered by Adler and Piran [11] in the case where only non-Abelian
electric fields $F^{a}_{0i}$ were present - it gave a
confining force between two point-like opposite charges, at least
in Abelian sector obtained by assuming that $A^{a}_{\mu}(x)
= \delta^{a}_{3} A_{\mu} (x)$. However,
it turns out that if one allows  for the presence of magnetic fields,
the corresponding energy density is not bounded from below.
Moreover, Lagrangian (\ref{Leff_2}) contains the standard kinetic term
$ (\partial_{\mu} A^{a}_{\nu} - \partial_{\nu} A^{a}_{\mu})^2$
hence the gluons could propagate freely. For these reasons, we conclude that
the logarithmic model obtained from (\ref{Leff_2}) is not satisfactory.

Let us turn to the model with $L_{eff}$ given by formula (\ref{Leff_3}).
When passing to Minkowski space-time we change notation a little bit.
In Minkowski space-time we take Lagrangian of the form
\begin{equation}
{\cal L} = - \frac{1}{2} {\cal F}_2
\left( \frac{{\cal F}_2^{2}}{\Lambda^{8}} \right)^{\delta},
\label{Leff_4}
\end{equation}
where $\delta>0,$ instead of Lagrangian (10).
The initial coupling constant $g_0$ has been included into $\Lambda$ which is
regarded as an empirically fixed energy scale in the model.


Components of energy-momentum tensor corresponding to Lagrangian
(\ref{Leff_4}) have the form
\begin{equation}
T_{00} = \frac{1}{2} \left[ \vec{B}^{2} + (1+ 4 \delta)
\vec{E}^{2} \right] \left(
\frac{ {\cal F}_2^{2}}{\Lambda^{8}} \right)^{\delta},
\label{T00}
\end{equation}
\begin{equation}
T_{0i} = - (1 + 2 \delta ) \varepsilon_{iks} E^{ak}B^{as}
\left( \frac{{\cal F}_2^{2}}{\Lambda^{8}} \right)^{\delta}
\label{T0i}
\end{equation}
and
\begin{equation}
T_{ik} = \left\{ \frac{1}{2} \left[ \vec{E}^{2} +
(1+4 \delta) \vec{B}^{2} \right] \delta_{ik} - (1 + 2 \delta)
\left[ E^{ai}E^{ak} + B^{ai}B^{ak} \right] \right\}
\left( \frac{{\cal F}_2^{2}}{\Lambda^{8}} \right)^{\delta}.
\label{Tik}
\end{equation}
We see that in this model  $T_{00} \geq 0$.

Lagrangian (\ref{Leff_4}) does not contain the standard
kinetic term for the gauge fields.
Therefore, it is by no means clear what are physical, propagating excitations
in the model. Let us introduce the non-Abelian dielectric induction
\begin{equation}
D^{ai} = \frac{\partial {\cal L}}{\partial E^{ai}} =
(1+2\delta) E^{ai}  \left( \frac{{\cal F}_2^{2}}{\Lambda^{8}} \right)^{\delta}
\label{D}
\end{equation}
and the non-Abelian magnetic field:
\begin{equation}
H^{ai} = - \frac{\partial {\cal L}}{\partial B^{ai}} =
(1 + 2 \delta) B^{ai} \left( \frac{{\cal F}_2^2}{\Lambda^{8}}
\right)^{\delta}.
\label{H}
\end{equation}
The modified Yang-Mills equations following
from Lagrangian (\ref{Leff_4}) can be written in the form
\begin{equation}
\nabla^a_{ib} D^{bi} = 0, \;\;\;  \nabla_{0b}^a D^{bk} - \varepsilon_{kir}
\nabla_{ib}^a H^{br} = 0,
\end{equation}
where
\[
\nabla^a_{\mu b} =  \delta^a_b \partial_{\mu} + f_{bca} A^c_{\mu}
\]
is the covariant derivative.
Of course in addition to these equations we have the
standard non-Abelian Bianchi identities for $E^{ai}$ and $B^{ai}$:
\begin{equation}
\nabla^a_{ib} B^{bi} =0, \;\;\;
\nabla^a_{0b} B^{bi}  + \varepsilon_{ijk} \nabla^a_{jb} E^{bk} =0.
\end{equation}
These identities are equivalent to formula (2).


It is clear that any fields such that
\begin{equation}
B^{ai}B^{ai} = E^{ai}E^{ai}
\label{EeqB}
\end{equation}
obey the field equations because then ${\cal F}_2=0$.
However, they have vanishing energy-momentum
tensor $T_{\mu \nu}$ and therefore they should be regarded as vacuum fields!
The model has enormously degenerate classical vacuum state, which includes,
in particular, plane waves.  In the standard Yang-Mills theory, small
amplitude classical plane waves correspond to the perturbative gluons. It
is encouraging that they do not belong to the spectrum of physical excitations
of Pagels-Tomboulis model. We plan to address the issue of physical 
excitations in Pagels-Tomboulis model in a forthcoming paper [14]. In the 
present paper we show that the model can describe the confinement of quarks.

Formula (16) implies that the dielectric induction is smaller than the 
electric field if $  \frac{{\cal F}_2^2}{\Lambda^{8}} < 
(1+2\delta)^{-1/\delta},$ that is the model implies antiscreening for weak
fields. For strong fields we have the usual screening. 


\section{Static solutions with external point charges}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf a) The case of non-vanishing total charge }\\
Equations (18), (19) are even more complicated than the standard Yang-Mills
equations, which are obtained for $\delta =0$. The fact that the classical
Yang-Mills theory is not confining can be seen already from an analysis of its
Abelian sector. For this reason, we check the Abelian sector of
Pagels-Tomboulis model ($\delta >0$). Let us remind that that sector is
constituted by gauge potentials $A^a_{\mu}$ with only one colour component,
identical for all $\mu = 0, 1, 2, 3. $  For example, we may take
\begin{equation}
A^a_{\mu} = \delta^a_3 A_{\mu}(x).
\end{equation}
In the Abelian sector, the covariant derivatives in Eqs. (18), (19) reduce to
the ordinary ones and we may omit the superscript $ ^a$. Nevertheless, the
equations remain nonlinear if $\delta>0$ -- the cases $\delta =0$
and $\delta >0$ are drastically different. In particular, if $\delta \geq 1/4$
the Abelian sector of Pagels-Tomboulis model is compatible with the
confinement of charges.

Let us consider first a smooth, spherically symmetric distribution
$j_0(r)$ of an external, static charge in a finite region around the origin.
The Gauss law has the form
\begin{equation}
\nabla \vec{D} =j_0(r),
\end{equation}
and the electric field obeys the condition
\begin{equation}
\nabla \times \vec{E} =0,
\end{equation}
with $\vec{D}$ and $\vec{E}$ related by formula (16) (we omit the superscript
$a=3$). Because of the nonlinearity, solutions of Eqs. (22), (23) can not
be obtained by superposition of solutions for point charges. Nevertheless,
spherically symmetric solutions follow easily from Gauss law (22).
Far away from the charge
\begin{equation}
\vec{D} =\frac{Q}{4\pi} \frac{\vec{r}}{r^3}, \;\;\;
\frac{\vec{E}}{\Lambda^2} =
sign(Q) (1+2\delta)^{-1/(1+4\delta)} \left( \frac{|Q|}{4 \pi\Lambda^2 r^2}
\right)^{\frac{1}{1+4\delta}} \frac{\vec{r}}{r},
\end{equation}
where
\[
Q= \int d^3r j_0(r).
\]
Thus, both fields vanish when $r \rightarrow \infty$, namely $|\vec{D}| \sim
r^{-2}, \;\; |\vec{E}| \sim r^{-2/(1+4\delta)}$.
Simple calculation shows that the total energy of the gauge field, $\int
d^3r T_{00}$, is infinite if $\delta \geq 1/4$,
due to the behaviour of $\vec{E}$
at large $r$. This fact is not changed by inclusion of the interaction energy
of the external charge with the gauge field, $\int d^3r j_0 A_0$, because
this integral is finite. For comparison, in the Yang-Mills case the energy
of the gauge field (24) (now with $\delta =0$) is finite.

This result suggests that in Pagels-Tomboulis model states with nonvanishing
total charge $Q$ are not physically feasible. In order to turn this
suggestion into a theorem one would have to find solutions of the non-Abelian
gauss law
\[
\nabla^a_{ib} D^{bi} = \delta^a_3 j_0(r),
\]
together with the remaining equations (18), (19), without the simplifying
assumption (21). Even in the Yang-Mills case this is a highly nontrivial task,
see e. g. \cite{15} for a review. In particular, Kiskis \cite{16} showed that
in the Yang-Mills case exist solutions with arbitrarily small (but positive)
energy. Such solutions in general are not static and have nonvanishing
magnetic fields. However, his reasoning is based on linearity of Yang-Mills
equations in the Abelian sector, and therefore it can not be repeated in
Pagels-Tomboulis model. This is encouraging, nevertheless the question whether
finite energy solutions with $Q \neq 0$ are completely excluded remains
open.

\vspace*{0.5cm}
\noindent
{\bf b) Dipol-like external charge density } \\
Effective model for QCD should allow for finite energy colour singlet
quark-antiquark states \footnote{In the case of SU(3) gauge group colour
singlets built of three quarks are related to the quark-antiquark states
because a diquark can be regarded as an antiquark as far as colour charge is
concerned.}.  Therefore, we shall now consider a dipole-like external charge.

The results of subsection 3a) are valid also when the external charge is
point-like, $j_0(r) = Q \delta(\vec{r})$. Then, the resulting energy density
has a singularity at $r=0$ which is integrable if $\delta >1/4$. Therefore,
it is not important for our purpose whether the two charges forming the
dipole are point-like or spatially extended. For simplicity, we assume that
they are point-like. Thus, we now take
\begin{equation}
j_{0} = q \delta(x) \delta(y) \left[ \delta( z + R/2 ) - \delta ( z - R/2 )
\right]
\label{ch_dens}
\end{equation}
where $q>0$, and again we look for solutions of Eqs. (22), (23) in the
Abelian sector. 

We would like to check that the corresponding total energy
$\cal E$ of the gauge field, given by the integral ${\cal E}=
\int d^3r T_{00}$, is finite. First, simple dimensional analysis shows that
\begin{equation}
{\cal E} =
c_0 |q|^{\frac{2+4\delta}{1+4\delta}} \Lambda^{\frac{8\delta}{1+4\delta}}
R^{\frac{4\delta-1}{4\delta+1}},
\label{lin_en}
\end{equation}
where $c_0$ is a numerical constant. This formula follows from the fact that
\[
{\cal E}
\sim |q|^{\frac{2+4\delta}{1+4\delta}} \Lambda^{\frac{8\delta}{1+4\delta}},
\]
as implied by Eq. (22) and formulas (13), (16). The exponent of $R$ is
dictated by the requirement that $\cal E$ has the dimension $cm^{-1}$.
Thus, it remains to show that the constant $c_0$ is finite. Because
singularities at the point-like charges are of integrable type, finiteness
of $c_0$ depends solely on behaviour of the fields at the spatial infinity.


Following Adler and Piran, \cite{11}, \cite{17}, \cite{18}, we express 
the dielectric induction $\vec{D}$ by the dual potential $\vec{C}$:
\begin{equation}
\vec{D} = \nabla \times \vec{C}.
\label{Phi_def}
\end{equation}
Then, at all points where $\vec{C}$ is sufficiently smooth, automatically
$\nabla  \vec{D} = 0$. However, the presence of source (25) on the r.h.s. of 
Eq. (22) implies that $\vec{C}$ can not be regular everywhere -- 
it has to have a singularity
akin to the Dirac string for magnetic monopoles. In our case it is natural to
assume that the string connects the two point charges.
To exploit the axial symmetry of the problem it is natural to introduce
the cylindrical coordinates $(\rho, \phi , z) $
instead of the Cartesian ones. The coordinates of the point sources are
then $\rho = 0, \; z = \pm R/2$. As in \cite{11}, \cite{17}, \cite{18} we
assume that the dual potential $\vec{C}$ can be expressed by a scalar flux 
function $\Phi (\rho, z)$, which is defined as follows
\begin{equation}
\vec{C} = \frac{\hat{\phi}  }{2 \pi \rho} \Phi,
\end{equation}
where $\hat{\phi}$ is the unit vector tangent to the $\phi$ coordinate line.


Equation (23) may be rewritten as
\begin{equation}
\nabla \times \left( \frac{\vec{D}}{\varepsilon } \right) = 0,
\label{fqes_E3}
\end{equation}
where
\begin{equation}
\varepsilon =
(1 + 2 \delta) \left( \frac{\vec{E}^2}{ \Lambda^4} \right)^{2 \delta }
\label{eps_E}
\end{equation}
is the dielectric function. The Ansatz (28) reduces Eq.(29) to
\begin{equation}
\nabla \left( \sigma  \nabla \Phi \right ) = 0,
\label{eq_s_phi}
\end{equation}
where
\begin{eqnarray}
\sigma = \left(  \frac{1}{\rho}  \right )^{ \frac{2+4 \delta}{1+4 \delta} }
\left( \frac{1}{\left| \nabla \Phi \right|^{2} }
\right)^{\frac{2 \delta}{1+4 \delta}}.
\label{sigma}
\end{eqnarray}


It remains to fix boundary conditions for $\Phi$.
Formulas (27), (28) imply that
\begin{eqnarray}
& & \Phi = 0 \; \;\; \mbox{for} 
\;\; \rho = 0,\;\; \left| z \right| > R/2, \nonumber \\
& & \Phi = q  \;\;\; \mbox{for} \;\; \rho = 0, \;\; \left| z \right| < R/2.
\label{bcond}
\end{eqnarray}
To obtain, for example, the second line in (33), consider a sphere surrounding
one of the charges with a small hole around the point at which the
sphere is punched by the segment of the z-axis connecting the charges.
The flux
of the $\vec{D}$ field through such surface is equal to line integral of
$\vec{C}$ over the boundary of the hole. Shrinking the hole to the point of
the intersection we find that the value of $\Phi$ at that point is equal to
the total flux $Q$ of $\vec{D}$ through the sphere.

We also assume that
\begin{equation}
\Phi \rightarrow 0 \;\;\; \mbox{for} \;\; \rho^{2} + z^{2} \rightarrow \infty.
\end{equation}
This condition is justified by the expectation that angular dependence of
$\Phi$ at the spatial infinity would increase the total energy. Because
it has been assumed that $\Phi$ vanishes on the z-axis if $|z| >R/2$, it has
to vanish in all other directions.


To summarize, the problem of solving the set of field equations for the
electric field $\vec{E}$ with given charge density (25) in three spatial
dimensions is reduced to one sourceless equation (31) in the region $\rho>0,
z \in(\infty, -\infty),$ which can be regarded as a cylindrical shell with
the outer radius going to $\infty$ and the inner one to 0. The boundary
conditions are given by (33), (34).

Equation (31) can be rederived from the condition that the flux function
minimizes the total field energy  ${\cal E}= \int d^3r T_{00}$ under the
boundary conditions (33), (34).  Formulas (13), (27) and (28) give
\begin{equation}
T_{00} = \frac{1}{2} (1+4\delta) \left[ 2\pi ( 1+4\delta ) \rho 
\right]^{\frac{-2-4\delta}{1+4\delta} } \Lambda^{\frac{8\delta}{1+4\delta}}
| \nabla \Phi |^{\frac{2+4\delta}{1+4\delta}},
\end{equation}
where $|\nabla \Phi | = \sqrt{(\nabla \Phi )^2}$.  One can easily construct
examples of the flux function which obey the boundary conditions and have 
finite total energy. For example, we may take
\begin{equation}
\Phi = \frac{q}{2} \left( \frac{z+R/2}{\sqrt{\rho^2 +(z+R/2)^2}} - 
\frac{z-R/2}{\sqrt{\rho^2 +(z-R/2)^2}} \right).
\end{equation}
This function has been obtained from the sum of the dual potentials for two
Dirac monopoles of opposite charges, and it implies that the Dirac string
just connects the charges $q, -q$ along the z-axis. The function (36) obeys 
the boundary conditions, but it does not obey Eq. (31) unless $\delta =0.$
Nevertheless, the finite value of $ {\cal E}$ corresponding to it may be taken
as an upper bound for the total field energy of the charges $q, -q$. 

Solutions of Eq.(31) can be found with the help of numerical computations. 
We use the standard
iterative procedure \cite{17}. Due to the symmetry of the problem it is
sufficient to restrict our approach to the region $ z \geq 0$, $\rho \geq 0$.
The continuous variables $\rho, z$ are replaced by the computational lattice
with $(n_{\rho}+1) \times (n_{z}+1) $ sites; the flux function $\Phi$ reduces 
to the discrete set of values on the lattice $\Phi_{i,j}$ where $i = 0\ldots
n_{\rho}$ and $j = 0\ldots n_{z}$.
The point charge $q$ is put on the site of the
computational lattice with $i = 0, j = n_{q}$ where $ 0 < n_{q} < n_{z}$.
The boundary conditions (33), (34) are replaced by:
\begin{eqnarray}
& & \Phi_{0,j} = q, \;\;\; 0 \leq j < n_{q}, \\
& & \Phi_{0,n_{q}} = q/2,  \nonumber \\
& & \Phi_{0,j} = 0, \;\;\; n_{q} < j \leq n_{z}, \nonumber \\
& & \Phi_{n_{\rho},j} = 0, \;\;\; 0 \leq j \leq n_{z}, \nonumber \\
& & \Phi_{i, n_{z}} = 0, \;\;\; 0 \leq i \leq n_{\rho}. \nonumber
\end{eqnarray}
In the first step, sites of the computational lattice
are populated by arbitrary values.
Next, new  $\Phi_{i,j}$ are computed using a discretized version of the
field equation (31). The iterative procedure stops when the difference 
between actual value of $\Phi_{i,j}$ and the one computed in
the previous step is less then a chosen accuracy. By assumption, such
approximate solution obeys the homogeneous boundary condition (34) already
at finite $\rho$ and $z$. 

In our computations we used $900 \times 900$ mesh.
Some sample results were also obtained for bigger meshes  but
the results changed insignificantly. We have repeated our procedure for
some values of the charge separation distance $R$ and fixed size of the 
lattice. The final results were obtained for $\delta = 0.75$, and $q = 1.1$.
The results are presented in Figs. $1$ -- $5$.  Fig. $1$ presents the flux
function $\Phi$,  Fig. $2$ the profile of $\Phi$ for $z = 0$.
From the definition of $\Phi$ the dielectric induction $\vec{D}$ as well 
as the energy density $\varepsilon$ may be derived. The rescaled 
(dimensionless) energy density
$\epsilon = T_{00}/\Lambda^{4}$ is depicted in Fig. $3$. Fig. $4$ presents
the energy density for one of the Coulomb peaks from Fig. $3$.
In Fig. 5 the energy density profiles ($\epsilon (\rho)$
for $\phi = z = 0$) for various values of $R$ are compared. In all figures we
use the rescaled spatial coordinates: $z \Lambda \rightarrow z,$ 
$\rho \Lambda \rightarrow \rho$ and $ R \rightarrow \Lambda R.$. It is clear
that the flux-tube connecting the two point charges becomes thicker when
the charge separation distance increases. We have obtained the same pictures 
starting from several different initial configurations for the iterative 
procedure. Our numerical procedure was also tested for $\delta = 0$.
In this case the analitical solution for $\Phi$ is given by formula (36). The
numerical procedure recovers it.


The energy density distribution was used to derive the total energy
of the considered field configuration. As the energy density decreases slowly
when $\rho^2 + z^2 \rightarrow \infty$ it is necessary to collect the energy
density from relatively large region around the charges. For this reason we
have used the lattice with link length varying from short near the charges
to long for large $\rho^2 + z^2$. The energy of configurations for various
$R$ was used to check the relation (\ref{lin_en}) - if it holds $\varepsilon$
should depend linearly on $\sqrt{R}$ for $\delta  = 0.75$. Using numerical
results for this linear relation one can estimate that $c_{0} \approx 4.1.$



\section{Summary and remarks}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We have investigated the Abelian sector of Pagels-Tomboulis model. If the 
total charge $Q$ is non-vanishing the resulting field has infinite energy,
while in the case of point-like charges $q, -q$ separated by the distance $R$
the energy is finite, proportional to $R^(4\delta -1)/ (4 \delta +1)$, 
provided that $\delta >1/4.$

There are several directions in which one could continue the present work. 
First, we have not investigated stability of our Abelian solutions against
fluctuations of gauge fields. Because the derivation of screening phenomenon
given in the standard Yang-Mills case by Kiskis, \cite{16}, does not work
if $\delta >0$, one may hope that a presence of the other components of 
the gauge field $(A_{\mu}^1, A_{\mu}^2)$ will increase the energy. Also, one
may ask about stability against a collapse of the field configuration in the 
Abelian sector. To answer this question one would have to investigate 
time-dependent solutions. It might turn out that additional terms would have
to be included in the Lagrangian in order to prevent such a collapse, in an
analogy to the well-known Skyrme term in a mesonic effective Lagrangian. 

Finally, one could use more refined forms of the $\beta(g)$ function in 
Eq. (4). The model considered in the present paper should be regarded as the
simplest one compatible with the asymptotic freedom.  

All problems mentioned above are important, but in our opinion the most 
important and interesting one is to describe
physical excitations in the models of the Pagels-Tomboulis type. 


\begin{thebibliography}{9}
\bibitem{1} M. Baker, J. S. Ball and F. Zachariasen, Phys. Rep.
\underline{209}, 73 (1991).
\bibitem{2} M. Baker, N. Brambilla, H. G. Dosch and A. Vairo, Phys. Rev.
\underline{D58}, 034010 (1998).
\bibitem{3} R. Friedberg and T. D. Lee, Phys. rev. \underline{D15}, 
1694 (1977); {\it ibidem} \underline{D18}, 2623 (1978). 
\bibitem{4} G. Chanfray, O. Nachtman and H.-J. Pirner, Phys. Lett. 
\underline{147B}, 249 (1984). 
\bibitem{5} H. G. Dosch and Yu. A. Simonov, Phys. Lett. \underline{B205}, 339
(1988).
\bibitem{6} Yu. A. Simonov, Nucl. Phys. \underline{B307}, 512 (1988). 
\bibitem{7} H. Pagels and E. Tomboulis,
Nucl. Phys. \underline{B43}, 485 (1978).
\bibitem{8} L. Motyka and K. Zalewski, Z. Phys. \underline{C69}, 343 (1996).
\bibitem{9} K. Zalewski, Acta Phys. Pol. \underline{B29}, 2535 (1998).
\bibitem{10} R. Mills, Phys. Rev. Lett. \underline{43 }, 549 (1979).
\bibitem{11} S.L Adler and T. Piran, Phys. Lett. \underline{B113}, 405 (1982).
\bibitem{12} M. Born and L. Infeld, Proc. Roy. Soc. (London) \underline{A144},
425 (1934).
\bibitem{13} See, e.g., A. Tseytlin, Nucl. Phys. \underline{B501}, 41 (1997);
D. V. Gal'tsov and R. Kerner, Phys. Rev. Lett. \underline{84}, 5955 (2000).
\bibitem{14} H. Arod\'z, M. \'Slusarczyk and A. Wereszczy\'nski, work in
progress.
\bibitem{15} H. Arod\'z, Acta Phys. Pol. \underline{B14}, 825 (1983).
\bibitem{16} J. Kiskis, Phys. Rev. \underline{D21}, 421 (1980).
\bibitem{17} S. L. Adler and
T. Piran, Rev. Mod. Phys. \underline{56}, 1 (1985).
\bibitem{18} S.L. Adler,  Phys.Rev. \underline{D20}, 3273 (1981).
\bibitem{19}  M. Grady, Phys. Rev. \underline{D50}, 6009 (1994).
\bibitem{20} R. Dick, Eur. Phys. J. \underline{C6}, 701 (1999).
\bibitem{21} M. \'Slusarczyk and A. Wereszczy\'nski, work in progress.
\end{thebibliography}

\pagebreak

\begin{figure}[ht]
      \unitlength 1cm
      \begin{picture}(0,10)
      \put(-3.5,-0.7){
        \special{
          psfile=Fig1.eps
          hscale=100
          vscale=100
        }
      }
      \end{picture}
    \caption{The flux function $\Phi$ for $\delta = 0.75$ and $ R = 20$.}
  \end{figure}
%
\begin{figure}[ht]
      \unitlength 1cm
      \begin{picture}(0,10)
      \put(0.1,-1.5){
        \special{
          psfile=Fig2.eps
          hscale=80
          vscale=80
        }
      }
      \end{picture}
    \caption{The flux function $\Phi$ 
    for $\delta = 0.75$, $R = 20$ and $z = 0$.}
  \end{figure}
%
\begin{figure}[ht]
      \unitlength 1cm
      \begin{picture}(0,10)
      \put(-3.5,-0.7){
        \special{
          psfile=Fig3.eps
          hscale=100
          vscale=100
        }
      }
      \end{picture}
    \caption{The energy density $\epsilon$ for $\delta = 0.75$
    and $R = 20$.}
  \end{figure}
%
\begin{figure}[ht]
      \unitlength 1cm
      \begin{picture}(0,10)
      \put(-3.5,-0.7){
        \special{
          psfile=Fig4.eps
          hscale=100
          vscale=100
        }
      }
      \end{picture}
    \caption{The energy density of the point charge  $\epsilon$
    for $\delta = 0.75$ and $R = 20$.}
  \end{figure}
%
%
\begin{figure}[ht]
      \unitlength 1cm
      \begin{picture}(0,10)
      \put(-1.0,-1.5){
        \special{
          psfile=Fig5.eps
           hscale=80
           vscale=80        }
      }
      \end{picture}
    \caption{The energy density $\epsilon(\rho)$ for $\delta = 0.75$ ,
    $z = \phi = 0$ and $R = 10,20,30$.}
  \end{figure}



\end{document}

