Brane Condensation and Confinement
We study the static quantum potential for a theory of anti-symmetric tensor fields that results from the condensation of topological defects, within the framework of the gauge-invariant but path-dependent variables formalism. Our calculations show that the interaction energy is the sum of a Yukawa and a linear potentials, leading to the confinement of static probe charges.
hep-th0411236.tex
\documentstyle[preprint,aps]{revtex}
\preprint{USM-TH-162}
\begin{document}
\title{Brane Condensation and Confinement}
\author{P. Gaete$^{1}$\thanks{
E-mail: patricio.gaete@usm.cl} and C. Wotzasek $^{2}$\thanks{
E-mail: clovis@if.ufrj.br}}
\address{$^1$Departamento de F\'{\i}sica, Universidad T\'{e}cnica
F. Santa Mar\'{\i}a, Casilla 110-V, Valpara\'{\i}so, Chile. \\
$^2$Instituto de F\'{\i}sica, Universidade Federal do Rio de
Janeiro, 21945, Rio de Janeiro, Brazil.} \maketitle

\begin{abstract}
We study the static quantum potential for a theory of
anti-symmetric tensor fields that results from the condensation of
topological defects, within the framework of the gauge-invariant
but path-dependent variables formalism. Our calculations show that
the interaction energy is the sum of a Yukawa and a linear
potentials, leading to the confinement of static probe charges.
\end{abstract}
\smallskip

\section{INTRODUCTION}

One of the fundamental issues of theoretical physics
%and long-standing whose solution has evaded complete comprehension despite an intense effort for already many decades
is that of the confinement for the fundamental constituents of
matter.
%On the other hand a great many deal of progress has been made towards making an unequivocal distinction .
In fact distinction between the apparently related phenomena of
screening and confinement is of considerable importance in our
present understanding of gauge theories. Field theories that yield
the linear potential are important to particle physics, since
those theories may be used to describe the confinement of quarks
and gluons and be considered as effective theories of quantum
chromodynamics.
%Due to intense interest mainly in string related topics, these studies have been extended out of the four dimensional domain and extended to theories of antisymmetric tensors of arbitrary ranks in arbitrary space-time dimensions that appear as low-energy effective field theories of strings. This has also helped us to gain insights over the mechanisms of confinement in different contexts.

We study the confinement versus screening properties of some
theories of massless antisymmetric tensors, magnetically and
electrically coupled to topological defects that eventually
condense, as a consequence of the Julia--Toulouse mechanism
(JTM)\cite{JT}. This mechanism is the dual to the Higgs mechanism
and has been shown
%by Quevedo and Trugenberg \cite{QT}
to lead to a concrete massive antisymmetric theory with a jump of
rank. We show that in the presence of two tensor fields the
condensation induces not only a mass term and a jump of rank but
also a BF coupling which will be responsible for the change from
the screening to the confining phase of the theory.

%In another circumstances antisymmetric tensor theories have been studied in the past because they are the natural extension of abelian gauge theories, our basic model for a gauge theory, with which they share very interesting properties, particularly the essential property of duality, leading to the strong--weak coupling mapping, known to exist in the electromagnetism. Such features are known to exist also in antisymmetric tensor theories in any dimension which will be essential for us in the sequel. Antisymmetric tensors also appear very naturally in supersymmetric field theories and in string theories where they play an important role in the realization of the various strong-weak coupling dualities among string theories.


%An antisymmetric tensor of rank $p$ couples naturally to an elementary extended $p-1$ dimensional object, a $(p-1)$-brane since its world-hypersurface is a $p$-dimensional object. However, if the antisymmetric tensors are compact fields, there may also appear defects or solitonic excitations in the underlying theory. Defects are classical solutions of the equations of motion and may be classified as topological or nontopological. Topological defects appear in models that support spontaneous symmetry breaking. They are important also in Cosmology and Condensed Matter Physics. Condensation of topological defects may drive phase transitions, particularly from the screening or Coulomb phases to the confining phase, which is our main interest in this paper.
%The prototype of this phenomenon is the well known Kosterlitz--Thouless transition in two space dimensions driven by vortex condensation. An important question in the analysis of phase transitions induced by topological defects regards the conditions for a topological defect to condense and for which values of parameters like temperature and coupling constants. Such questions have been tackled mostly by lattice techniques.


An important issue here is the nature of the phase transition in
the presence of a finite condensate of topological defects. It is
this aspect, in $D=d+1$ dimensions for generic antisymmetric
tensors theories, that is of importance for us. This issue was
discussed long time ago
%by Julia and Toulouse
\cite{JT} in the framework of ordered solid-state media and more
recently in the relativistic context
%by Quevedo and Trugenberg
\cite{QT}. The basic idea in Ref.\cite{JT} was to consider models
with non-trivial homotopy group able to support stable topological
defects characterized by a length scale $r=1/M$, where the mass
parameter $M$ is a cut-off for the low-energy effective field
theory.
%\footnote{ This scale may be written in terms of a mass parameter $M$ to be considered as a cut-off for the low-energy effective field theory as $r=1/M$.}.
The long wavelength fluctuations of the continuous distribution of
topological defects are the new hydrodynamical modes for the
%low-energy
effective theory that appear when topological defects condense. In
\cite{JT} there is an algorithm to identify these modes in the
framework of ordered solid-state media. However, due to the
presence of non-linear terms, the lack of relativistic invariance
and the need to introduce dissipation terms it becomes difficult
to write down an action for the phase with a condensate of
topological defects. In the relativistic context none of the above
problems is present.
%This allowed Quevedo and Trugenberg to show that the Julia--Toulouse prescription can be made much more precise, leading to
In \cite{QT} an explicit form for the action in the finite
condensate phase, for generic compact antisymmetric field theories
was found. In this context the JTM is the natural generalization
of the confinement phase for a vector gauge field.

In this paper we make use of the JTM, as presented in \cite{QT},
to study the low-energy field theory of a pair of massless
anti-symmetric tensor fields, say $A_p$ and $B_q$ with $p+q+2=D$,
coupled electrically and magnetically to a large set of
$(q-1)$-branes, characterized by charge $e$ and a Chern-Kernel
$\Lambda_{p+1}$ \cite{HL}, that eventually condense. It is shown
that the effective theory that results displays the confinement
property by computing explicitly the effective potential for a
pair of static, very massive point probes.
%
Basically, we are interested in studying the JTM in model field
theories involving $B_q$ and $A_p$ coupled to a $(q-1)$-brane,
according to the following action
\begin{eqnarray} \label{R10}
{\cal S} &=&  \int \frac 12 \frac{(-1)^q}{(q+1)!}
\left[H_{q+1}(B_q) \right]^2 + e\, B_q J^{q}(\Lambda) +  \frac 12
\frac{(-1)^p}{(p+1)!}  \left[F_{p+1}(A_p) - e
\Lambda_{p+1}\right]^2,
\end{eqnarray}
and consider the condensation phenomenon when $\Lambda_{p+1}$
becomes the new massive mode of the effective theory. Our compact
notation here goes as follows. The field strength reads
$F_{p+1}\left(A_p\right)= F_{\mu_1 \mu_2 \ldots
\mu_{p+1}}=\partial_{[\mu_1}A_{\mu_2\cdots\mu_{p+1}]}$ and
$\Lambda_{p+1}=\Lambda_{\mu_1\cdots\mu_{p+1}}$ is a totally
anti-symmetric object of rank ($p+1$).
%and $\langle\cdots\rangle$ denotes integration over the $D$ dimensional
The conserved current $J^q(\Lambda)$ is given by a delta-function
over the world-volume of the ($q-1$)-brane \cite{Kleinert:kx}.
This conserved current may be rewritten in terms of the kernel
$\Lambda_{p+1}$ as
\begin{equation} J^q(\Lambda) = \frac
1{(p+1)!} \epsilon^{q,\alpha ,p+1}\partial_\alpha \Lambda_{p+1}\;
, \end{equation} and $\epsilon^{q,\alpha ,p+1} =
\epsilon^{\mu_1\ldots\mu_q,\alpha ,\nu_1\ldots\nu_{p+1}}$. This
notation will be used in the discussion of the JTM in the next
section as long as no chance of confusion occurs.
%We will show in the next section how the Quevedo-Trugenberg prescription is used to constructed the effective interacting action,  in the condensed phase, between the anti-symmetric tensor field $B_q$ and the degrees of freedom of the condensate $\Lambda_{p+1}$. In Section III we study the confinement properties of this effective action, after the condensate is integrate out, by computing the effective quantum potential for a pair of static probe charges within the framework of the gauge-invariant but path-dependent variables formalism. In particular, we shall be interested in the dependence of the confinement properties with the condensation parameters coming from the Julia--Toulouse mechanism.



%\section{The Effective Action}

\section{The Julia--Toulouse Mechanism and the Action in the Condensed Phase}

%We begin with a discussion of the Julia--Toulouse mechanism \cite{JT} in the relativistic context as originally developed in Ref.\cite{QT} which we follow closely. Consider a generic field theory in a $D$-dimensional space-time whose symmetry group $G$ is spontaneously broken down to $H$. Topological defects may arise in this theory according to the values taken by the homotopy group of the quotient-space $\Pi_h\left(G/H\right)$. For $h<d$, a non-trivial $\Pi_h\left(G/H\right)$ will lead to $(d-h-1)$--dimensional solitons  while instantons appears in the Euclidean version of the model when $h=d$. The characteristic sizes of these extended classical solutions with finite-energy are $r_i=1/M_i$, where $M_i$ are mass parameters associated with the spontaneous symmetry breaking process. For ordered solid-state media \cite{JT} the low-energy excitations are generically described by field theories for some order parameter as in the Ginzburg--Landau theory. For the case at hand, the effective low-energy theory that has symmetry group $H$, is meaningful only on scales much bigger than $max\{r_i \}$ and, most important, experiences topological defects essentially as $d-h-1$--dimensional singularities in the space $R^d$ (for solitons) or point singularities in $R^{d+1}$ (for instantons). To our interest here the important fields are the anti-symmetric tensors of rank $(d-h)$ and $(h-1)$ that are able to couple electrically and magnetically, respectively, to the singularities. Besides, the most important point to stress here is that the effective action for this low-energy theory is then well-defined only outside these singularities.

%As discussed in the introduction we wish to focus on anti-symmetric tensor field theories which are simple, yet highly non-trivial generalization of the usual Maxwell theory. Furthermore, we consider compact antisymmetric field theories, which are the generalizations of the compact QED, as put forward in \cite{Polyakov,PO}, to higher-rank tensor theories. In compact antisymmetric field theories $p$-branes appear as topological defects of the original theory and their world-volume can be viewed as closed $(d - h)$-dimensional singularities excluded from the (space-time) domain of the model. They constitute the charges of the effective theory with strength $e_i$ and Chern-Kernels $\Lambda_h^{(i)}$ such that $J^{d-h}_i=\epsilon^{d-h,1,h}\,\partial \Lambda_h^{(i)}$. These charges together with the anti-symmetric tensors are the basic building blocks in the construction of the low-energy effective field theory outside these singularities.


%To understand the mechanism proposed by Julia and Toulouse one has to observe that the presence of these singularities induces non-trivial homology cycles for the antisymmetric tensor fields. There are then topological quantum number given in terms of the fields and the singularities, that are essentially $(d-h-1)$--dimensional holes defined on $D$--dimensional Minkowski space-time while instantons would correspond to a model on $(d+1)$--dimensional Euclidean space with holes at the location of the instantons. To realize the significance of this we consider the intersection of these singularities with an $(h + 1)$-dimensional hyperplane $\Sigma_{h+1}$ perpendicular to it. Consider a sphere $S_h$ on $\Sigma_{h+1}$ such that it isolates one of the two intersection points with the singularities world-volume (this is a $(d - h)$--dimensional object). To see the topological quantum numbers, given in terms of $S_h$, generated by stable topological defects, for which the relevant homotopy groups are $\Pi_h \left(G/H\right)=Z$, we proceed as follows. Let us define an $h$-form $\Omega_h$ which is exact outside $S_h$ whose components read (except for normalization) $\Omega_{\mu_1\cdots\mu_h} =
%\frac 1{(h+1)!} \partial_{[\mu_1}\phi_{\mu_2\cdots\mu_{h}]}$ where $\phi_{h-1}$ is an $(h-1)$-anti-symmetric tensor field. Then there exists topological invariants that can be written as $\Phi = {\int_{S_h}\Omega_h}d\sigma^h$ where the compact notation of the preceding section has been invoked.

%Let us consider next the effect of many topological defects on the low-energy effective theory since topological defects can condense leading to drastic changes in the infra-red structure of the underlying theory \cite{PO}. Therefore, the relevant question to address regards the change of the physical scenario when the number of topological defects grows making the manifold on which the low-energy effective action is defined to become very complicated. It may happens then that, for a certain range of parameters, dynamics favors the formation of a macroscopic number of topological excitations (the condensate) with a finite density. The condensate is then described by new long-lived modes. To identify the additional hydrodynamical modes of a solid state medium due to the continuous distribution of topological defects one uses the Julia--Toulouse theory, while in the framework of relativistic field theories, Quevedo and Trugenberg have provided a prescription to show that these modes are manifest as additional fields in the new phase of the low-energy theory at finite density of topological defects.

%The new massive low-energy modes in the phase with a finite density of topological defects are obtained by promoting the closed form $\Omega_h$ defined outside the complicated manifold that includes the plethora of defects to a fundamental form defined everywhere on $R^{d+1}$. After condensation the original potential $\phi_{h-1}$ becomes meaningless. In particular, for the model discussed in Section I, this role is played by the Chern-Kernel $\Lambda_h$, now promoted to the category of propagating tensor, that becomes the smooth field on $R^{d+1}$ describing the conserved fluctuations of the continuous distribution of topological defects which constitute the new hydrodynamical modes.

Although the JT algorithm becomes problematic in the ordered solid
state media,
%due to non-linearities and the need of introducing dissipation terms,
Quevedo and Trugenberg have shown that it leads to simple demands
in the study of compact antisymmetric tensor, where it produces
naturally the effective action for the new phase. They observed
that when the $(d-h-1)$-branes condense this generates a new scale
$\Delta$ related to the average density $\rho $ of intersection
points of the $(d-h)$-dimensional world-hypersurfaces of the
condensed branes with any $(h+1)$-dimensional hyperplane. The four
requirements
%made by Quevedo and Trugenberg over the action candidate
to describe effectively the dense phase are: (i) an action built
up to two derivatives in the new field possessing (ii) gauge
invariance, (iii) relativistic invariance  and, most important,
(iv) the need to recover the original model in the limit $\Delta
\to 0$.  One is therefore led to consider the action for the
condensate as
\begin{eqnarray} \label{RM15} {\cal S}_{\Omega}=\int
\frac {(-1)^h }{2 \Delta^2(h+1)!} \left[{F_{h+1}(\Omega_h)}
\right]^2 -\frac {(-1)^h\, h!}{2\, e^2} \left[\Omega_h - H_h
(\phi_{h-1})\right]^2,
\end{eqnarray}
where $H_{\mu_1\cdots\mu_h} =
%\frac 1{(h+1)!}
\partial_{[\mu_1}\phi_{\mu_2\cdots\mu_{h}]}$ and
the underlying gauge invariance is manifest by the simultaneous
transformations $\Omega_{\mu_1\cdots\mu_h} \to
\Omega_{\mu_1\cdots\mu_h} +
\partial_{[\mu_1}\psi_{\mu_2\cdots\mu_h]}$ and
$\phi_{\mu_1\cdots\mu_{h-1}} \to \phi _{\mu_1\cdots\mu_{h-1}}+
\psi_{\mu_1\cdots\mu_{h-1}}$. Upon fixing this invariance one can
drop all considerations over $\phi_{h-1}$ after absorbing
$H_h(\phi_{h-1})$ into $\Omega_h$, so that the action describes
the exact number of degrees of freedom of a massive field whose
mass parameter reads $m=\Delta/e$. This process, named as JTM, is
%by Quevedo and Trugenberg, is the process
dual to the Higgs mechanism. Here on the other hand, the new modes
generated by the condensation of magnetic topological defects
absorbs the original variables of the effective theory, thereby
acquiring a mass while in the Higgs mechanism it is the original
field that incorporates the degrees of freedom of the electric
condensate to acquire mass. This difference explains the change of
rank in the JT mechanism that is not present in the Higgs process.
In the limit $\Delta \to 0$ the only relevant field configurations
are those that satisfy the constraint $F_{h+1}(\Omega_h) =0$ whose
solution reads $\Omega_{\mu_1\cdots\mu_h} =
%\frac 1{(h+1)!}
\partial_{[\mu_1}\psi_{\mu_2\cdots\mu_{h}]}$ where $\psi_{h-1}$ is an
$(h-1)$-anti-symmetric tensor field. The field $\psi_{h-1} $ can
then be absorbed into $\phi_{h-1} $ this way recovering the
original low-energy effective action before condensation.




%\section{The Action in the Condensed Phase}

%We are now ready to discuss the consequences of the JT mechanism over the action (\ref{R10}).
The distinctive feature of the JT mechanism is that after
condensation $\Lambda_{p+1}$ is elevated to the condition of
propagating field. The new degree of freedom absorbs the degrees
of freedom of the tensor $A_p$ this way completing its
longitudinal sector. The new mode is therefore explicitly massive.
Since $A_p\to \Lambda_{p+1}$ there is a change of rank with
dramatic consequences. The last term in (\ref{R10}), displaying
the magnetic coupling between the field-tensor $F_{p+1}(A_p)$ and
the $(q-1)$-brane, becomes the mass term for the new effective
theory in terms of the tensor field $\Lambda_{p+1}$ and a new
dynamical term is induced by the condensation.
%It is important to notice the change of rank due to the condensation mechanism.Another important feature is that
The minimal coupling of the $B_q$ tensor becomes responsible for
another contribution for the mass, this time of topological
nature. Indeed the second term (\ref{R10}) becomes a
%``$BF$--term" in the form
``$B\wedge F(\Lambda)$" term between the remaining propagating
modes, inducing topological mass, in addition to the induced
condensed mass,
%The final result reads
\begin{eqnarray}
\label{RM10} {\cal S}_{cond} &=& \int  \frac{(-1)^q}{2(q+1)!}
\left[H_{q+1}(B_q) \right]^2 + e\, B_q
\epsilon^{q,\alpha,p+1}\partial_\alpha \Lambda_{p+1} + \nonumber\\
&+& \int \frac {(-1)^{p+1}}{2 (p+2)!} \left[F_{p+2}(\Lambda_{p+1})
\right]^2 -\frac {(-1)^{p+1}\, (p+1)!}2 m^2 \Lambda_{p+1}^2.
\end{eqnarray}
%where $m = \Delta/ e$.
Recall that the theory before condensation displayed two
independent fields coupled to a $(q-1)$--brane. The nature of the
two couplings were however different. The $A_p$ tensor, that was
magnetically coupled to the brane, was then absorbed by the
condensate after phase transition. On the other hand, the electric
coupling, displayed by the $B_q$ tensor, became a ``$B\wedge
F(\Lambda)$" topological term after condensation. There has been a
drastic change in the physical scenario.
%To show that the new systems displays a confining phase is the goal of this work.
We want next to obtain an effective action for the $B_q$ tensor.
To this end we shall next integrate out the field $\Lambda$
describing the condensate to obtain,
%The implications of the resulting effective action will be studied in the next section.
%We integrate out the condensate field $\Lambda$ to obtain
%we rewrite this sector of the action as, \begin{eqnarray} \label{RM100} {\cal S}_{\Lambda} &=& \int   \frac {(-1)^{p+1}}{2 (p+2)!} \left[F_{p+2}(\Lambda_{p+1}) \right]^2 -\frac {(-1)^{p+1}\, (p+1)!}2 m^2 \Lambda_{p+1}^2 +  e\, B_q \epsilon^{q,\alpha,p+1}\partial_\alpha \Lambda_{p+1}\nonumber \\ &=& \int \frac{(-1)^{p+2}(p+1)!}{2}\, \Lambda_{p+1}\left(\Delta^2 +m^2\right)\Lambda^{p+1} +  e\, B_q \epsilon^{q,\alpha,p+1}\partial_\alpha \Lambda_{p+1} \end{eqnarray} where we have made use of (i) the identity \begin{eqnarray} \label{identity} \epsilon_{\mu_1\cdots\mu_p\nu_1\cdots\nu_q}\,\epsilon^{\mu_1\cdots\mu_p\nu_1\cdots\nu_q} = (-1)^{D+1}\, p! \, \delta^{\tilde\nu_1}_{[\nu_1}\cdots\delta^{\tilde\nu_q}_{\nu_q]} \end{eqnarray} which in our compact notation reads \begin{eqnarray} \label{compactidentity}     \epsilon_{p,q} \,\epsilon^{p,q}=     (-1)^{D+1}\, p! \, \delta^{\;\tilde q}_{[q]} \end{eqnarray} and (ii) of an integration by parts, such that \begin{eqnarray} \left(\epsilon^{q,\alpha,p+1}\partial_\alpha\Lambda_{p+1}\right)^2 = (-1)^D\, q!\, p!\, \Lambda_{p+1} \,(\Delta^2) \,\Lambda^{p+1} \end{eqnarray} Next we solve the equations of motion \begin{eqnarray}     \frac{\delta{\cal S}_{\Lambda}}{\delta\Lambda_{p+1}} = 0 \end{eqnarray} to obtain \begin{eqnarray} \label{solution} \Lambda_{p+1} = \frac{e \,(-1)^{p+1}\, \wp(\epsilon\partial)}{(p+1)!} \frac{1}{\Delta^2 +  m^2}\;\epsilon_{p+1,\alpha,q}\,\partial^\alpha \, B^q \end{eqnarray} where $\wp(\epsilon\partial)= (-1)^{p(q+1)}$ is the parity of the generalized curl operator, in the sense that \begin{eqnarray} \int \psi_p\,\epsilon^{p,\alpha,q}\partial_\alpha \, \phi_q =\wp(\epsilon\partial) \int \phi_q \,\epsilon^{q,\alpha,p}\partial_\alpha \,\psi_p\, . \end{eqnarray} Substituting (\ref{solution}) back into the action (\ref{RM100}) and using that, \begin{eqnarray} \left(\epsilon_{p+1,\alpha,q}\partial^\alpha B^q\right)^2 = (-1)^{D+1}  \frac{(p+1)!}{(q+1)1} H_{q+1}(B_q) H^{q+1}(B_q) \end{eqnarray} gives, after the inclusion of the free term for the $B_q$ tensor,
our final effective theory as
\begin{eqnarray}
\label{acaoeffetiva} {\cal S}_{eff} =\int \frac{(-1)^{q+1}}{2\,
(q+1)!} H_{q+1}(B_q) \left(1 +\frac{e^2}{\Delta^2 + m^2}
\right)H^{q+1}(B_q).
\end{eqnarray}

\section{Interaction Energy}

Next we examine the screening versus confinement issue. We shall
consider a specific example involving two Maxwell tensors coupled
electrically and magnetically to a point-charge such that after
condensation we end up with a Maxwell and a massive Kalb-Ramond
field (the condensate) coupled topologically to each other.
%besides the presence of an explicit mass term for the Kalb-Ramond.
We shall calculate the interaction energy for the effective theory
between external probe sources by computing the expectation value
of the energy operator $H$ in the physical state $\left| \Phi
\right\rangle$ describing the sources, denote by $ \left\langle H
\right\rangle _\Phi$.
%It is worth mentioning that our starting point is Eq.(\ref{RM10}) with $p=q=1$.
%is the Lagrangian obtained in \cite{Deguchi}: \begin{equation} {\cal L} = \frac{1}{{12}}H_{\mu \nu \rho } H^{\mu \nu \rho }  - \frac{1}{4}m^2 \Lambda_{\mu \nu }\Lambda^{\mu \nu } - \frac 14 F_{\mu \nu } F^{\mu \nu } - \frac 12\, e \Lambda_{\mu \nu }F^{\mu \nu }  - A_0 J^0, \label{KR10} \end{equation} where
The Kalb-Ramond field $\Lambda_{\mu\nu}$ carrying the degrees of
freedom of the condensate is integrate out leading to
%as discussed at the end of the last Section. Here $H_{\mu \nu \rho }  = \partial _\mu  \Lambda_{\nu \rho }  + \partial _\rho  \Lambda_{\mu \nu }  + \partial _\nu  \Lambda_{\rho \mu }$,  As stated, our objective will be to calculate the potential energy for this theory. As in the previous subsection, the first step in this direction is to carry out the integration over $\Lambda_{\mu \nu }$ in Eq.(\ref{KR10}). This allows us to write
%the following effective Lagrangian
\begin{equation}
{\cal L} =  - \frac{1}{4}F_{\mu \nu } \left( {1 + \frac{{e^2
}}{{\triangle^2 + m^2 }}} \right)F^{\mu \nu }  - A_0 J^0 ,
\label{KR15}
\end{equation}
where
%$F_{\mu \nu }  = \partial _\mu  A_\nu   - \partial _\nu  A_\mu$ and
$J^0$ is an external current.
%which is a particular case of (\ref{acaoeffetiva}).
We observe
%from (\ref{KR15}),
that the limits $e\to 0$ or $m\to 0$ are well defined and lead
%(from the point of view of the probe charges)
to a pure Maxwell theory or to a (topologically) massive model.
Since the probe charges only couple to the Maxwell fields, the
Kalb- Ramond condensate will not contribute to their interaction
energy in the first case because in the limit where the parameter
$e\to 0$ the Maxwell field and the condensate decouple. The second
limit means that we are back to the non-condensed phase. As so the
confinement of the probe charges are expected to disappear being
taken over by an screening phase controlled by the parameter $e$
playing the role of topological mass.

Once this is done, the canonical quantization of this theory from
the Hamiltonian point of view follows straightforwardly. The
canonical momenta read $\Pi ^\mu   = - \left( {1 + \frac{{e^2
}}{{\Delta ^2 + m^2 }}} \right)F^{0\mu }$ with the only
nonvanishing canonical Poisson brackets being $\left\{ {A_\mu
\left( {t,x} \right),\Pi ^\nu  \left( {t,y} \right)} \right\} =
\delta _\mu ^\nu  \delta \left( {x - y} \right)$. Since $\Pi_0$
vanishes we have the usual primary constraint $\Pi_0=0$, and $\Pi
^i  = \left( {1 + \frac{{e^2 }}{{\Delta ^2 + m^2 }}}
\right)F^{i0}$. The canonical Hamiltonian is thus
\begin{equation}
H_C  = \int {d^3 } x\left\{ { - \frac{1}{2}\Pi ^i \left( {1 +
\frac{{e^2 }}{{\Delta ^2  + m^2 }}} \right)^{ - 1} \Pi _i  + \Pi
^i \partial _i A_0  + \frac{1}{4}F_{ij} \left( {1 + \frac{{e^2
}}{{\Delta ^2  + m^2 }}} \right)F^{ij}  + A_0 J^0 }.
\right\}.\label{KR25}
\end{equation}
Time conservation of the primary constraint $ \Pi _0$ leads to the
secondary Gauss-law constraint $\Gamma _1 \left( x \right) \equiv
\partial _i \Pi ^i - J^0 = 0$. The preservation of $\Gamma_1$ for
all times does not give rise to any further constraints. The
theory is thus seen to possess only two constraints, which are
first class, therefore the theory described by $(\ref{KR15})$ is a
gauge-invariant one. The extended Hamiltonian that generates
translations in time then reads $H = H_C  + \int {d^3 } x\left(
{c_0 \left( x \right)\Pi _0 \left( x \right) + c_1 \left( x
\right)\Gamma _1 \left( x \right)} \right)$, where $c_0 \left( x
\right)$ and $c_1 \left( x \right)$ are the Lagrange multiplier
fields. Moreover, it is straightforward to see that $\dot{A}_0
\left( x \right)= \left[ {A_0 \left( x \right),H} \right] = c_0
\left( x \right)$, which is an arbitrary function. Since $ \Pi^0 =
0$ always, neither $ A^0 $ nor $ \Pi^0 $ are of interest in
describing the system and may be discarded from the theory.
%Then, the Hamiltonian takes the form \begin{equation} H = \int {d^3 x\left\{ { - \frac{1}{2}\Pi _i \left( {1 + \frac{{e^2 }}{{\Delta ^2  + m^2 }}} \right)^{ - 1} \Pi ^i  + \frac{1}{4}F_{ij} \left( {1 + \frac{{e^2 }}{{\Delta ^2  + m^2 }}} \right)F^{ij}  + c\left( x \right)\left( {\partial _i \Pi ^i  - J^0 } \right)} \right\}}, \label{KR35} \end{equation} where $c(x) = c_1 (x) - A_0 (x)$.

The quantization of the theory requires the removal of nonphysical
variables, which is done by imposing a gauge condition such that
the full set of constraints becomes second class. A convenient
choice is found to be \cite{Pato} $\Gamma _2 \left( x \right)
\equiv \int\limits_{C_{\xi x} } {dz^\nu } A_\nu \left( z \right)
\equiv \int\limits_0^1 {d\lambda x^i } A_i \left( {\lambda x}
\right) = 0$, where  $\lambda$ $(0\leq \lambda\leq1)$ is the
parameter describing the spacelike straight path $ x^i = \xi ^i  +
\lambda \left( {x - \xi } \right)^i $, and $ \xi $ is a fixed
point (reference point). There is no essential loss of generality
if we restrict our considerations to $ \xi ^i=0 $. In this case,
the only nonvanishing equal-time Dirac bracket is
\begin{equation}
\left\{ {A_i \left( x \right),\Pi ^j \left( y \right)} \right\}^ *
=\delta{ _i^j} \delta ^{\left( 3 \right)} \left( {x - y} \right) -
\partial _i^x \int\limits_0^1 {d\lambda x^j } \delta ^{\left( 3
\right)} \left( {\lambda x - y} \right). \label{KR45}
\end{equation}
%In passing we recall that the transition to quantum theory is made by the replacement of the Dirac brackets by the operator commutation relations according to $\left\{ {A,B} \right\}^ * \to \left( { - i} \right)\left[ {A,B} \right]$.

We now turn to the problem of obtaining the interaction energy
between pointlike sources in the model under consideration. The
state $\left| \Phi \right\rangle$ representing the sources is
obtained by operating over the vacuum with creation/annihilation
operators. We want to stress that, by construction, such states
are gauge invariant. In the case at hand we consider the
gauge-invariant stringy $\left|{\overline \Psi  \left( \bf y
\right)\Psi \left( {\bf y^ \prime } \right)} \right\rangle$, where
a fermion is localized at ${\bf y}\prime$ and an antifermion at $
{\bf y}$ as follows \cite{Dirac2},
\begin{equation}
\left| \Phi  \right\rangle  \equiv \left| {\overline \Psi  \left(
\bf y \right)\Psi \left( {\bf y}\prime \right)} \right\rangle  =
\overline \psi \left( \bf y \right)\exp \left(
{iq\int\limits_{{\bf y}\prime}^{\bf y} {dz^i } A_i \left( z
\right)} \right)\psi \left({\bf y}\prime \right)\left| 0
\right\rangle, \label{KR60}
\end{equation}
where $\left| 0 \right\rangle$ is the physical vacuum state and
the line integral appearing in the above expression is along a
spacelike path starting at ${\bf y}\prime$ and ending $\bf y$, on
a fixed time slice. It is worth noting here that the strings
between fermions have been introduced in order to have a
gauge-invariant function $\left| \Phi  \right\rangle $. In other
terms, each of these states represents a fermion-antifermion pair
surrounded by a cloud of gauge fields sufficient to maintain gauge
invariance. As we have already indicated, the fermions are taken
to be infinitely massive (static).



From our above discussion, we see that $\left\langle H
\right\rangle _\Phi$ reads
\begin{equation}
\left\langle H \right\rangle _\Phi   = \left\langle \Phi
\right|\int {d^3 } \left\{ { - \frac{1}{2}\Pi _i \left( {1 -
\frac{{e^2 }}{{\nabla ^2  - m^2 }}} \right)^{ - 1} \Pi ^i }
\right\}\left| \Phi  \right\rangle, \label{KR65}
\end{equation}
where, in this static case, $\Delta ^2 = - \nabla ^2$. Observe
that when $e=0$ we obtain the pure Maxwell theory, as mentioned
after (\ref{KR15}).  From now on we will suppose $e\neq 0$.

Next, from the foregoing Hamiltonian analysis, $\left\langle H
\right\rangle _\Phi$ becomes $\left\langle H \right\rangle _\Phi =
\left\langle H \right\rangle _0  + V^{\left( 1 \right)}  +
V^{\left( 2 \right)}$, where $\left\langle H \right\rangle _0  =
\left\langle 0 \right|H\left| 0 \right\rangle$. The $V^{\left( 1
\right)}$ and $V^{\left( 2 \right)}$ terms are given by:
\begin{equation}
V^{\left( 1 \right)}  =  - \frac{{q^2 }}{2}\int {d^3 x} \int_{\bf
y}^{\bf y^\prime} {dz^\prime_i } \delta ^{\left( 3 \right)} \left(
{x - z^\prime} \right)\frac{1}{{\nabla _x^2  - M^2 }}\nabla _x^2
\int_{\bf y}^{\bf y^\prime} {dz^i } \delta ^{\left( 3 \right)}
\left( {x - z} \right), \label{KR80}
\end{equation}
and
\begin{equation}
V^{\left( 2 \right)}  =   \frac{{q^2 m^2}}{2}\int {d^3 x}
\int_{\bf y}^{\bf y^\prime} {dz^\prime_i } \delta ^{\left( 3
\right)} \left( {x - z^\prime} \right)\frac{1}{{\nabla _x^2  - M^2
}} \int_{\bf y}^{\bf y^\prime} {dz^i } \delta ^{\left( 3 \right)}
\left( {x - z} \right), \label{KR85}
\end{equation}
where $ M^2\equiv{m^2+ e^2} $ and the integrals over $z^i$ and
$z^\prime_i$ are zero except on the contour of integration.

The $V^{\left( 1 \right)}$ term may look peculiar, but it is
nothing but the familiar Yukawa interaction plus self-energy
terms. In effect, as was explained in Ref. \cite{GG2}, the
expression (\ref{KR80}) can also be written as
\begin{equation}
V^{\left( 1 \right)}  = \frac{{e^2 }}{2}\int_{\bf y}^{{\bf
y}^{\prime}  } {dz_i^{\prime}}\partial _i^{z^{\prime}} \int_{\bf
y}^{{\bf y}^{\prime}} {dz^i }\partial _z^i G\left( {{\bf
z}^{\prime},{\bf z}} \right)=   - \frac{{q^2 }}{{4\pi }}\frac{{e^{
- M|{\bf y} - {\bf y}^ {\prime}| } }}{{|{\bf y} - {\bf
y}^{\prime}|}}, \label{KR90}
\end{equation}
where we used that the Green function $G({\bf z}^{\prime},{\bf z})
= \frac{1}{{4\pi }}\frac{{e^{ - M|{\bf z}^{\prime} - {\bf z}|}
}}{{|{\bf z}^{\prime} - {\bf z}|}}$ and remembered that the
integrals over $z^i$ and $z_i^{\prime}$ are zero except on the
contour of integration. The expression
%(\ref{KR90})
then reduces to the Yukawa-type potential after subtracting the
self-energy terms.
%, that is, \begin{equation} V^{\left( 1 \right)}  =  - \frac{{q^2 }}{{4\pi }}\frac{{e^{ - M|{\bf y} - {\bf y}^ {\prime}| } }}{{|{\bf y} - {\bf y}^{\prime}|}}. \label{KR100} \end{equation}

We now turn our attention to the calculation of the $V^{\left( 2
\right)}$ term, which is given by
\begin{equation}
V^{\left( 2 \right)}  = \frac{{q^2 m^2 }}{2}\int_{\bf y}^{{\bf
y}^{\prime}  } {dz^{{\prime} i} } \int_{\bf y}^{{\bf y}^{\prime} }
{dz^i } G({\bf z}^{\prime} ,{\bf z}). \label{KR105}
\end{equation}
It is appropriate to observe here that the above term is similar
to the one found for the system consisting of a gauge field
interacting with a massive axion field \cite{GG2}.
Notwithstanding, in order to put our discussion into context it is
useful to summarize the relevant aspects of the calculation
described previously \cite{GG2}. In effect, as was explained in
Ref. \cite{GG2}, by using the Green function in momentum space,
that is, $\frac{1}{{4\pi }}\frac{{e^{ - M|{\bf z}^{\prime}   -
{\bf z}|} }}{{|{\bf z}^{\prime} - {\bf z}|}} = \int {\frac{{d^3
k}}{{\left( {2\pi } \right)^3 }}\frac{{e^{i{\bf k} \cdot \left(
{{\bf z}^{\prime}- {\bf z}} \right)} }}{{{\bf k}^2  + M^2 }}}$,
the expression (\ref{KR105}) can also be written as
\begin{equation}
V^{\left( 2 \right)}   = q^2 m^2 \int {\frac{{d^3 k}}{{\left(
{2\pi } \right)^3 }}} \left[ {1 - \cos \left( {{\bf k} \cdot {\bf
r}} \right)} \right]\frac{1}{{({\bf k}^2  + M^2)
}}\frac{1}{{\left( {{\bf {\hat n}} \cdot {\bf k}} \right)^2 }},
\label{KR115}
\end{equation}
where ${\bf {\hat n}}\equiv \frac{{{\bf y} - {\bf
y}^{\prime}}}{{|{\bf y} - {\bf y}^{\prime}| }}$ is a unit vector
and ${\bf r}={\bf y}-{\bf y^{\prime}}$ is the relative vector
between the quark and antiquark. Since ${\bf {\hat n}}$ and ${\bf
r}$ are parallel, we get accordingly $V^{\left( 2 \right)} =
\frac{{q^2 m^2 }}{{8\pi ^3 }}\int\limits_{ - \infty }^\infty
{\frac{{dk_r }}{{k_r^2 }}} \left[ {1 - \cos \left( {k_r r}
\right)} \right]\int\limits_0^\infty  {d^2 k_T \frac{1}{{(k_r^2 +
k_T^2  + M^2) }}}$, where $k_T$ denotes the momentum component
perpendicular to ${\bf r}$. Integration over $k_T$ yields $
V^{\left( 2 \right)}  = \frac{{q^2 m^2 }}{{8\pi ^2 }}\int\limits_{
- \infty }^\infty  {\frac{{dk_r }}{{k_r^2 }}} \left[ {1 - \cos
\left( {k_r r} \right)} \right]\ln \left( {1 + \frac{{\Lambda ^2
}}{{k_r^2  + M^2 }}} \right)$, where $\Lambda$ is an ultraviolet
cutoff. We also observe at this stage that similar integral was
obtained independently in Ref.\cite{Suganuma} in the context of
the dual Ginzburg-Landau theory by an entirely different approach.

We now proceed to compute the previous integral. For this purpose
we introduce a new auxiliary parameter $\varepsilon$ by making in
the denominator of the previous integral the substitution
$k_r^2\rightarrow k_r^2+\varepsilon^2$. Thus it follows that $
V^{\left( 2 \right)} \equiv \lim _ {\varepsilon  \to 0}
{\widetilde V}^{\left( 2 \right)}= \lim _{\varepsilon \to
0}\frac{{q^2 m^2 }}{{8\pi ^2 }}\int\limits_{ - \infty }^\infty
{\frac{{dk_r }}{{(k_r^2  + \varepsilon ^2) }}} \left[ {1 - \cos
\left( {k_r r} \right)} \right]\ln \left( {1 + \frac{{\Lambda ^2
}}{{k_r^2  + M^2 }}} \right)$. We further note that the
integration on the $k_r$-complex plane yields ${\widetilde
V}^{\left( 2 \right)} = \frac{{q^2 m^2 }}{{8\pi }}\left( {\frac{{1
- e^{ - \varepsilon |{\bf y} - {{\bf y}^\prime}|  } }}{\varepsilon
}} \right)\ln \left( {1 + \frac{{\Lambda ^2 }}{{M^2 - \varepsilon
^2 }}} \right)$. Taking the limit $\varepsilon  \to 0$, this
expression then becomes $ V^{\left( 2 \right)}  = \frac{{q^2 m^2
}}{{8\pi }}|{\bf y} - {{\bf y}^\prime}| \ln \left( {1 +
\frac{{\Lambda ^2 }}{{M^2 }}} \right)$.

 This, together with Eq.(\ref{KR90}), immediately shows
that the potential for two opposite charges located at ${\bf y}$
and ${\bf y^\prime}$ is given by
\begin{equation}
V(L) =  - \frac{{q^2 }}{{4\pi }}\frac{{e^{ - ML} }}{L} +
\frac{{q^2 m^2 }}{{8\pi }}L\ln \left( {1 + \frac{{\Lambda ^2
}}{{M^2 }}} \right), \label{KR145}
\end{equation}
where $L\equiv|{\bf y}-{\bf {y^\prime}}|$.
%In this context it may be recalled the calculation reported in Ref. \cite{Kondo} by taking into account topological nontrivial sectors in $U(1)$ gauge theory is given by \begin{equation} V\left( L \right) =  - \frac{{q^2 }}{{4\pi }}\frac{1}{L} + \sigma L. \label{KR155} \end{equation} Notice that the result (\ref{KR145}) agrees with (\ref{KR155}) in the limit of large $M$. Thus one is led to the conclusion that, although both calculations lead to confinement, the physical mechanism of obtaining a linear potential is quite different. In other terms, our result it may be considered as a physical realization of the topological nontrivial sectors studied in \cite{Kondo}. Let us also mention here that the result (\ref{KR145}) is exactly the one obtained in Ref. \cite{Suganuma} in the context of the dual Landau-Ginzburg theory. But we do not think that the agreement is an accidental coincidence. As we mentioned before, a gauge theory in the presence of external fields and axions displays the same behavior \cite{GG2}.
%It seems a challenging work to extend to a class of models the above analysis, which can predict the same interaction energy. We expect to report on progress along these lines soon.

\section{Final Remarks}

We have studied the confinement versus screening issue for a pair
of antisymmetric tensors coupled to topological defects that
eventually condense, giving a specific realization of the
Julia--Toulouse phenomenon. We have seen that the Julia--Toulouse
mechanism for a couple of massless antisymmetric tensors is
responsible for the appearance of mass and the jump of rank in the
magnetic sector while the electric sector becomes a BF--type
coupling. The condensate absorbs and replaces one of the tensors
and becomes the new massive propagating mode but does not couple
directly to the probe charges.  The effects of the condensation
are however felt through the BF coupling with the remaining
massless tensor.  It is therefore not surprising that they become
manifest in the interaction energy for the effective theory.  We
have obtained the effective theory for the condensed phase in
general and computed the interaction energy between two static
probe charges, in a specific example, in order to test the
confinement versus screening properties of the effective model.
Our results show that the interaction energy in fact contains a
linear confining term and an Yukawa type potential. It can be
observed that confinement completely disappears in the limit $m\to
0$ while the screening takes over controlled by the topological
mass parameter instead. Although we have considered the case where
the effective model consists of the BF--coupling between a
Kalb-Ramond field (that represents the condensate) and a Maxwell
field, our results seem to be quite general.  A direct calculation
for tensors of arbitrary rank in the present approach is however a
quite challenging problem that we hope to be able to report in the
future.

\begin{thebibliography}{}

\bibitem{JT} B. Julia and G. Toulouse, J. Physique Lett. {\bf 40} L395 (1979).

\bibitem{QT} F. Quevedo, C. A. Trugenberger, Nucl.Phys.{\bf B501} 143 (1997).

\bibitem{HL} [7] F.R. Harvey and H.B. Lawson Jr., A theory of characteristic
currents associated with a singular connection, Asterisque 213
(1993), Soci\`et\`e Math\`ematique de France.

\bibitem{Kleinert:kx}
H.~Kleinert, ``Gauge Fields In Condensed Matter. Vol. 1: Superflow
And Vortex Lines, Disorder Fields, Phase Transitions,'' World
Scientific Publishing Co., Singapore 1989.

\bibitem{Polyakov} A.M. Polyakov, ``Gauge Fields and Strings", Harwood
Academic Publishers (1987).

\bibitem{PO} P. Orland, Nucl. Phys. {\bf B205} 107 (1982).
\bibitem{Deguchi} S. Deguchi and Y. Kokubo,  Mod. Phys. Lett. {\bf
A17}, 503 (2002).

\bibitem{Pato} P. Gaete, Z. Phys. {\bf C76}, 355 (1997); Phys. Lett.
{\bf B515}, 382 (2001); Phys. Lett. {\bf B582}, 270 (2004).

\bibitem{Dirac2} P. A. M. Dirac, {\it The Principles of Quantum Mechanics}
(Oxford University Press, Oxford, 1958); Can. J. Phys. {\bf 33},
650 (1955).

\bibitem{GG2} P. Gaete and E. I. Guendelman, ``Confinement in the
presence of external fields and axions", hep-th/0404040.

\bibitem{Suganuma} H. Suganuma, S. Sasaki and H. Toki,  Nucl. Phys. {\bf B435}, 207 (1995).

%\bibitem{Kondo} K.-I. Kondo, Phys. Rev. {\bf D58}, 085013 (1998).
\end{thebibliography}
\end{document}

hep-th0411236.extracted_bib
\begin{thebibliography}{}

\bibitem{JT} B. Julia and G. Toulouse, J. Physique Lett. {\bf 40} L395 (1979).

\bibitem{QT} F. Quevedo, C. A. Trugenberger, Nucl.Phys.{\bf B501} 143 (1997).

\bibitem{HL} [7] F.R. Harvey and H.B. Lawson Jr., A theory of characteristic
currents associated with a singular connection, Asterisque 213
(1993), Soci\`et\`e Math\`ematique de France.

\bibitem{Kleinert:kx}
H.~Kleinert, ``Gauge Fields In Condensed Matter. Vol. 1: Superflow
And Vortex Lines, Disorder Fields, Phase Transitions,'' World
Scientific Publishing Co., Singapore 1989.

\bibitem{Polyakov} A.M. Polyakov, ``Gauge Fields and Strings", Harwood
Academic Publishers (1987).

\bibitem{PO} P. Orland, Nucl. Phys. {\bf B205} 107 (1982).
\bibitem{Deguchi} S. Deguchi and Y. Kokubo,  Mod. Phys. Lett. {\bf
A17}, 503 (2002).

\bibitem{Pato} P. Gaete, Z. Phys. {\bf C76}, 355 (1997); Phys. Lett.
{\bf B515}, 382 (2001); Phys. Lett. {\bf B582}, 270 (2004).

\bibitem{Dirac2} P. A. M. Dirac, {\it The Principles of Quantum Mechanics}
(Oxford University Press, Oxford, 1958); Can. J. Phys. {\bf 33},
650 (1955).

\bibitem{GG2} P. Gaete and E. I. Guendelman, ``Confinement in the
presence of external fields and axions", hep-th/0404040.

\bibitem{Suganuma} H. Suganuma, S. Sasaki and H. Toki,  Nucl. Phys. {\bf B435}, 207 (1995).

%\bibitem{Kondo} K.-I. Kondo, Phys. Rev. {\bf D58}, 085013 (1998).
\end{thebibliography}