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\markboth{\quad\small \em M.N.~Chernodub, M.I.~Polikarpov and 
V.I.~Zakharov\hfill}{\hfill\small \em
Topics on the magnetic monopoles in QCD \quad}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%BEGINNING OF TEXT
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\sloppy

\title{TOPICS ON THE MAGNETIC MONOPOLES IN QCD}

\author{M.N.~CHERNODUB,$^{a,b}$ M.I.~POLIKARPOV$^{\,a}$ and V.I.~ZAKHAROV$^{\,c}$\\[-1mm]~\\ }

\address{
${}^a$\,ITEP, B.Cheremushkinskaya 25, Moscow, 117259, Russia\\\vspace*{1mm}
${}^b$\,ITP, Kanazawa University, Kanazawa 920-1192, Japan\\\vspace*{1mm}
${}^c$\,Max-Planck Institut f\"ur Physik, F\"ohringer Ring 6, 80805 M\"unchen, Germany}

\maketitle\abstracts{We overview the issue of the monopole condensation in
non-Abelian theories. We emphasize the simplicity of the (lattice)
experimental picture and some difficulties of its interpretation
in theoretical terms. Some features of  the polymer picture
of the monopole cluster and of the corresponding field theory
are discussed.
%The review is written for the M.S. Marinov Memorial volume.
}

\vspace{0.4cm}

\tableofcontents

\newpage

\section{Well understood monopoles}


There are actually quite a few types of monopoles discussed: say,
Dirac monopoles, non-Abelian monopoles, lattice monopoles. Their
theoretical status is in fact very different. Let us begin with
the cases well understood theoretically (and we confine ourselves
to pure gauge theories, without Higgs fields). Emphasizing, of
course, the points relevant to the later discussion of the
``realistic case'' of the non-Abelian monopoles.

\subsection{Compact $U(1)$}

The show case of the monopole condensation is the compact $U(1)$.
As is noted first by Polyakov\,\cite{polyakov} the Dirac string
does not cost any energy because of the compactness of the lattice
QED. The action is defined as a sum over the plaquette actions and
for the latter, roughly speaking one has,
\beq
\label{wilson}
S_{plaq}~=~-{1\over 2e^2}{\mathrm{Re}}\left({\rm e}^{i\!\oint\!
A_{\mu}dx_{\mu}}-1\right)\,,
\eeq
where the contour integral is over the plaquette boundary. For the
soft fields (on the lattice spacing scale) the action is proportional
to $F^2_{\mu\nu}$ and reduces to the standard one. While for the Dirac
string the action vanishes provided that the Dirac quantization
condition, $e \, g_m = 2 \pi n$ is satisfied.

As for the self energy due to the magnetic field, it is linearly
divergent at small distances:
\beq
\label{linear}
M_{mon}~=~{1\over 8\pi}\int{\bf B}^2d^3r~\sim~{1\over 8e^2}{1\over
\aaa}\,,
\eeq
where
$\aaa$ is the lattice spacing, $e$ is the electric charge and the
magnetic charge is\,\footnote{\,The notation $g$ is
reserved for the non-Abelian coupling, the magnetic coupling is
denoted as $g_m$.} $g_{m}=2 \pi /e$\,. Thus, the monopoles are still
infinitely heavy
and, at first sight, this precludes any condensation since the
probability to find a monopole trajectory of the length $L$ is
suppressed as
\beq
\label{action}
\exp(-S)~=~\exp\left(-{c\over e^2}\cdot {L\over \aaa}\right)\,.
\eeq
Note that the constant $c$ depends on the details of the lattice
regularization but can be found explicitly for any concrete case.

However there is an exponentially large enhancement factor due to
the entropy. Namely, the trajectory of the length $L$ can be
realized on a cubic lattice in $N_L=7^{L/\aaa}$ various ways. Here
$7$ stands for the number of directions in 4D in which the
monopole can go at each step. The eight direction looks exactly
backward with respect to the trajectory and if this eight
direction is chosen the trajectory would be cancelled against the
previous step. The entropy factor,
\beq
N_L~=~\exp\left(\ln 7 \cdot {L\over \aaa}\right)\,,
\eeq
cancels the suppression due to the action
(\ref{action}) if the coupling $e^2$ satisfies the condition
\beq
\label{critical}
e^2_{crit}~=~c/\ln7~\approx~1\,,
\eeq
where we quote the numerical value of $e^2_{crit}$ for the Wilson action
and cubic lattice. At $e^2_{crit}$  any monopole trajectory length
$L$ is allowed and the monopoles condense.


This simple theory works within about one percent as far as the value of $e^2_{crit}$
is concerned.\cite{suzuki1} Note that the energy-entropy balance above does not
account for interaction with the neighbors.

\subsection{Non-Abelian monopoles in the classical limit}

Consider now the non-Abelian theory with the Lagrangian
$$
L~=~-{1\over 4 g^2}(F_{\mu\nu}^a)^2\,,
$$
where
%$a$ is the color index and --> obvious and mixes with "a" -- lattice spacing.
for simplicity we confine ourselves to the
$SU(2)$ case. Also, we will not include quarks.
%\subsection{Heavy external monopoles in non-Abelian theories}

On the classical level, theory of  non-Abelian monopoles turns to
be extremely simple as well (for review and further references
see, e.g. Refs.\,\protect\citebk{coleman}, \protect\citebk{alive}). 
First, there are no specific
non-Abelian classical solutions and all the monopoles are gauge
rotations of the Dirac monopoles with the corresponding magnetic
charge. Moreover, presence of the massless particles of spin 1,
gluons, affects the monopoles drastically. Indeed, the $U(1)$ charge of
the gluons is $e_{gl}=g$. Thus the minimal magnetic charge allowed by
the Dirac quantization condition for the gluons is
\beq
\label{minimal}
(g_m)_{min}~=~{2 \pi \over g}\;.
\eeq
The Wilson action however is formulated in terms of the adjoint
representation. For the quarks
the $U(1)$ charge is $e_{q}=g \slash 2$ and  the minimal magnetic charge is
twice as big as (\ref{minimal}). Thus, the action (\ref{wilson})
is not vanishing but, to the contrary, is equal to its maximal
possible value, $S_{plaq}^{string}=1/2g^2$. In the continuum
limit, the string energy per unit length is quadratically
divergent in the ultraviolet, $E_{string}\sim (L/\aaa^2)$.

As a result, the monopoles with the charge $Q_m=1$ are infinitely heavy.
Not only because of the energy of the radial field (see Eq.\,(\ref{linear})), but even more so because of the energy of the
Dirac string attached. Therefore the $Q_m=1$ monopoles can be introduced
only as external probes of
the vacuum of the non-Abelian theory (here $Q_m$ is the magnetic charge
in units of the minimal magnetic charge (\ref{minimal}))
Moreover, there is no space
for dynamical quarks since the Dirac string would be visible to
the quarks.

All other monopoles with other magnetic charges, $|Q_m|\neq 1$, are
unstable: charged gluons fall onto the center.

There is a remarkably (conceptionally) simple way to introduce the
external $Q_m=1$ monopoles on the lattice through the 't~Hooft
loop.\cite{loop} Namely, the monopoles are visualized as
end-points of the corresponding Dirac strings which in turn are
defined as piercing negative plaquettes. The trick to introduce
the negative plaquettes on the lattice is to formally change the
sign of the square of the coupling $g^2$ on a manifold of
plaquettes. Then these plaquettes become negative in the limit
$|g^2|\to 0$.

Proceeding to more detailed definitions, the 't~Hooft loop is
formulated\,\cite{susskind} in terms of the action
\beq
S(\beta,- \beta) =
\beta \sum\limits_{p\notin M} \;{\mathrm Tr}\;U_p
- \beta\sum\limits_{p\in M}\;{\mathrm Tr}\;U_p\,,
\label{S1}
\eeq
where $M$ is a manifold which is dual to the surface spanned on the monopole
world-line $j$.  Introducing the corresponding partition function,
$Z(\beta,-\beta)$ and considering a planar rectangular $T\times R$, $T\gg R$
contour $j$ one can define
\beq
V_{m\bar{m}}(R) ~\equiv ~ -\frac{1}{T} \ln{ Z(\beta,-\beta)\over Z(\beta,\beta)}\,.
\label{energy}
\eeq
By the analogy with expectation value of the Wilson loop
the quantity $V_{m\bar{m}}(R)$ is referred to as monopole-antimonopole,
or heavy monopole
potential.

\subsection{Heavy monopole potential at short distances}

There is no difficulty to predict the heavy monopole potential at
short distances. Indeed, as is mentioned above, all the
non-Abelian monopoles classically are in fact the same Abelian
monopoles. Therefore, classically the potential is:
\beq
\label{vmmm} V_{m\bar{m}}(r)~=~-{\pi \over g^2}{1\over r}\,.
\eeq
The classical expression is a reliable zero-order
approximation at short distances, $r\Lambda_{QCD}\ll 1$. Moreover,
the effect of the quantum correction is also fixed on general
grounds: the coupling $g^2$ is to be replaced by its  running
counterpart $g^2(r)$. Thus:
\beq
\label{vmm} \lim\limits_{r\to 0}{V_{m\bar{m}}(r)}~=~
-{\pi \over g^2(r)}{1\over r}\,.
\eeq
The result seems absolutely safe theoretically. Its actual derivation
is still full of paradoxes, for a review see Ref.\,\protect\citebk{alive}.

Equation (\ref{vmm}) reveals a double-face nature of the heavy, $g_m=1$
monopoles.  On one hand, they are Abelian objects as is testified to by
the overall coefficient in front of $1/r$. On the other hand, the full
non-Abelian gluon interactions is responsible for the renormalization.

In other words, Eq.\,(\ref{vmm}) poses a paradox which theory
should resolve since it concerns short distances
in QCD.\cite{main} Namely, the $1/r$ behavior implies one-gluon exchange
at short distances. Indeed, Eq.\,(\ref{vmm}) holds down to
academically small distances so that no language of ``effective infrared QCD''
can apply. On the other hand, one-gluon exchange
implies change of the monopole color. Which is not possible since
the monopoles are in fact Abelian-like and there are no monopoles belonging to a
representation of $SU(2)$.

\section{Symmetries in the presence of magnetic charges}

\subsection{Short distances: $U(1)_{el}\times U(1)_{magn}$, $SU(2)_{color}\times U(1)_{magn}$}


The paradox would look absolutely unresolvable if it were not so that
there existed experience of dealing with somewhat similar paradoxes in
case of QED. Indeed, already in the QED case once we introduce both
electric and magnetic charges
there are in fact two kinds of charges which cannot annihilate each other.
Which means that we do not have simply a $U(1)$ symmetry but a product of 
two $U(1)^{'}$s:
\beq
U(1)~\rightarrow~U(1)_{el}\times U(1)_{magn}
\eeq
instead. Moreover, both types of charges are clearly separated only as far as
they are at rest. Once there is relative motion, the magnetic and electric
charges interact according to the laws of the classical electrodynamics. In the quantum
field theory language, the interaction is due
to an exchange of one and the same gauge boson,
that is photon.

Generically, a dual gauge boson is understood as a field interacting with the
magnetic current $j_m$:
  \beq
L_{int}~=~Q_m B\cdot j_m~~.
\eeq
One of our basic points is that the field $B$ should be treated as
fundamental in gluodynamics since the external monopoles introduced via the
't~Hooft loop are point like in the continuum limit, see above.  A well known example
of a theoretical framework
to introduce a dual gauge boson is the Zwanziger Lagrangian describing
the $U(1)_e\times U(1)_{magn}$ electrodynamics.\cite{zwanziger}  Formally one introduces two vector fields, the
``standard'' photon $A_{\mu}$ and the dual photon $B_{\mu}$, so that
$L_{int}=Q_eA\cdot j_e+Q_m B\cdot j_m$.  However, the number of the degrees
of freedom is not changed since there is a constraint that the field
strength tensor constructed on the potential $A$ coincides with the dual
field strength tensor constructed on the potential $B_{\mu}$.  More
precisely:
\beq\label{zwanziger}
m_{\mu}F_{\mu\nu}(B)~=~m_{\mu}\dual{F_{\mu\nu}(A)}~~
\eeq
where $m_{\mu}$ is an arbitrary space-like vector. The choice of the vector
$m_{\mu}$ is a kind of new gauge freedom. Physically,
$m_{\mu}$ is the vector directed along the Dirac strings.
Explicitly:
\bea
L_{Zw}(A,B) & = & 
\frac{1}{2}\left(m\cdot[\diff\wedge A]\right)^2 + \frac{1}{2}\left(m\cdot[\diff\wedge
B]\right)^2 \nonumber\\
& & 
+ \frac{i}{2}\left(m\cdot[\diff\wedge A])(m\cdot\dual{[\diff\wedge B]}\right) \label{Zw-action}\\
& &  - \frac{i}{2}\left(m\cdot[\diff\wedge B])(m\cdot\dual{[\diff\wedge
A]}\right) + i\,j_e\cdot A + i\,j_m\cdot B\,, \nonumber
\eea
where $j_e$, $j_m$ are electric and magnetic
currents, respectively, $m_{\mu}$ is a constant vector, $m^2=1$ and
$$
[A\wedge B]_{\mu\nu} = A_\mu B_\nu - A_\nu B_\mu\,, \qquad
(m \cdot [A\wedge B])_\mu = m_\nu [A\wedge B]_{\mu\nu}\,,
$$
$$
\dual{[A\wedge B]}_{\mu\nu} ~=~ \frac{1}{2}\,\varepsilon_{\mu\nu\lambda\rho}\,
[A\wedge B]_{\lambda\rho} .
$$

Similarly, upon introduction of the (heavy) monopoles the symmetry
of the gluodynamics becomes:\,\cite{main}
\beq
SU(2)~\rightarrow~SU(2)_{color}\times U(1)_{magn} 
\eeq 
where the
corresponding groups are used for classification of the charges,
non-Abelian and magnetic, respectively. The number of gauge fields
(gluons) is of course not increased. The way out of this apparent
paradox is  a generalization of the constraint (\ref{zwanziger}):
\beq\label{zwanziger1}
m_{\mu}F_{\mu\nu}(B)~=~m_{\mu}\dual{(n^aF^a_{\mu\nu}(A))} \eeq
where $F_{\mu\nu}^a$ is the non-Abelian field strength tensor and
$n^a$ is an arbitrary vector in the color space. Again, the choice
of $n^a$ is a matter of gauge fixing and the physical results do
not depend on this choice.  The vector $n^a$ fixes
the direction in the color space of the magnetic fields
transported along the Dirac strings attached to the monopoles.

One can derive a Zwanziger-type Lagrangian which ensures the validity of the
constraint (\ref{zwanziger1}).\cite{main} This
formulation allows then to make predictions for the heavy monopole potential
(\ref{vmm}). In particular, at short distances the potential can be derived
now from the first principles. However, there is a specific problem
that standard Feynman rules do not respect the Dirac veto for virtual particles
and they interact with the string. Thus, special rules of UV regularization are
to be introduce to remove these effects.\cite{main}

To summarize, introduction of the monopoles extends the
classification group to $SU(2)_{color}\times U(1)_{magn}$.
The choice of a particular gauge boson which is shared between
the groups is a matter of gauge fixing which in a way violates the $SU(2)$.

\subsection{Spontaneous breaking of the magnetic $U(1)$}

 At larger distances, there are further points related
to the symmetry of problem which should be discussed. Let us
begin with the fundamental (point-like) monopoles introduced  via
the 't Hooft loop. The monopoles  are in fact $Z_2$ monopoles, so
that two monopoles of ``the same charge'' can annihilate each
other. Classically, the $Z_2$ monopoles interact still as $U(1)$
monopoles.\cite{main} If one starts from short distances then the
first sign of the $Z_2$ nature of the monopoles is a possible
linear correction to the monopole potential due to a quantum
transition between the states with magnetic charge $Q_M=0$ (the
ground state) and $Q_M=2$ (an excited state):\,\cite{mm}
\beq
\delta V_{\bar{m}m}~\sim M_{0,2}\,{1\over E_0-E_2}\,M_{2,0}~\sim ~g^2 r\,.
\eeq
Unfortunately, little can be said about the corresponding
matrix elements $M_{0,2}$. At larger distances,
the differences between $Z_2$ and $U(1)$ monopoles is even more
open question. We will ignore this problem and stick to the $U(1)$
description of the monopoles.

At large distances it is also natural to expect that the monopoles condense and
 the magnetic $U(1)$ is spontaneously broken.  Note that
no breaking of the (global) $SU(2)$
is introduced at this point as far as the group is indeed $SU(2)_{color}\times
U(1)_{magn}$. Moreover since the $U(1)$ boson
is actually one of the same $SU(2)$ gluons
which interact with quarks,
acquiring the mass for the $U(1)$ boson implies
confinement of the quarks. In the sense,
that forming a tube-like structure for
the quarks  field is the only way to satisfy the Gauss law.

On the theoretical side, the symmetries of the Lagrangian are becoming
difficult to implement in practical calculations
once the spontaneous breaking of the magnetic
$U(1)$ is introduced.
The problems can be illustrated on the example
of the Zwanziger Lagrangian (\ref{zwanziger}) for ordinary QED.
 Indeed, let us assume that a
charged scalar field acquires a nonvanishing vacuum expectation value
or, even simpler, a mass term ${m_V^2\over 2}A_{\mu}^2$ is added to
the Lagrangian.  Then a straightforward diagonalization of the
bilinear terms in the Lagrangian results in the following propagators
of $A$- and $B$-fields (see, e.g., Ref.\,\protect\citebk{balachandran}):
\bea
\label{bb}
\langle B_{\mu} B_{\nu}\rangle (k) &=& {1\over k^2+m_V^2}\left(\delta_{\mu\nu}+ {m_V^2\over
(km)^2}(\delta_{\mu\nu}-m_{\mu}m_{\nu})+ \cdots \right), \\
\langle A_{\mu}A_{\nu}\rangle (k) &=& {1\over k^2+m_V^2}\left(\delta_{\mu\nu}+ \cdots \right)
\eea
where the dots stand for terms proportional to $k_{\mu}$ and which
can eventually be omitted because of the current conservation.
If we evaluate the interaction energy of a monopole
pair due to the (massive) photon exchange then the
$m_{\mu}$ dependence does not drop off. Also, the double pole in
$(km)$ causes infrared problems.

These inconsistencies are not a consequences of a trivial mistake but
deeply rooted in the formalism. The point is that the independence
on the Dirac strings assumes that the Dirac veto is observed.
However, if the charges are condensed, then the strings are ``everywhere''
and this results in the inconsistencies of the propagator (\ref{bb}).
The difficulty was resolved only recently\,\cite{gubarev}
and only in case of static sources (or $k_0=0$).



\section{Monopoles, as they are seen}

\subsection{Monopole condensation in non-Abelian case: expectations}

If we try to adjust the lessons from the compact $U(1)$
to the non-Abelian case we run into painful questions.
The only good news, to begin with, is that all the $U(1)$ subgroups
of $SU(2)$ are indeed compact, this is no problem.

Let us therefore try to work out a simple dynamical picture
for monopole condensation. Dynamics of any subgroup of the $SU(2)$
is governed by the same running coupling $g^2(r)$.
Then -- we would conclude -- starting from small size lattices
we would not see monopoles because $g^2(\aaa)$
falls below $e^2_{crit}$. However, we could hope that going to a
coarser lattice a la Wilson we could come to the point
where $g^2(\aaa^2)\approx e^2_{crit}$. Then one could hope
to apply the entropy-energy balance which works so well in case
of the compact $U(1)$.


The first shock comes if we try to estimate, what the ``small
$\aaa$''means numerically, in the present context. Now, the $U(1)$
critical coupling is well known, $e^2_{crit}\sim 1$. In the QCD case we can
rewrite the condition (\ref{critical}) as a condition on the critical scale.
In the realistic case
we have at the LEP energies $E^2 \sim (100~\mbox{GeV})^2$, $\alpha \approx
0.1$, $g\approx 1$. Then
\beq\label{huge}
M_{crit}~\sim~100~\mbox{GeV}
\eeq
and, remarkably enough,
we are getting rather weak interactions scale than $\sim\Lambda_{QCD}$.
On the other hand the
lattice data give another estimation of  $M_{crit}$. It is well
known that in $SU(3)$ gluodynamics $\beta=6$ corresponds to the lattice
spacing $a\approx 0.1$ fm and the scale is:
\beq\label{nothuge}
M_{crit}~\sim~2~\mbox{GeV}\,.
\eeq

In other words, according to the first estimate, for any presently
available lattice we are deep in the ``strong-coupling'' region despite of
the asymptotic freedom.  Then, naively, we could expect that
approximating the total energy by self-energy does not work, the
monopole population is very dense so that the interaction energy
is comparable to the self energy. Moreover, the scale $100~\mbox{GeV}$
looks rather as a headache than an enlightenment since there is no
independent evidence for relevance of such a mass scale to QCD.

This formidable perspective is balanced only by the doubts that
the monopoles are at all relevant. The point is that intrinsically
the monopoles are a $U(1)$ object and there is no unique way to
choose the $U(1)$ subgroup from the $SU(2)$.

Mostly, monopoles are defined in the non-Abelian case as
pure topological objects,\cite{thooft} with no direct relation to the
full non-Abelian action. Moreover, it appears  obvious that,
no matter which $U(1)$ we choose, the minimal action for such
a configuration should collapse to zero. At least, on the classical level so.

There is no general proof of this conjecture but as a support for
 this let us quote the solution for an open Dirac string which costs
no non-Abelian action at all.\cite{main}
One can show\,\cite{main}  such a configuration is generated
from the vacuum
by the following gauge rotation matrix:
\beq
\label{example}
\Omega~=~\left(\matrix{
{\rm e}^{i\varphi}\sqrt{A_D} & \sqrt{1-A_D}\cr &\cr 
-\sqrt{1-A_D}          &  e^{-i\varphi}\sqrt{A_D}\cr
}\right)\,,
\eeq
where $\varphi$ is the angle of rotation around the axis connecting the
monopoles and $A_D$ is the $U(1)$ potential representing pure Abelian
monopole -- antimonopole pair:
\beq
A_{\mu}dx_{\mu}~=~
{1\over 2}\left({z_+\over r_+}-{z_-\over r_-}\right)d\varphi
~\equiv~ A_D(z,\rho)d\varphi\,,
\eeq
where $z_{\pm}=z\pm R/2$, $\rho^2=x^2+y^2$, $r_{\pm}^2=z_{\pm}^2+\rho^2$.
Note that the action associated with the Dirac string is considered in this
case zero, in accordance with  the lattice version of the theory
(for details see Ref.\,\protect\citebk{main}).

To summarize, we are failing to work out a simple dynamical picture
mainly because we cannot specify reasonable monopole-like configurations.
Our problem is that, first, the action  is too low to match $\ln7$.
To fight this, we would rather argue that there is no such object as point-like
 monopole at all so that the entropy is counted also wrong.
 But then nothing is left of the idea of the monopole condensation
in the non-Abelian case.

\subsection{Monopole dominance}

On the background of the theoretical turmoil, the data
on the monopoles indicate  a very simple and solid picture.
We will constrain ourselves to the monopoles in the so
called Maximal Abelian gauge and the related projection
(MAP). We just mention some facts, a review and further
references can be found, e.g., in Ref.\,\protect\citebk{review}.

 Since the monopoles of the non-Abelian theory are expected
to actually be $U(1)$ objects one first uses the gauge freedom
to bring the non-Abelian fields as close to the Abelian
ones as possible. The gauge is defined by maximization of a
functional which in the continuum limit corresponds to $R(\hat{A})$ where
\beq
R(\hat{A})~=~-\int d^4x\big[(A_{\mu}^1)^2+ (A_{\mu}^2)^2\big]
\eeq
where $1,2$ are color indices.

As the next step, one projects the  non-Abelian fields generated on the lattice
into their Abelian part, essentially, by putting $A^{1,2}\equiv 0$.
In this Abelian projection one defines the monopole currents
$k_{\mu}$ for each field configuration.

The relation of the monopoles to the confinement is revealed
through evaluation of the Wilson loop for the quarks in the
fundamental representation. Namely it turns out, first, that the
string tension in the Abelian projection is close to the string
tension in the original $SU(2)$ theory:\,\cite{mondom}
\beq
\sigma_{U(1)}~\approx~ \sigma_{SU(2)}\,.
\eeq
Moreover, one can
define also the string tension which arises due to the monopoles
alone. To this end, one calculates the field created by a monopole
current:
\beq
\label{laplasian}
A_{\mu}^{mon}(x) = \frac{1}{2} \varepsilon_{\mu\nu\alpha\beta}
\sum\limits_y \Delta^{-1}(x-y) \, \partial_\nu  m_{\alpha\beta}[y;k]\,,
\eeq
where $\Delta^{-1}$ is the inverse Laplacian, and sums up (numerically)
over the Dirac surface, $m[k]$, spanned on the monopole
currents $k$. The resulting string tension is again close to
that of the un-projected theory:
\beq
\sigma_{mon}~\approx~ \sigma_{SU(2)}\,.
\eeq

It might worth mentioning that these basic features remain also true
upon inclusion of the dynamical fermions in $SU(3)$ case (full lattice
QCD).\cite{latest}

\subsection{Invariant properties of the monopoles.}

Despite of the apparent gauge-dependence of the monopoles
introduced within the MAP, they encode gauge-invariant information.
In particular,we would mention two points: scaling of the monopole
density and full non-Abelian action associated with the monopoles.

According to the measurements (see Ref.\,\protect\citebk{bornyakov2} and 
references therein)
the mo\-no\-po\-le density $\rho_{mon}$ in three-dimensional volume (that is, at
any given time) is expressible in the physical units.
In other words, the density scales according to the
renormgroup as a quantity of dimension 3. Numerically:
\beq
\label{scaling}
\rho_{mon}~=~ 0.65(2) ~(\sigma_{SU(2)})^{3/2}\,.
\eeq
One important remark is in order here. While discussing the
monopole density one should
distinguish between what is sometimes called ultraviolet (UV)  and
infrared (IR) clusters.\cite{teper} The infrared,
or percolating  cluster fills in the whole lattice while the UV clusters
are short. There is a spectrum of the UV clusters, as a function of
their length, while the percolating cluster is in a single copy.
The statement on the scaling (\ref{scaling})
refer only to the IR cluster. The UV clusters are seemingly a
lattice artifact.

Also, upon identification of the monopoles in the Abelian
projection, one can measure the non-Abelian action associated with
these monopoles. For practical reasons, the measurements refer to
the plaquettes closest to the center of the cube containing the
monopole. Since the self energy is UV divergent, it might be a
reasonable approximation. The importance of such measurements is
that we expect that it is the non-Abelian action which enters the
energy-entropy balance for the monopoles.

The results of one of the latest measurements of this type
are reproduced in Fig.\,1 (see Ref.\,\protect\citebk{bornyakov3}).
What is plotted here is the average excess of the action on
the plaquettes closest to the monopole (monopoles
are positioned at the centers of the cubes).
The action is the lattice units. In other words, the
straightforward continuum would imply divergent mass of the
monopole of order $1/\aaa$ if the action of order unit.
\begin{figure}[h]
\epsfxsize=7.5cm
\centerline{\epsfbox{nabel1.eps}}
\caption{The dependence of the average excess of the action on
the plaquettes closest to the monopole, as a function of $\beta$.}
\label{nonabeact}
\end{figure}

The conclusions which can be drawn\,\cite{bornyakov3}
from the measurements are apparently as follows:

i) The IR and UV monopoles are distinguishable through their
non-Abelian actions. For the UV monopoles the action is larger, in
accordance with the fact that they do not percolate (condense).

ii) The excess of the action for the monopoles falls close
to $\ln7$,
\beq
S_{mon}~\sim~\ln7 \cdot {L\over \aaa}\,,
\eeq
quite dramatically confirming
in this way the reality of the (Abelian-like) energy-entropy
balance. Actually evidence to this effect, less direct,
was obtained already much earlier, see.\cite{action}

iii) We are talking actually about small distances, by all the
standards of QCD. Thus, the action for the IR and UV monopoles are
definitely different only at the monopole sizes (radii) about
\beq
R_{mon}~\approx~0.06~\mbox{fm}\,,
\label{Rmon}
\eeq
And even at smallest available
distances the monopoles are still very ``hot''.
The corresponding monopole mass (see Fig.\,1) is, roughly, proportional to $1/a
\sim 2~\mbox{GeV}$. We cannot exclude the case $M_{mon}\sim 1/a $ when we
approach the continuum limit.
The inside  region of the monopoles where the
asymptotic freedom sets in has not been reached yet even at $\aaa=0.04fm$.
Which indicates that the surprising estimate (\ref{huge})
might indeed be relevant.

\section{Interpretation and phenomenology}

\subsection{Monopole cluster as a polymer}

After learning that the reality of the monopole condensation looks
much simpler than our confused expectations we can try to adjust
our views.  Let us therefore try to summarize, what we have
actually learned.

First, our fears that the monopole action would fall too much
below the entropy-related ln7 are not realized. To the contrary,
at least the first look at Fig.\,1  rather suggests  that the
action is too high. Moreover it has been known from analysis of
the monopoles in the Abelian projection that the entropy-energy
balance is similar in the non-Abelian and Abelian cases.\cite{action} 
Which, reinforced now with  the data in Fig.\,1,
means that in the zero approximation $
S_{mon}^{phen}~\approx~\ln7\cdot{L\over\aaa}$.
Which means, in
turn, that the $SU(2)$ monopole action is higher than its Abelian
counterpart. Indeed, phenomenological fits suggest:\,\cite{action}
\beq
\label{mass}
M_{mon}~\approx~M_{mon}^{Coul}+ \mathrm{const}\,,
\eeq
where
by $M_{mon}$ we understand the action associated with the monopole
trajectory of length $L$ divided by $\aaa$. Note also that the
Coulombic part of the mass, $M_{mon}^{Coul}$ is of order $1/\aaa$.

As it follows from the discussion in section 1,
the mass (\ref{mass}) is not so much meaningful
and one is to consider the difference between the mass
and $\ln7/\aaa$. The  existing data are not precise enough
to provide this difference directly. However, it seems natural to
speculate that
\beq
\label{assumption}
-\mu~=~{\ln7\over \aaa}-M_{mon}~\sim ~\Lambda_{QCD}\,.
\eeq
It is a pure assumption but otherwise
it would be difficult to understand the scaling of the
monopole density. Indeed, if we view the monopole trajectory as
a polymer, then there are essentially two entries to describe monopoles,
that is the  chemical potential and the interaction which
is presumably Coulomb-like:
\beq\label{nou1}
S~=~L\mu + g^2_{m}\, {\sum_{a,b}}' {\aaa^2\over(r_a-r_b)^2}\,,
\eeq
where the primed sum,  $\Sigma^{'}_{a,b}$, does not include the self-energy.
Thus, if $1/\aaa$ survives in $\mu$
it would be very difficult to understand the scaling of the $\rho_{mon}$.



The central point about the polymer action (\ref{nou1}) is that it
is written in $SU(2)$ invariant terms alone and, if we can indeed go ahead with
such a formulation, we are getting closer
to the ideal  of describing the magnetic monopoles
within $SU(2)_{color}\times U(1)_{magn}$ (see Section 2.2).

\subsection{Monopole cluster in the field-theoretical language}

The appearance of large masses in the polymer language, $M_{crit} \sim
1/a$ might look quite scary. Moreover, there is no indication to the
relevance of such mass to other observables in QCD.  Thus, let us
address this issue in more detail.  To this end, it is desirable to
develop a field theoretic language within which the masses can be
interpreted more directly.  As is well known, the condensation of the
monopoles can be described in terms of a (dual) Abelian Higgs model (for
review and references see Ref.\,\protect\citebk{review}).  However, usually 
this model
is formulated within an Abelian-projected action. Now, we would like to
continue with the $SU(2)$ invariant description (\ref{nou1}).


The transition from the polymer to the field theoretical language
is common in the statistical physics (see, 
e.g., Ref.\,\protect\citebk{parisi}).
The first applications to the monopole physics are due to the
authors in Ref.\,\protect\citebk{stone}. Here we, again, emphasize only
a few points.


The monopole trajectory represented as a random walk and the
corresponding partition function is:
\bea
Z = \int \dd^4 \, x \, \sum^\infty_{N=1} \frac{1}{N} \, e^{ - \mu N}
\, Z_N(x,x)\,,
\label{Z}
\eea
where $\mu$ is the chemical
potential and $Z_N(x_0,x_f)$ is the partition function of a polymer
broken into $N$ segments:
\beq
Z_N(x_0,x_f) = \Bigl[\prod\limits^{N-1}_{i=1} \int \dd^4 x_i\Bigr] \,
\prod^{N}_{i=1} \Biggl[\frac{\delta(|x_i - x_{i-1}|-a)}{2\pi^2
a^3}\Biggr]\, \exp\Bigl\{ - \sum\limits^N_{i=1} g
V(x_i)\Bigr\}\,.
\label{ZN}
\eeq
This partition function represent a summation over all atoms of
the polymer weighted by the Boltzmann factors.
The $\delta$--functions in (\ref{ZN}) ensure that each bond in
the polymer has length $a$. The starting point of the polymer
(\ref{ZN})is $x_0$ and the ending point is $x_f \equiv x_N$.

In the limit $a \to 0$ the partition function (\ref{ZN})
can be treated analogously to a Feynman integral.
The crucial step is the a coarse--graining: the $N$--sized polymer
is divided into $m$ units by $n$ atoms ($N = mn$), and the limit is considered
when both $m$ and $n$ are large while $a$ and $\sqrt{n} a$ are
small. We get,
\beq
\label{constraint}
\prod\limits^{(\nu+1)n-1}_{i=\nu n} {1\over
2\pi^2\aaa^3}\delta(|x_i-x_{i+1}|-\aaa) \to
{\Bigl(\frac{2}{\pi n a^2}\Bigr)}^2
\,\exp\Bigl\{ - \frac{2}{n \, a^2} {(x_{(\nu + 1) n} - x_{\nu n
})}^2 \Bigr\} \,,
\eeq
where the index $i$, $i=\nu n \cdots (\nu+1)n-1$, labels the atoms in
$\nu^{\mathrm{th}}$ unit. The polymer partition function
becomes:\,\cite{stone}
\bea
\label{polymer}
Z_N(x_0,x_f) & = & {\mathrm{const}} \cdot
\left[
%\sum^\infty_{N=1}
\prod^{m-1}_{\nu =1} \dd^4 x \right] \Biggl[
{\Bigl(\frac{2}{\pi n a^2}\Bigr)}^{2 m} \exp\Bigl\{
\sum_{\nu=1}^m{(x_{\nu}-x_{\nu-1})^2\over n\aaa^2}\Bigr\}\Biggr]
\nonumber \\
& & \cdot \exp\Bigl\{ - \sum_{\nu=1}^m n (\mu + V(x_\nu)) \Bigr\}\,.
\eea
The $x_i$'s have been re-labeled so that $x_{\nu}$ is the average
value of $x$ in at the coarser cell. Using the variables:
\beq
\label{related}
s={1\over 8}\,n\aaa^2\nu\,,\quad \tau={1\over 8}\aaa^2\,N\,, \quad m_0^2=
{8\mu \over \aaa^2}\,,
\eeq
one can rewrite the partition function (\ref{Z}) as
\beq
\label{m0}
Z = {\mathrm{const}}\cdot \!\int\limits_0^{\infty}
\frac{\dd\tau}{\tau} \! \!\!\int\limits_{x(0) = x(\tau)=x}\!\!\!\!\!\!\!\!\!
D x\, \exp\Biggl\{-\int\limits_0^\tau \Bigr[{1\over 4}\dot{x}^2_\mu(s)
+ m_0^2 + g_0 V(x(s))
\Bigr] \, \dd s \Biggr\}\,.
\eeq
The next step is to rewrite the integral over trajectories $x(\tau)$ as
the standard path integral representation for a free scalar field.
For us it is important only that the $m_0^2$
term in the Eq.\,(\ref{m0}) is becoming the standard mass term
in the field theoretical language:
%\bea
%\cZ & = & \sum\limits_{M=0}^\infty \frac{1}{M!} Z^M \nonumber\\
%& = & {\mathrm{const}}\cdot  \int D \phi \, \exp\Bigl\{
%- \int \dd^4 x \, \Bigl[ (\partial_\mu \phi)^2 + m^2_0 \, \phi^2
%+ g_0 V(x) \phi^2 \Bigr] \Bigr\}\,.
%\label{Zphi}
%\eea
\beq
\cZ = \sum\limits_{M=0}^\infty \frac{Z^M}{M!}
= {\mathrm{const}}\cdot \! \int\!\! D \phi \, \exp\Biggl\{\!
-\!\! \int\!\! \dd^4 x \, \Bigl[ (\partial_\mu \phi)^2 + m^2_0 \, \phi^2
+ g_0 V(x) \phi^2 \Bigr] \Biggr\}.
\label{Zphi}
\eeq
The whole machinery can be easily generalized to the case of charged
particles (monopoles) with Coulomb-like interactions. Below we
assume that
we are in the Higgs phase, or that the chemical potential
is negative, $\mu~<0~$,
and the vacuum is stabilized through a repulsive
$\lambda\phi^4$ interaction.\cite{stone}

Here, it is the relation between $\rho_{mon}$ and
$\langle \phi^2\rangle$ which is most important for us.
To derive the relation  differentiate first
the partition function in the polymer
representation:
\beq
\label{l}
\langle L \rangle = \frac{\partial}{\partial \mu} \ln \cZ\,.
\eeq
Since  the density $\rho_{mon}$ scales:
\beq
\langle L \rangle = \rho_{mon}\cdot V_4\,,
\eeq
where $V_4$ is the 4-volume occupied by the lattice.
On the other hand, differentiating the same partition function but
in the field theoretical representation (\ref{m0}) with respect to
$m_0^2$ we get the vacuum condensate:
\beq\label{phi}
\langle \phi^2\rangle = \frac{\partial}{\partial m^2_0} \ln \cZ\,.
\eeq
It is worth emphasizing that in the both cases (\ref{l}) and (\ref{phi})
we keep only the contribution of the IR monopole cluster corresponding
to the condensing Higgs field in the field-theoretic language.

Finally, since the parameters $\mu$ and $m_0^2$ are directly related, see
Eq.\,(\ref{related}), we get:
\beq\label{mainn}
\langle \phi^2\rangle~=~{1\over 8}\,\rho_{mon}\cdot \aaa~,
\eeq
which is one of our main results\,\footnote{\,Let us remind the reader
that by $\rho_{mon}$ and $\langle\phi^2\rangle$ we understand
in fact the contributions of the IR cluster to these quantities. The
v.e.v.\ $\langle \phi^2\rangle$ contains the quadratic divergent piece,
$a^{-2}$, due to a perturbative contribution. On the other hand the
contribution to the total monopole density from the UV--clusters which
should diverge as $a^{-3}$ according to dimensionality arguments. These
divergences match each other in Eq.\,(\ref{mainn}).}.
Note that,
up to an overall numerical factor,
Eq.\,(\ref{mainn}) is quite obvious on the dimensional grounds.

\subsection{Naive limit $\aaa\to 0$}

Thus, let us assume that the scaling of the monopole density $\rho_{mon}$
continues to be true for smaller lattice spacings as well,
at least until we reach the mass scale sensitive to
the non-local structure of the monopoles,
see discussion above. Then we have the following simple picture:
\beq\label{a.f.}
\lim\limits_{\aaa\to 0}{m_0^2}\sim {\mu\over \aaa}\rar\infty\,,\quad
\lim\limits_{\aaa\to 0}{\langle \phi^2\rangle}\sim\rho_{mon}\,\aaa~\rar~0\,,
\quad
\lim\limits_{\aaa\to 0}{m_V^2}\sim g^{-2}\rho_{mon}\,\aaa\rar 0\,.
\eeq
It is worth emphasizing that the masses we are discussing here are
gauge invariant since we started from the
non-Abelian action per unit length. And we see
that existence of the huge mass scale (\ref{huge}) might in fact be in
no contradiction with the asymptotic freedom. Indeed, only the
chemical potential has physical meaning
and the scaling of the $\rho_{mag}$ indicates that
it is of order $\Lambda_{QCD}$. Moreover,
the effect of the condensate on the gluon mass goes away as a power of $\aaa$.

It is worth emphasizing that Eq.\,(\ref{a.f.}) implies that
\beq
\lim\limits_{\aaa\to 0}{m_0^2\cdot \langle \phi^2\rangle}~\sim~\mbox{const}\,.
\eeq
In other words, the potential energy behaves smoothly as $\aaa\to 0$.
And this is, in fact, the most adequate formulation of the
emerging picture. It was possible to find the $\aaa$-dependence for $m_0^2$
and  $\langle \phi^2\rangle$ separately only because of normalizing
the kinetic energy to unit, as usual.

Note that the scaling laws (\ref{a.f.}) are still consistent with
$\rho_{mon}=\mathrm{const}$. Moreover, this seems to be sufficient
to ensure the monopole dominance and
\beq
\lim\limits_{\aaa\to 0}{\sigma_{mon}}~\sim ~\mbox{const}\,,
\eeq
where the monopole string tension is calculated with the use
of Eq.\,(\ref{laplasian}). Which means in turn that the parameters
used to describe the structure of the string within the Abelian
projection can be stable in the limit $\aaa~\to ~0$.
Moreover, say,
\beq
\lim_{\aaa\to 0} {(m_V^2)_{Ab. proj.}}~\sim~ \mbox{const}\,,
\eeq
is in no  direct contradiction with (\ref{a.f.}) since the masses
determined in terms of the Abelian-projected action are not directly
related to the masses (\ref{a.f.}) determined
in terms of the non-Abelian action.

Thus, the picture which emerges if we start with assumption (\ref{assumption})
has some attractive features. However the connection of the string tension with
the classical solutions is lost. Moreover, the data in Fig.\,1
suggest that the assumption (\ref{assumption})
is not valid at $a$ larger than, say, $a_{crit}\sim 0.15~{\mathrm{fm}}$.
One might hope for a matching at these distances of the polymer picture
with a negative chemical potential $\mu$
and of a more phenomenological approach of Refs.\,\protect\citebk{action}.

Although little can be said at this time about the $\lambda\phi^4$--type
theory corresponding to the monopole ``polymer'',
the whole issue seems to be very interesting.
Indeed, it is well known that it is very difficulty, if not impossible,
to construct a non-trivial $\lambda\phi^4$ theory and
the polymer regularization is commonly discussed in this 
connection.\cite{caracciolo} The scaling law (\ref{scaling})
implies that the QCD, when projected onto the properties of the
monopole polymer, corresponds to a
non-trivial $\lambda\phi^4$--type theory.


\subsection{Phenomenology}

It is quite clear that the theory of the monopole condensation is
far from being complete and numerical predictions are difficult
to make. Indeed, we discussed mostly symmetry properties.
Nevertheless, let us try to formulate some consequences
grouping them into three categories:

{(\it i}) We speculated that the polymer formulation (\ref{assumption})
encodes the gauge invariant information on the monopoles in the most direct
way. Therefore, studies of the monopole trajectories (IR cluster)
might be most important. One expects:

a) The monopole trajectories are random walk for any $\aaa$
in the sense that there is no correlation between the vectors
tangent to the monopole trajectory;

b) monopole density scales, $\rho_{mon}=\mathrm{const}$ and is independent
of $\eps$ at least as far as the monopole action exceeds the average
in the lattice units (and not in $\Lambda_{QCD}^4$);

c) as is known (see, e.g., Ref.\,\protect\citebk{intersect})
the monopole trajectories intersect.
It is natural to speculate that the distance between the self-intersections
also scales, reflecting the scaling of the potential energy;

d) the intersections correspond in the field theoretical language to the
$\lambda\phi^4$ interaction:
\beq
V(\phi)~=~-m_0^2\phi^2 + \lambda\, \phi^4
\eeq
As we argued, one expects that the potential
energy is $\aaa$-independent. This would imply that the effective scalar mass
defined in terms of the second derivative of the potential at
the minimum is also $\aaa$-independent. Which could be checked through
measurements.


{(\it ii}) We argued that the actual symmetry, in the
presence of the magnetic charges,
is $SU(2)_{color}\times U(1)_{magn}$.
However, direct use of the Zwanziger-type Lagrangian
(see Eq.\,(\ref{zwanziger1})) is not possible already
in the $U(1)$ case, see subsection 2.2. Thus, we can
rely on the symmetry considerations alone. Then:

a) effective Higgs-like theories introduce dimension d=2
condensate. The resulting leading ``effective'' correction
to the original Lagrangian which is consistent
with the symmetry, is the gluon mass. Moreover,
generation of the effective mass is a prerequisite of the confinement.
There exist independent evidence supporting introduction
of the corresponding power correction.\cite{quadratic}

b) Explanation of the Casimir scaling is a challenge
to any phenomenological model
of the confinement.\cite{debbio}
Within our approach, we cannot directly evaluate the string tension.
Indeed, even for the fundamental monopoles at short distances
one should account for the full non-Abelian interaction to account
for the running of the coupling. Generally speaking, the non-Abelian
nature should be even more manifest in large-distance interaction.
However, there is a phenomenological observation that the string tension
for quarks in the fundamental representation one can use
pure Abelian interaction (\ref{laplasian}). It is quite natural to assume
that the same will be true for quarks with the highest value of
$T_3$.\,\cite{poulis} Because of the intrinsic $SU(2)$ invariance it
should be then true for the invariant string tension $\sigma_{SU(2)}$.
Phenomenologically one can explain existing data on the Casimir scaling
along these lines.\cite{poulis}


{(\it iii)} The picture of the monopoles dominated vacuum
suggests the so called two-step QCD (see the last 
paper\,\cite{quadratic} and references therein).
Namely, even for non-perturbative fluctuations we may
have rather hard fields which, however, are not related
directly to the confinement and/or resonance properties.
In particular, the condensate $\langle G^2\rangle_{non\mbox{-}pert}$ receives the following
contribution from the monopoles:
\beq\label{div}
\langle G^2 \rangle_{mon}~\sim~{\rho_{mon}\over \aaa}
\eeq
where $a$ is the lattice spacing and one can use the estimate (\ref{div})
as long as the inside region of the monopole is not reached.

\section*{Acknowledgments}
\addcontentsline{toc}{section}{\numberline{}Acknowledgments}

One of the authors (V.Z.) acknowledges with great gratitude deep
friendly relations with M.S.~Marinov which lasted more than 40 years.
During all these years M.S.~Marinov was a high moral authority for him.
It is an honorable and sad event, to contribute to this volume.

The authors are grateful to S.~Caracciolo and T.~Suzuki  for discussions.
One of the authors (V.Z.) would like to thank Prof.\ R.~Barbieri
for the hospitality during his visit to the Scuola Normale
Superiore at Pisa. M.N.~Ch. is supported by the JSPS Fellowship P01023.

%\newpage
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