$SU(2)$ Abelian projected action for renormali- sation group flows
Filipe FREIRE
The dual Meissner effect scenario of confinement is analysed using exact renormalisation group (ERG) equations. In particular, the low energy regime of $SU(2)$ Yang-Mills (YM) is studied in a maximal Abelian gauge (MAG). It is shown that under general conditions the effective action derived when integrated using ERG methods contains the relevant degrees of freedom for confinement. In addition, the physics in the confining regime is dual to that of the broken phase of an Abelian Higgs model.
hep-th0205032.tex
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\def\tr{{\rm tr}}
\def\Tr{{\rm Tr}}
\def\ep{\epsilon}
\def\half{\frac{1}{2}}
\def\quarter{\frac{1}{4}}
\def\fracpow#1#2#3{\frac{#1}{#2}{}_{{}_{\!\!#3}}}
\def\mad{M_{{\rm\scriptscriptstyle\!A\!D}}}
\def\qcd{\Lambda_{\scriptscriptstyle\rm{Q\hspace{-.02cm}C\hspace{-.02cm}D}}}
\def\pg#1#2{P_{\!\!{\scriptscriptstyle#1}\!{\scriptscriptstyle#2}}}
\def\sym{S^{}_{{\rm \scriptscriptstyle Y\!M}}}
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\begin{opening}
\title{$SU(2)$ Abelian projected action for renormali- sation group flows}
\author{Filipe FREIRE}
\institute{School of Theoretical Physics, Dublin Inst. for Adv. Studies\\
           and\\
           Dept. of Mathematical Physics, NUI Maynooth, Ireland}
\end{opening}
\begin{document}
\begin{abstract}
    The dual Meissner effect scenario of confinement is analysed using
    exact renormalisation group (ERG) equations. In particular,
    the low energy regime of $SU(2)$ Yang-Mills (YM) is studied in a
    maximal Abelian gauge (MAG). 
    It is shown that under general conditions the effective
    action derived when integrated using ERG methods contains 
    the relevant degrees of freedom for confinement. In addition, 
    the physics in the confining regime is dual to that of the broken
    phase of an Abelian Higgs model.
\end{abstract}

\part{Introduction}
In the dual Meissner effect picture the vacuum of a YM gauge theory,
with mass gap, consists of a magnetic condensate which leads to
a linear potential that confines quarks \cite{Nambu:1974zg}. 
This has been studied in the context of dual models of
superconductivity where magnetic degrees of
freedom are treated as particle fluctuations. However, it is unlikely that 
this scenario alone works because monopoles are heavy making it unlikely
for a magnetic Higgs mechanism to take place. 
An alternative approach using the notion of Abelian projection 
has been developed in theories without Higgs fields
\cite{Mandelstam:1976pi,'tHooft:1981ht} such as QCD. 
This approach is taken in the present work.
Abelian projection uses gauge fixing conditions to attenuate
non-Abelian degrees of freedom while the maximal Abelian subgroup of
the gauge group remains unbroken. The residual invariance is
fixed by using Lorentz conditions for each remnant Abelian invariance.

Here a MAG condition is used for the gauge fixing. On the lattice Abelian
dominance \cite{Suzuki:1990gp} has been observed in this gauge
indicating the existence of an intermediate scale below which
non-Abelian gauge field variables are suppressed. This separation of
scales is a good feature for working in a MAG.
The ERG for gauge theories \cite{Reuter:1994kw,Litim:1998qi} provides 
an ideal framework for investigating the present scenario because of this
splitting of scales at low energies. The use of ERG has already been 
highlighted by Ellwanger \cite{Ellwanger:1998wv} who has done similar
studies although using a Landau gauge, therefore not having the
benefit of Abelian dominance.
The ERG has been formulated for general linear gauges \cite{Litim:1998nf}.
Though the MAG is non-linear it poses no new problems as it can be
seen as a particular case of a background field gauge and therefore
it is easy to transfer the results of \cite{Freire:1996db}.

In this talk I apply this procedure to the pure $SU(2)$ YM and derive
an effective Abelian action for this theory. Then, under 
general conditions, it is shown, using a functional analysis based on ERG
methods, that the effective action derived here contains the
prerequisites for confinement. Moreover,
it is also shown that the physics in the confining phase is dual to 
that of a type II superconductor. Technical details can be
found in \cite{Freire:2001nd}.\\[.4cm]
\part{The $SU(2)$ YM action} Consider pure $SU(2)$ YM in a four dimensional
Euclidean space. In conventional variables its action is
$\sym\!=\frac{1}{4}\int_x F_{\mu\nu}^a\,F_{\mu\nu}^a$,
where $\int_x$ is a short-hand for $\int\!d^4x$ and $F_{\mu\nu}^{a}$
is the field strength. When expressed in terms of the field variables
$ A_\mu=A_\mu^3$, $\phi_\mu = \frac{1}{\sqrt{2}}(A_\mu^1-iA_\mu^2)$ and
$\phi^\dagger_\mu$, respectively neutral and charged potentials, this
action takes the form
\begin{equation}
  \label{newymaction}
  \sym = \frac{1}{4} \int_x(f^{}_{\mu\nu}+C^{}_{\mu\nu})\,
  (f^{}_{\mu\nu}+C^{}_{\mu\nu}) +\half \int_x \Phi^\dagger_{\mu\nu}
  \Phi^{}_{\mu\nu}.
\end{equation}
Here $f_{\mu\nu}$ and $\Phi_{\mu\nu}$ are the field strengths for
the Abelian and the charged gauge fields, respectively,
$f_{\mu\nu} = \partial_\mu A_\nu - \,\partial_\nu A_\mu\,$ and
$\Phi_{\mu\nu} = D_\mu \phi_\nu - \,D_\nu \phi_\mu$, with 
$D_\mu = \partial_\mu -igA_\mu$ the Abelian covariant derivative.
Finally, the antisymmetric tensor $C_{\mu\nu}$ is a real quadratic 
combination of the off-diagonal fields,
$C_{\mu\nu} = ig\,(\phi_\mu^\dagger\ \phi_\nu -
  \phi_\nu^\dagger\ \phi_\mu)$.

Clearly the action~(\ref{newymaction}) is explicitly invariant under a
$U(1)$ subgroup of $SU(2)$. The gauge is fixed by the condition,
$F^\pm[\phi,\,A]\!:=\!(\partial_\mu\pm ig\!A_\mu)\phi_\mu\!=\!0$.
This MAG leaves the $U(1)$ invariance in~(\ref{newymaction}) unbroken. The 
remnant gauge freedom is fixed with the Lorentz condition,
$F[A]\!:=\!\partial_\mu A_\mu\!=\!0$.
Gribov copies exist for this particular choice of gauge fixing but
their effects are not sizable \cite{Hart:1997ar} 
and shall be neglected at this stage.\\[.4cm]
\part{Abelian dominance}
The suppression of the off-diagonal fields observed on the lattice in a 
MAG \cite{Suzuki:1990gp} suggests that they acquire a mass dynamically.
Some mass generation mechanisms have been investigated in MAGs
\cite{Schaden:1999ew,Kondo:2000ey}. See also
\cite{Gubarev:2000nz,Dudal:2002aj}. 
It is expected that this mass sets a scale of Abelian dominance, say
$M_{\!{}_{\rm{A\!D}}}$, below which a qualitative change of the
relevant dynamical variables takes place. At a much lower scale this
will lead to confinement. Preliminary results from the lattice give
$\mad\!\simeq\!1.2$ GeV \cite{Amemiya:1999zf} which indicates that the
scale of Abelian dominance is indeed larger than the confining scale
$\qcd$. Though the mass of the off-diagonal
gauge field is gauge dependent, it does not imply that this energy
scale itself is a gauge artifact.  The value of $\mad$ obtained in
\cite{Amemiya:1999zf} is, within error margins, not
far from the lightest glueball mass in $SU(2)$ YM,
$m^{0^{\scriptscriptstyle ++}}_{\scriptscriptstyle\rm G}\!\!\sim\!1.67$ 
GeV \cite{Langfeld:2000ej}. In $SU(3)$ YM
the value of $m^{0^{\scriptscriptstyle ++}}_{\scriptscriptstyle\rm G}$ 
is in the same range,
$m^{0^{\scriptscriptstyle ++}}_{\scriptscriptstyle\rm G}\!\!=\!1.6\!-\!1.8$ 
GeV \cite{Morningstar:1999rf}.
It would be interesting to look for signs of Abelian dominance at
scales below $m^{0^{\scriptscriptstyle ++}}_{\scriptscriptstyle\rm G}$.

The first step towards an effective Abelian theory is to integrate
over the off-diagonal vector fields $\phi_\mu$. Though there are
vertices with four of these fields, the $C_{\mu\nu}C_{\mu\nu}$ term
in~(\ref{newymaction}), the integration can be performed
by introducing an auxiliary antisymmetric tensor field, $B_{\mu\nu}$,
via the replacement, $\quarter \int_x C_{\mu\nu} C_{\mu\nu}\to - \quarter
  \int_x \tilde B_{\mu\nu} \tilde B_{\mu\nu} + \half \int_x \tilde
  B_{\mu\nu} C_{\mu\nu}$.
It follows that the equation of motion for $B_{\mu\nu}$ is
$\tilde B_{\mu\nu} = C_{\mu\nu}$. The resulting action is now
quadratic in the off-diagonal fields. By fixing the MAG parameter
$\xi^\prime = 1$, these quadratic terms in $\phi_\mu$ are
\begin{equation}
  \label{eq:matrix}
  S_{\phi^2} = \int_x \left(\phi_\mu^\dagger\ 
  a_{\mu\nu}^{-+}\ \phi^{}_\nu + \phi^{}_\mu\ a_{\mu\nu}^{+-}\
  \phi_\nu^\dagger + \phi^{}_\mu\ a_{\mu\nu}^{++}\ \phi^{}_\nu +
  \phi_\mu^\dagger\ a_{\mu\nu}^{--}\ \phi_\nu^\dagger\right),
\end{equation}
where $a_{\mu\nu}^{-+}\!=\!- \half g^{}_{\mu\nu} (D_{\rho}
  D_{\rho} + g^2\ \bar c_+ c_+) + \frac{ig}{2} (2f_{\mu\nu} 
  + \tilde B_{\mu\nu}) = 
  [a_{\mu\nu}^{+-}]^\dagger$ and $a_{\mu\nu}^{++} = g^2\,g^{}_{\mu\nu}\,
  \bar c_- c_+\, = \,[a_{\mu\nu}^{--}]^\dagger$.
After a Gaussian integration the effective action receives a
contribution of the form, $\half\,{\rm Tr} \ln\,a_{\mu\nu}^{SS'}$, with
${\scriptstyle S}, {\scriptstyle S'} = {\scriptstyle\pm}$\,. In 
a Schwinger proper-time representation using a complete basis of plane
waves and performing a Taylor series expansion on the proper-time $t$,
the trace takes the form \cite{Freire:2001nd}
\begin{eqnarray}
  \label{trace4}
  {\rm Tr}\ln\,a_{\mu\nu}^{SS'} =
  - \lim_{s \to0}\ \frac{d}{ds}\
   \frac{\mu^{2s}}{\Gamma(s)}{\displaystyle\int_0^\infty}\!\!\!dt\,t^{s-3}
  {\displaystyle\int_x}
  \sum_{n=0}^{\infty}\frac{\scriptstyle{(-1)}}{n!}^n\!\!\int_k e^{-\half k^2}
   \!\times\hspace{1cm}\nonumber\\
   {\rm tr}
  \Big[\left({\scriptstyle{t\,a_{\mu\nu}^{SS'} - i\,\sqrt{t}\,
  g^{}_{\mu\nu}(\,\bar\delta^{S+}\bar\delta^{S'\!\!-}\,k_\rho D_\rho 
  +\,\bar\delta^{S-}\bar\delta^{S'\!\!+}\,k_\rho
  D^\dagger_\rho)}}\right)^n\Big]\Big\vert_{{\rm reg.}}\!,
\end{eqnarray}
where {\rm tr} denotes a trace over all indices and $\bar\delta^{SS'}$
is the inverse Kronecker delta,
$\bar\delta^{SS'} = 1$ if ${\scriptstyle S} \neq {\scriptstyle S'}$ and
$\bar\delta^{SS'} = 0$ if ${\scriptstyle S} = {\scriptstyle S'}$.

The momentum integration in~(\ref{trace4}) is convergent but not
solvable. In order to proceed the Taylor series is truncated by
keeping only the leading terms up to $n=3$ inclusively in a small $t$
expansion. This provides a systematic 
approach for computing the leading effective vertices for short range
interactions by decreasing importance in their UV relevance.\\[.4cm]
\part{The Abelian effective action}
On the right-hand side of~(\ref{trace4}) only the integer powers of 
$t$ do not vanish after the integration over
the momenta, because the integrand for the half integer powers of $t$
is odd in the momenta. The $t$ integral is now splitted into two parts,
$\int_{_0}^{1/\Lambda^{\!2}}\!+\int_{1/\Lambda^{\!2}}^{^\infty}$.
The scale $\Lambda$ is a UV scale larger than $\mad$ chosen so that the
integration for $t > 1/\Lambda^{\!2}$ is suppressed by powers of
$\mad^2/\Lambda^{\!2}$.  Therefore, in the spirit of Abelian dominance
only the integration for $t < 1/\Lambda^{\!2}$ is kept.
After the integration, the expansion in powers of $t$ emerges
as an expansion in vertex operators of decreasing UV relevance. Here
only the terms up to the first non-relevant ones, i.e. ${\cal
  O}(1/\Lambda^{2})$ are kept. The resulting $U(1)$ invariant effective action,
where the renormalisation scale is set $\mu=\Lambda$, is
\begin{eqnarray}
  \label{adea}
  S_{\rm{eff}}\!\!\!&=&\!\!\!\quarter\left(1 - 
  \fracpow{5g^2\gamma}{12\pi}{2}\right)
  \int_x f_{\mu\nu}f_{\mu\nu} -
  \frac14 \int_x B_{\mu\nu} \left(\fracpow{g^2}{96\pi}{2}
  \fracpow{\Box}{\Lambda}{2} +
  1 + \fracpow{g^2\gamma}{8\pi}{2} \right)B_{\mu\nu} -\nonumber\\
&&-\ \fracpow{g^2\gamma}{8\pi}{2}\int_x\!\tilde
  f_{\mu\nu}\,B_{\mu\nu} + \frac{1}{2\xi}\int_x\!
  \left(\partial_\mu A_\mu\right)^2\!
  -\!\int_x \bar c_+\!\left(D_\mu^\dagger D_\mu^\dagger
  - \frac{g^2\Lambda^2}{2\pi^{{}^{\scriptscriptstyle2}}}\right) c_+ 
  -\nonumber\\
&&-\ \int_x \bar c_-\left(D_\mu D_\mu -
  \frac{g^2\Lambda^2}{2\pi^{{}^{\scriptscriptstyle2}}}\right)c_- 
  +\fracpow{g^4\gamma}{2\pi}{2}\int_x \bar c_+ c_+ \bar c_- c_-
   -\nonumber\\
&&-\ \fracpow{1}{\Lambda}{2}\ \fracpow{g^2}{96\pi}{2}\left(
  \int_x \tilde f_{\mu\nu}\Box B_{\mu\nu} +\
% + \frac{11}{10}\int_x
%   f_{\mu\nu}\Box f_{\mu\nu}\ + \right.\nonumber\\[.5cm]
%   & &+\ g^2 \int_x \left(\bar c_+ c_+\!+ \bar c_- c_-\right) 
%   (f_{\mu\nu}f_{\mu\nu} + \half B_{\mu\nu}B_{\mu\nu} +
%   2 \tilde f_{\mu\nu}B_{\mu\nu})\ +\nonumber\\[.5cm] 
%   & &\left.+\ 2g^2 \int_x (\bar c_- c_+)
%   D^{(2)}_\mu D^{(2)}_\mu\,(\bar c_+ c_-)
  \cdots\ \right) + 
  {\cal O}\left(\fracpow{1}{\Lambda}{4},\,B^4\right).
\end{eqnarray}
The one-loop coupling and wave function corrections are the terms 
proportional to the Euler gamma. Note that the   
${\cal O}(\Lambda^{2})$ terms correspond to a mass renormalisation. In a
more complete treatment where the hard modes of all fields
are integrated out to one-loop order down to $\Lambda$, the 
ghost mass terms are canceled in accordance to BRS invariance.

There are two new relevant vertices in~(\ref{adea}): the coupling
between the dual of the Abelian field strength and the tensor field,
and the 4-ghosts vertex. Kondo \cite{Kondo:1998pc} has shown by
using a Hodge decomposition of $B_{\mu\nu}$ that the
$\tilde f_{\mu\nu}\,B_{\mu\nu}$ term encapsulates the coupling between
a gauge field potential and a magnetic current
$J^{M}_\nu=\partial_\mu \tilde f_{\mu\nu}$. An effective theory of
this type reproduces the standard $\beta$-function for $g$
\cite{Kondo:1998pc,Quandt:1997rg}. Note that no term
involving $f_{\mu\nu}\,B_{\mu\nu}$ is generated at this order.

Finally, one of the UV irrelevant vertices, ${\cal O}(1/\Lambda^{2})$,
has been singled out. This corresponds to the term governing the
dynamics of the tensor field. In~(\ref{adea}) it was 
included alongside the relevant terms in the first line. This vertex
corresponds to the leading ghost free term in lowest order in a
derivative expansion. It will be discussed at greater length presently.

The effective action~(\ref{adea}) is in the spirit of
\cite{Weinberg:1980wa} an appropriate initial condition at the scale
$\Lambda$ for an ERG analysis of the low energy theory. Remember
that it has been required for $\Lambda$ to be in the perturbative
region of $SU(2)$ YM to ensure a reliable calculation of the
coefficients in~(\ref{adea}). In line with Abelian dominance it has
also been required that $\Lambda >> \mad$. There is another aspect to this last
requirement. It can be seen that when an explicit mass $\mad$ is 
introduced in the effective action the coefficients in~(\ref{adea}) 
receive corrections in powers of $\mad^{\,2}/\Lambda^2$ and therefore 
it is necessary to have $\Lambda \gg \mad$ for reliability.
Note that the scale $\mad$ is not to be confused with $\qcd$.
This are distinct though expected to be related scales and 
$\mad > \qcd$.\\[.4cm]
\part{Exact renormalisation group and confinement}
Now it is shown that qualitatively the Abelian effective
action~(\ref{adea}) contains the relevant degrees of freedom for
confinement. This is done within a functional analysis using ERG
methods. The action is now truncated in two ways.
First, the ghost sector is omitted as its inclusion would not change the
present qualitative results. Second, only terms up to the
quadratic order in a derivative expansion are retained, and therefore
the kinetic term for the tensor field is kept.

The ERG equations are flow equations for an effective action
$\Gamma_k$ with respect to the scale $k$. The $k$ dependence comes
from IR cut-off 
functions $R_k$ explicitly inserted in the action \cite{Reuter:1994kw}.
The resulting flow equation has the structure of a one-loop equation
\begin{eqnarray}
  \label{flow-eqn}
  \partial_t \Gamma_k = \half\,\Tr\left[\,{\partial_t R_k}
  \left(\Gamma_k^{(2)} + R_k\right)^{-1}\right],\hspace{.2cm}t=\ln k\,,
\end{eqnarray}
where $\Tr$ stands for the sum over all fields and group indices 
as well as the integration over the coordinate space, 
whilst $\Gamma^{(2)}_k$ is the fully dressed 1PI functional. 
The effective action~(\ref{adea}) suggests the following Ansatz,
\begin{eqnarray}
  \label{ansatz}
  \Gamma_k = \int_x\!\Big\{\frac{Z_{\!A}}{4}f_{\mu\nu}^2 +
  \frac{1}{4}(-Z_{\!B}\,\Box + M_{\!B}^2)\,B_{\mu\nu}^2 +
  \frac{Y}{2}\,\tilde f_{\mu\nu} B_{\mu\nu}
  +\frac{1}{2\xi}\left(\partial_\mu A_\mu \right)^2 \Big\},
\end{eqnarray}
where $Z_{\!A}$, $Z_{\!B}$, $M_{\!B}^2$, $Y$ and $\xi$ are
$k$ dependent functions. The use of this Ansatz reduces the complex
equation~(\ref{flow-eqn}) to a set of finite treatable coupled equations.
The flow connects $\Gamma_{\Lambda}$ to $\Gamma_0$, the full
effective action. Of course, a necessary condition is that the IR cut-off 
functions vanish as $k\to0$.

Next, it is shown that this Ansatz contains enough information to
exhibit a confining phase at low energies. A signature for confinement
is sought in the form of a $1/p^4$ singularity in the gauge field 
propagator, a feature that has been shown to
lead to an area law in a Wilson loop \cite{West:1982bt}.

The initial condition for the ERG flow is assigned by taking
$\Gamma_{\Lambda} \approx S_{\rm eff}$. For the new coefficient
$Z_{\!B}$ this reduces to $Z_{\!B}(\Lambda) \approx 0$ which
simply reflects the fact that $B_{\mu\nu}$ was introduced as an
auxiliary field.  Note also from~(\ref{adea}) that $M_{\!B}^2(\Lambda) < 0$.
Only a more complete treatment of the flow equations including the
ghost sector can provide information on whether $M_{\!B}^2$ becomes
positive at low energies. For the present purpose, it is sufficient to
assume that near the confinement scale $M_{\!B}^2 > 0$.

{From} the Ansatz~(\ref{ansatz}) the
propagators can be expressed as functionals of the renormalisation
functions. The gauge field propagator is
\begin{equation}
  \label{eq:propagators}
  \left(\pg{A}A\right)_{\mu\nu} =
  \left(\delta_{\mu\nu} - \frac{p_\mu p_\nu}{p^2}\right)
  \frac{Z_{\!B}\,p^2 + M_{\!B}^2}{Z_{\!A}Z_{\!B}\,p^4 +
  (M_{\!B}^2 Z_{\!A} - Y^2)\,p^2} +
  \xi\ \frac{p_\mu p_\nu}{p^4}\,.
\end{equation}
As $\xi\to0$ at low energies, the denominator of the
propagator~(\ref{eq:propagators}) is dominated by the
$p^4$ term if
\begin{equation}
  \label{fixedpoint}
  M_{\!B}^2(k)\,Z_{\!A}(k) = Y(k)^2.
\end{equation}
Then the gauge field propagator has a $1/p^4$ singularity when
\begin{eqnarray}
  \label{eq:qcd}
  Z_{\!B}(k_c)\,p^2 \ll M_{\!B}^2(k_c) \ \ \ \Rightarrow \ \ \ \sqrt{p^2} \ll
  \frac{M_{\!B}(k_c)}{\sqrt{Z_{\!B}(k_c)}}:= \qcd.
\end{eqnarray}
It is expected that~(\ref{fixedpoint}) is an IR stable quasi-fixed
point as in the approximations considered in \cite{Ellwanger:1998wv}.

In a recent work, Ellwanger and Wschebor \cite{Ellwanger:2001yy}, have
found that for an effective gauge theory where it is assumed that the
charged fields as well as the ghosts have been integrated over, the
equivalent to condition~(\ref{fixedpoint}) is relaxed to an inequality
which in the present case would read 
$M_{\!B}^2(k)\,Z_{\!A}(k) - Y(k)^2 < 0$, for $k>0$. The equality is
recovered in the IR limit $k\to0$.\\[.4cm]
\part{Confinement and the dynamics of the $B_{\mu\nu}$ field}
The IR $1/p^4$ singularity occurs because:
(a) there is a term proportional to $p^4$ in the denominator of the
propagator that becomes prominent when the condition~(\ref{fixedpoint}) holds; 
(b) at small momentum, when~(\ref{eq:qcd}) is fulfilled, the
numerator becomes $p^2$ independent. The proportionality factor to
$p^4$ is $Z_{\!A} Z_{\!B}$.  Clearly, for this factor to be non zero
the $B_{\mu\nu}$ field must become dynamical at low energies. 
Therefore, somewhere in between the
deep UV region $k\simeq\Lambda$ and the confinement scale $\qcd$,
the tensor field kinetic vertex must undergo a crossover that will
make it relevant in the IR. This crossover scale is expected to be
linked to the Abelian dominance scale $\mad$. 
Above this scale the dynamics of $B_{\mu\nu}$ 
is protected by the still unsuppressed $\phi_{\mu}$ fields by
${\cal{O}}(1/\Lambda^{2})$. However, below $\mad$ the off-diagonal fields 
are suppressed and are counterbalanced by the Abelian tensor
field.  Consequently, $Z_{\!B}$ becomes equally relevant in the
Abelian dominated regime when compared with the other renormalisation
functions which makes the above scenario leading
to a $1/p^4$ behaviour plausible. Polonyi's view of confinement as
an irrelevant-to-relevant crossover \cite{Polonyi:1995jz} of some UV
irrelevant operator is in line with the present scenario.\\[.4cm]
\part{Duality in the confinement phase}
In the confining regime when conditions~(\ref{fixedpoint})
and~(\ref{eq:qcd}) are satisfied the effective action reduces to
\begin{equation}
  \label{eq:conf-action}
  \Gamma_{\rm conf} = \int_x \Big\{\frac{1}{4Z_{\!A}}(Z_{\!A} f_{\mu\nu} +
  Y\tilde B_{\mu\nu})^2 + \frac{Z_{\!B}}{4}
  (\partial_{\rho} \tilde B_{\mu\nu})^2\Big\}.
\end{equation}
Now consider the transformation $(a_\mu,B_{\mu\nu}) \to (b_\mu,\theta)$
defined implicitly by
\begin{eqnarray}
  \label{eq:dual-transform}
  b_{\mu\nu}=\frac{\ep_{\mu\nu\rho\sigma}}{2\sqrt{Z_{\!A}}}\,
  (Z_{\!A}f_{\rho\sigma} + Y \tilde B_{\rho\sigma}),\,\,
  \partial_\sigma \theta + \sqrt\frac{3\,Y^2}{Z_{\!A}Z_{\!B}}\,b_{\sigma}
  = \sqrt\frac{Z_{\!B}}{12}\,\ep_{\mu\nu\rho\sigma}\, 
  \partial_\mu \tilde B_{\nu\rho},
\end{eqnarray}
where $b_{\mu\nu} = \partial_\mu b_\nu - \partial_\nu b_\mu$.
In terms of the new variables the confining
action~(\ref{eq:conf-action}) becomes
\begin{equation}
  \label{eq:dual-action}
  \tilde \Gamma_{\rm conf} = \int_x \Big\{\frac14\,b_{\mu\nu}^2 + \half\,
  \Big(\partial_\mu \theta + \sqrt\frac{3\,Y^2}{Z_{\!A}Z_{\!B}}\,\,b_{\mu}
  \Big)^2 \Big\}.
\end{equation}
By substituting $\Phi = \rho\,e^{-i\theta}$ with $\rho$ frozen above, the
confining action~(\ref{eq:dual-action}) is shown to describe
a type-II dual superconductor with dual London penetration depth
$\lambda_b \sim \sqrt{Z_{\!A}Z_{\!B}}/Y$. This action is analogue to
the one found in \cite{Ellwanger:1998wv}.

The change of variables~(\ref{eq:dual-transform}) is a dual transformation
in the usual sense that the equations of motion for $(a_\mu,B_{\mu\nu})$
become Bianchi identities for $(b_\mu,\theta)$ and vice-versa.
Note that the mismatch in degrees of freedom arises from an
invariance in the transformation~(\ref{eq:dual-transform}) with 
respect to the shift
\begin{equation}
  \label{eq:shift}
  a_\mu \to a_\mu + \ep_\mu,\,\,\tilde B_{\mu\nu} \to \tilde
  B_{\mu\nu} - \frac{Z_{\!A}}{Y}\,(\partial_\mu\ep_\nu - \partial_\nu\ep_\mu)
\end{equation}
for arbitrary $\ep_\mu$. This type of shift is used to generate
monopole configurations in Abelian theories on the lattice.\\[.4cm]
\part{Summary and discussion}
The analytic approach described here provides a promising framework
to study the low energy confining regime of non-Abelian gauge
theories. To bring it to a more quantitative level further research 
is necessary. ERG methods have been
combined with the benefits of working in a MAG to study the monopole
mechanism for confinement in $SU(2)$ YM. As Abelian dominance has been
observed on the lattice with a MAG gauge fixing \cite{Suzuki:1990gp}
it is expected that that an intermediate Abelian dominance scale
$\mad > \qcd$ is dynamically generated. Then starting from a pure 
$SU(2)$ YM theory a perturbative Abelian effective action was derived. 
Because there is a scale separation in a MAG this action is suitable 
to be integrated using ERG equations. It is shown that under certain conditions
confinement occurs if the ERG flow at low energies settles about an IR
stable quasi-fixed point. Confinement hinges on an
irrelevant-to-relevant crossover for the kinetic term of the tensor
field and it is expected that the crossover takes place at the scale of
Abelian dominance $\mad$. Finally, it was shown that in the confining 
phase the effective action describes the physics of a type II magnetic
Abelian Higgs system in the broken phase.\\[.4cm]
\part{Acknowledgements} The author is grateful to Jan
Pawlowski and Denjoe O'Connor for
useful discussions and \v{S}tefan Olejn\'{\i}k and Jeff Greensite for putting
together such an interesting workshop.\\[-.5cm]
 
\begin{thebibliography}{99}
%\cite{Nambu:1974zg}
\bibitem{Nambu:1974zg}
Y.~Nambu,
%``Strings, Monopoles, And Gauge Fields,''
Phys.\ Rev.\ D {\bf 10} (1974) 4262.
%%CITATION = PHRVA,D10,4262;%%

%\cite{Mandelstam:1976pi}
\bibitem{Mandelstam:1976pi}
S.~Mandelstam,
%``Vortices And Quark Confinement In Nonabelian Gauge Theories,''
Phys.\ Rept.\  {\bf 23} (1976) 245.
%%CITATION = PRPLC,23,245;%%

%\cite{'tHooft:1981ht}
\bibitem{'tHooft:1981ht}
G.\,'t\,Hooft,
%``Topology Of The Gauge Condition And New Confinement Phases In
%Nonabelian Gauge Theories,''
Nucl.\ Phys.\ B {\bf 190} (1981) 455.
%%CITATION = NUPHA,B190,455;%%

%\cite{Suzuki:1990gp}
\bibitem{Suzuki:1990gp}
T.~Suzuki and I.~Yotsuyanagi,
%``A Possible Evidence For Abelian Dominance In Quark Confinement,''
Phys.\ Rev.\ D {\bf 42} (1990) 4257;
%%CITATION = PHRVA,D42,4257;%%
%\cite{Hioki:1991ai}
%\bibitem{Hioki:1991ai}
S.~Hioki, S.~Kitahara, S.~Kiura, Y.~Matsubara, O.~Miyamura, S.~Ohno
and T.~Suzuki, 
%``Abelian dominance in SU(2) color confinement,''
Phys.\ Lett.\ B {\bf 272} (1991) 326
[Erratum-ibid.\ B {\bf 281} (1991) 416].
%%CITATION = PHLTA,B272,326;%%

%\cite{Reuter:1994kw}
\bibitem{Reuter:1994kw}
M.~Reuter and C.~Wetterich,
%``Effective average action for gauge theories and exact evolution equations,''
Nucl.\ Phys.\ B {\bf 417} (1994) 181;
%\cite{Ellwanger:1994iz}
%\bibitem{Ellwanger:1994iz}
U.~Ellwanger,
%``Flow equations and BRS invariance for Yang-Mills theories,''
Phys.\ Lett.\ B {\bf 335} (1994) 364
[hep-th/9402077];
%\cite{Bonini:1994sj}
%\bibitem{Bonini:1994sj}x
M.~Bonini, M.~D'Attanasio and G.~Marchesini,
%``Renormalization group flow for SU(2) Yang-Mills theory and 
% gauge invariance,'' 
Nucl.\ Phys.\ B {\bf 421} (1994) 429
[hep-th/9312114].
%%CITATION = HEP-TH 9312114;%%

%\cite{Litim:1998qi}
\bibitem{Litim:1998qi}
D.~F.~Litim and J.~M.~Pawlowski,
%``Flow equations for Yang-Mills theories in general axial gauges,''
Phys.\ Lett.\ B {\bf 435} (1998) 181
[hep-th/9802064].
%%CITATION = HEP-TH 9802064;%%

%\cite{Ellwanger:1998wv}
\bibitem{Ellwanger:1998wv}
U.~Ellwanger,
%``Confinement, monopoles and Wilsonian effective action,''
Nucl.\ Phys.\ B {\bf 531} (1998) 593
[hep-ph/9710326];
%%CITATION = HEP-PH 9710326;%%
%\cite{Ellwanger:1999th}
%\bibitem{Ellwanger:1999th}
%U.~Ellwanger,
%``Field strength correlator and an infrared fixed point of the
%Wilsonian  exact renormalization group equations,''
Eur.\ Phys.\ J.\ C {\bf 7} (1999) 673
[hep-ph/9807380];
%%CITATION = HEP-PH 9807380;%%
%\cite{Ellwanger:1999vc}
%\bibitem{Ellwanger:1999vc}
%U.~Ellwanger,
%``Monopole condensation and antisymmetric tensor fields: Compact QED
%and  the Wilsonian RG flow in Yang-Mills theories,''
Nucl.\ Phys.\ B {\bf 560} (1999) 587
[hep-th/9906061].
%%CITATION = HEP-TH 9906061;%%

\bibitem{Litim:1998nf}
D.~F.~Litim and J.~M.~Pawlowski,
%``On gauge invariant Wilsonian flows''
in Proceedings of the Workshop on the Exact Renormalisation
Group, Faro, Portugal, Sep 1998, pgs. 168-185 [hep-th/9901063];
%%CITATION = HEP-TH 9901063;%%
%\cite{Litim:1999wk}
%\bibitem{Litim:1999wk}
%D.~F.~Litim and J.~M.~Pawlowski,
%``On gauge invariance and Ward identities for the Wilsonian
% renormalisation group,''
Nucl.\ Phys.\ Proc.\ Suppl.\  {\bf 74} (1999) 325
[hep-th/9809020].
%%CITATION = HEP-TH 9809020;%%
%\cite{Litim:1998nf}

%\cite{Freire:1996db}
\bibitem{Freire:1996db}
F.~Freire and C.~Wetterich,
%``The Ward Identity from the Background Field Dependence of the
%Effective Action,''
Phys.\ Lett.\ B {\bf 380} (1996) 337
[hep-th/9601081];
%\cite{Freire:2000bq}
%\bibitem{Freire:2000bq}
F.~Freire, D.~F.~Litim and J.~M.~Pawlowski,
%``Gauge invariance and background field formalism in the exact
% renormalisation group,''
Phys.\ Lett.\ B {\bf 495} (2000) 256
[hep-th/0009110].
%%CITATION = HEP-TH 0009110;%%

%\cite{Freire:2001nd}
\bibitem{Freire:2001nd}
F.~Freire,
%``Abelian projected SU(2) Yang-Mills action for renormalisation 
% group  flows,''
Phys.\ Lett.\ B {\bf 526} (2002) 405
[hep-th/0110241]
%%CITATION = HEP-TH 0110241;%%

%\cite{Hart:1997ar}
\bibitem{Hart:1997ar}
A.~Hart and M.~Teper,
%``Gribov copies in the maximally Abelian gauge and confinement,''
Phys.\ Rev.\ D {\bf 55} (1997) 3756
[hep-lat/9606007].
%%CITATION = HEP-LAT 9606007;%%

%\cite{Schaden:1999ew}
\bibitem{Schaden:1999ew}
M.~Schaden,
``Mass generation in continuum SU(2) gauge theory in covariant
Abelian  gauges'',
hep-th/9909011;
%%CITATION = HEP-TH 9909011;%%
%\cite{Schaden:2001xu}
%\bibitem{Schaden:2001xu}
%M.~Schaden,
``Mass generation, ghost condensation and broken symmetry: $SU(2)$ in
covariant Abelian gauges'',
hep-th/0108034.
%%CITATION = HEP-TH 0108034;%%

%\cite{Kondo:2000ey}
\bibitem{Kondo:2000ey}
K.~I.~Kondo and T.~Shinohara,
%``Abelian dominance in low-energy gluodynamics due to dynamical mass
%generation,''
Phys.\ Lett.\ B {\bf 491} (2000) 263
[hep-th/0004158].
%%CITATION = HEP-TH 0004158;%%

%\cite{Gubarev:2000nz}
\bibitem{Gubarev:2000nz}
F.~V.~Gubarev and V.~I.~Zakharov,
%``On the emerging phenomenology of <(A(a)(mu))**2(min)>,''
Phys.\ Lett.\ B {\bf 501} (2001) 28
[hep-ph/0010096].
%%CITATION = HEP-PH 0010096;%%

%\cite{Dudal:2002aj}
\bibitem{Dudal:2002aj}
D.~Dudal, K.~van Acoleyen and H.~Verschelde,
%``Dynamical mass generation in quantum field theory : some methods with application to the Gross-Neveu model and Yang-Mills theory,''
in this Proceedings [hep-th/0204216].
%%CITATION = HEP-TH 0204216;%%

%\cite{Amemiya:1999zf}
\bibitem{Amemiya:1999zf}
K.~Amemiya and H.~Suganuma,
%``Off diagonal gluon mass generation and infrared Abelian dominance
%in the maximally Abelian gauge in lattice QCD,''
Phys.\ Rev.\ D {\bf 60} (1999) 114509
[hep-lat/9811035];
%%CITATION = PHRVA,D60,114509;%%
%\cite{Amemiya:2000yk}
%\bibitem{Amemiya:2000yk}
%K.~Amemiya and H.~Suganuma,
%``Effective mass generation of off-diagonal gluons and Abelian
%dominance  in the maximally Abelian gauge in QCD,''
Nucl.\ Phys.\ Proc.\ Suppl.\  {\bf 83} (2000) 419
[hep-lat/9909096].
%%CITATION = HEP-LAT 9909096;%%

%\cite{Langfeld:2000ej}
\bibitem{Langfeld:2000ej}
K.~Langfeld and A.~Sch\"afke,
%``Vortex dominance of the 0+ and 2+ glueball mass in SU(2) lattice
%gauge  theory,''
Phys.\ Lett.\ B {\bf 493} (2000) 350
[hep-lat/0008023].
%%CITATION = HEP-LAT 0008023;%%

%\cite{Morningstar:1999rf}
\bibitem{Morningstar:1999rf}
C.~Morningstar and M.~Peardon,
%``The glueball spectrum from an anisotropic lattice study,''
Phys.\ Rev.\ D {\bf 60} (1999) 034509
[hep-lat/9901004];
%%CITATION = HEP-LAT 9901004;%%
%\cite{Vaccarino:1999ku}
%\bibitem{Vaccarino:1999ku}
A.~Vaccarino and D.~Weingarten,
%``Glueball mass predictions of the valence approximation to lattice QCD,''
Phys.\ Rev.\ D {\bf 60} (1999) 114501
[hep-lat/9910007].
%%CITATION = HEP-LAT 9910007;%%

%\cite{Kondo:1998pc}
\bibitem{Kondo:1998pc}
K.~Kondo,
%``Abelian-projected effective gauge theory of {QCD} with asymptotic
%freedom  and quark confinement,''
Phys.\ Rev.\ D {\bf 57} (1998) 7467
[hep-th/9709109];
%%CITATION = HEP-TH 9709109;%%
%\cite{Kondo:2001zv}
%\bibitem{Kondo:2001zv}
K.~Kondo and T.~Shinohara,
%``Renormalizable Abelian-projected effective gauge theory derived
%from  quantum chromodynamics,''
Prog.\ Theor.\ Phys.\ {\bf 105} (2001) 649
[hep-th/0005125];
%%CITATION = HEP-TH 0005125;%%
%\cite{Shinohara:2001cw}
%\bibitem{Shinohara:2001cw}
T.~Shinohara, T.~Imai and K.~Kondo,
%``The most general and renormalizable maximal Abelian gauge,''
hep-th/0105268.
%%CITATION = HEP-TH 0105268;%%

%\cite{Quandt:1997rg}
\bibitem{Quandt:1997rg}
M.~Quandt and H.~Reinhardt,
%``Field strength formulation of SU(2) Yang-Mills theory in the
%maximal  Abelian gauge: Perturbation theory,'' 
Int.\ J.\ Mod.\ Phys.\ A {\bf 13} (1998) 4049
[hep-th/9707185].
%%CITATION = HEP-TH 9707185;%%

%\cite{Weinberg:1980wa}
\bibitem{Weinberg:1980wa}
S.~Weinberg,
%``Effective Gauge Theories,''
Phys.\ Lett.\ B {\bf 91} (1980) 51.
%%CITATION = PHLTA,B91,51;%%

%\cite{West:1982bt}
\bibitem{West:1982bt}
G.~B.~West,
%``Confinement, The Wilson Loop And The Gluon Propagator,''
Phys.\ Lett.\ B {\bf 115} (1982) 468.
%%CITATION = PHLTA,B115,468;%%

%\cite{Ellwanger:2001yy}
\bibitem{Ellwanger:2001yy}
U.~Ellwanger and N.~Wschebor,
%``Solvable confining gauge theories at large N,''
Phys.\ Lett.\ B {\bf 517} (2001) 462
[hep-th/0107093];
%%CITATION = HEP-TH 0107093;%%
%\cite{Ellwanger:2001yr}
%\bibitem{Ellwanger:2001yr}
%U.~Ellwanger and N.~Wschebor,
%``Confinement with Kalb-Ramond fields,''
JHEP {\bf 0110} (2001) 023
[hep-th/0107196].
%%CITATION = HEP-TH 0107196;%%

%\cite{Polonyi:1995jz}
\bibitem{Polonyi:1995jz}
J.~Polonyi,
``Confinement and renormalization'', hep-ph/9511243;
%%CITATION = HEP-PH 9511243;%%
%\cite{Polonyi:2000mc}
%\bibitem{Polonyi:2000mc}
%J.~Polonyi,
``Confinement as crossover'', hep-ph/0012265.
%%CITATION = HEP-PH 0012265;%%
\end{thebibliography}
\end{document}

%%% Local Variables: 
%%% mode: latex
%%% TeX-master: t
%%% End: 




hep-th0205032.extracted_bib
\begin{thebibliography}{99}
%\cite{Nambu:1974zg}
\bibitem{Nambu:1974zg}
Y.~Nambu,
%``Strings, Monopoles, And Gauge Fields,''
Phys.\ Rev.\ D {\bf 10} (1974) 4262.
%%CITATION = PHRVA,D10,4262;%%

%\cite{Mandelstam:1976pi}
\bibitem{Mandelstam:1976pi}
S.~Mandelstam,
%``Vortices And Quark Confinement In Nonabelian Gauge Theories,''
Phys.\ Rept.\  {\bf 23} (1976) 245.
%%CITATION = PRPLC,23,245;%%

%\cite{'tHooft:1981ht}
\bibitem{'tHooft:1981ht}
G.\,'t\,Hooft,
%``Topology Of The Gauge Condition And New Confinement Phases In
%Nonabelian Gauge Theories,''
Nucl.\ Phys.\ B {\bf 190} (1981) 455.
%%CITATION = NUPHA,B190,455;%%

%\cite{Suzuki:1990gp}
\bibitem{Suzuki:1990gp}
T.~Suzuki and I.~Yotsuyanagi,
%``A Possible Evidence For Abelian Dominance In Quark Confinement,''
Phys.\ Rev.\ D {\bf 42} (1990) 4257;
%%CITATION = PHRVA,D42,4257;%%
%\cite{Hioki:1991ai}
%\bibitem{Hioki:1991ai}
S.~Hioki, S.~Kitahara, S.~Kiura, Y.~Matsubara, O.~Miyamura, S.~Ohno
and T.~Suzuki, 
%``Abelian dominance in SU(2) color confinement,''
Phys.\ Lett.\ B {\bf 272} (1991) 326
[Erratum-ibid.\ B {\bf 281} (1991) 416].
%%CITATION = PHLTA,B272,326;%%

%\cite{Reuter:1994kw}
\bibitem{Reuter:1994kw}
M.~Reuter and C.~Wetterich,
%``Effective average action for gauge theories and exact evolution equations,''
Nucl.\ Phys.\ B {\bf 417} (1994) 181;
%\cite{Ellwanger:1994iz}
%\bibitem{Ellwanger:1994iz}
U.~Ellwanger,
%``Flow equations and BRS invariance for Yang-Mills theories,''
Phys.\ Lett.\ B {\bf 335} (1994) 364
[hep-th/9402077];
%\cite{Bonini:1994sj}
%\bibitem{Bonini:1994sj}x
M.~Bonini, M.~D'Attanasio and G.~Marchesini,
%``Renormalization group flow for SU(2) Yang-Mills theory and 
% gauge invariance,'' 
Nucl.\ Phys.\ B {\bf 421} (1994) 429
[hep-th/9312114].
%%CITATION = HEP-TH 9312114;%%

%\cite{Litim:1998qi}
\bibitem{Litim:1998qi}
D.~F.~Litim and J.~M.~Pawlowski,
%``Flow equations for Yang-Mills theories in general axial gauges,''
Phys.\ Lett.\ B {\bf 435} (1998) 181
[hep-th/9802064].
%%CITATION = HEP-TH 9802064;%%

%\cite{Ellwanger:1998wv}
\bibitem{Ellwanger:1998wv}
U.~Ellwanger,
%``Confinement, monopoles and Wilsonian effective action,''
Nucl.\ Phys.\ B {\bf 531} (1998) 593
[hep-ph/9710326];
%%CITATION = HEP-PH 9710326;%%
%\cite{Ellwanger:1999th}
%\bibitem{Ellwanger:1999th}
%U.~Ellwanger,
%``Field strength correlator and an infrared fixed point of the
%Wilsonian  exact renormalization group equations,''
Eur.\ Phys.\ J.\ C {\bf 7} (1999) 673
[hep-ph/9807380];
%%CITATION = HEP-PH 9807380;%%
%\cite{Ellwanger:1999vc}
%\bibitem{Ellwanger:1999vc}
%U.~Ellwanger,
%``Monopole condensation and antisymmetric tensor fields: Compact QED
%and  the Wilsonian RG flow in Yang-Mills theories,''
Nucl.\ Phys.\ B {\bf 560} (1999) 587
[hep-th/9906061].
%%CITATION = HEP-TH 9906061;%%

\bibitem{Litim:1998nf}
D.~F.~Litim and J.~M.~Pawlowski,
%``On gauge invariant Wilsonian flows''
in Proceedings of the Workshop on the Exact Renormalisation
Group, Faro, Portugal, Sep 1998, pgs. 168-185 [hep-th/9901063];
%%CITATION = HEP-TH 9901063;%%
%\cite{Litim:1999wk}
%\bibitem{Litim:1999wk}
%D.~F.~Litim and J.~M.~Pawlowski,
%``On gauge invariance and Ward identities for the Wilsonian
% renormalisation group,''
Nucl.\ Phys.\ Proc.\ Suppl.\  {\bf 74} (1999) 325
[hep-th/9809020].
%%CITATION = HEP-TH 9809020;%%
%\cite{Litim:1998nf}

%\cite{Freire:1996db}
\bibitem{Freire:1996db}
F.~Freire and C.~Wetterich,
%``The Ward Identity from the Background Field Dependence of the
%Effective Action,''
Phys.\ Lett.\ B {\bf 380} (1996) 337
[hep-th/9601081];
%\cite{Freire:2000bq}
%\bibitem{Freire:2000bq}
F.~Freire, D.~F.~Litim and J.~M.~Pawlowski,
%``Gauge invariance and background field formalism in the exact
% renormalisation group,''
Phys.\ Lett.\ B {\bf 495} (2000) 256
[hep-th/0009110].
%%CITATION = HEP-TH 0009110;%%

%\cite{Freire:2001nd}
\bibitem{Freire:2001nd}
F.~Freire,
%``Abelian projected SU(2) Yang-Mills action for renormalisation 
% group  flows,''
Phys.\ Lett.\ B {\bf 526} (2002) 405
[hep-th/0110241]
%%CITATION = HEP-TH 0110241;%%

%\cite{Hart:1997ar}
\bibitem{Hart:1997ar}
A.~Hart and M.~Teper,
%``Gribov copies in the maximally Abelian gauge and confinement,''
Phys.\ Rev.\ D {\bf 55} (1997) 3756
[hep-lat/9606007].
%%CITATION = HEP-LAT 9606007;%%

%\cite{Schaden:1999ew}
\bibitem{Schaden:1999ew}
M.~Schaden,
``Mass generation in continuum SU(2) gauge theory in covariant
Abelian  gauges'',
hep-th/9909011;
%%CITATION = HEP-TH 9909011;%%
%\cite{Schaden:2001xu}
%\bibitem{Schaden:2001xu}
%M.~Schaden,
``Mass generation, ghost condensation and broken symmetry: $SU(2)$ in
covariant Abelian gauges'',
hep-th/0108034.
%%CITATION = HEP-TH 0108034;%%

%\cite{Kondo:2000ey}
\bibitem{Kondo:2000ey}
K.~I.~Kondo and T.~Shinohara,
%``Abelian dominance in low-energy gluodynamics due to dynamical mass
%generation,''
Phys.\ Lett.\ B {\bf 491} (2000) 263
[hep-th/0004158].
%%CITATION = HEP-TH 0004158;%%

%\cite{Gubarev:2000nz}
\bibitem{Gubarev:2000nz}
F.~V.~Gubarev and V.~I.~Zakharov,
%``On the emerging phenomenology of <(A(a)(mu))**2(min)>,''
Phys.\ Lett.\ B {\bf 501} (2001) 28
[hep-ph/0010096].
%%CITATION = HEP-PH 0010096;%%

%\cite{Dudal:2002aj}
\bibitem{Dudal:2002aj}
D.~Dudal, K.~van Acoleyen and H.~Verschelde,
%``Dynamical mass generation in quantum field theory : some methods with application to the Gross-Neveu model and Yang-Mills theory,''
in this Proceedings [hep-th/0204216].
%%CITATION = HEP-TH 0204216;%%

%\cite{Amemiya:1999zf}
\bibitem{Amemiya:1999zf}
K.~Amemiya and H.~Suganuma,
%``Off diagonal gluon mass generation and infrared Abelian dominance
%in the maximally Abelian gauge in lattice QCD,''
Phys.\ Rev.\ D {\bf 60} (1999) 114509
[hep-lat/9811035];
%%CITATION = PHRVA,D60,114509;%%
%\cite{Amemiya:2000yk}
%\bibitem{Amemiya:2000yk}
%K.~Amemiya and H.~Suganuma,
%``Effective mass generation of off-diagonal gluons and Abelian
%dominance  in the maximally Abelian gauge in QCD,''
Nucl.\ Phys.\ Proc.\ Suppl.\  {\bf 83} (2000) 419
[hep-lat/9909096].
%%CITATION = HEP-LAT 9909096;%%

%\cite{Langfeld:2000ej}
\bibitem{Langfeld:2000ej}
K.~Langfeld and A.~Sch\"afke,
%``Vortex dominance of the 0+ and 2+ glueball mass in SU(2) lattice
%gauge  theory,''
Phys.\ Lett.\ B {\bf 493} (2000) 350
[hep-lat/0008023].
%%CITATION = HEP-LAT 0008023;%%

%\cite{Morningstar:1999rf}
\bibitem{Morningstar:1999rf}
C.~Morningstar and M.~Peardon,
%``The glueball spectrum from an anisotropic lattice study,''
Phys.\ Rev.\ D {\bf 60} (1999) 034509
[hep-lat/9901004];
%%CITATION = HEP-LAT 9901004;%%
%\cite{Vaccarino:1999ku}
%\bibitem{Vaccarino:1999ku}
A.~Vaccarino and D.~Weingarten,
%``Glueball mass predictions of the valence approximation to lattice QCD,''
Phys.\ Rev.\ D {\bf 60} (1999) 114501
[hep-lat/9910007].
%%CITATION = HEP-LAT 9910007;%%

%\cite{Kondo:1998pc}
\bibitem{Kondo:1998pc}
K.~Kondo,
%``Abelian-projected effective gauge theory of {QCD} with asymptotic
%freedom  and quark confinement,''
Phys.\ Rev.\ D {\bf 57} (1998) 7467
[hep-th/9709109];
%%CITATION = HEP-TH 9709109;%%
%\cite{Kondo:2001zv}
%\bibitem{Kondo:2001zv}
K.~Kondo and T.~Shinohara,
%``Renormalizable Abelian-projected effective gauge theory derived
%from  quantum chromodynamics,''
Prog.\ Theor.\ Phys.\ {\bf 105} (2001) 649
[hep-th/0005125];
%%CITATION = HEP-TH 0005125;%%
%\cite{Shinohara:2001cw}
%\bibitem{Shinohara:2001cw}
T.~Shinohara, T.~Imai and K.~Kondo,
%``The most general and renormalizable maximal Abelian gauge,''
hep-th/0105268.
%%CITATION = HEP-TH 0105268;%%

%\cite{Quandt:1997rg}
\bibitem{Quandt:1997rg}
M.~Quandt and H.~Reinhardt,
%``Field strength formulation of SU(2) Yang-Mills theory in the
%maximal  Abelian gauge: Perturbation theory,'' 
Int.\ J.\ Mod.\ Phys.\ A {\bf 13} (1998) 4049
[hep-th/9707185].
%%CITATION = HEP-TH 9707185;%%

%\cite{Weinberg:1980wa}
\bibitem{Weinberg:1980wa}
S.~Weinberg,
%``Effective Gauge Theories,''
Phys.\ Lett.\ B {\bf 91} (1980) 51.
%%CITATION = PHLTA,B91,51;%%

%\cite{West:1982bt}
\bibitem{West:1982bt}
G.~B.~West,
%``Confinement, The Wilson Loop And The Gluon Propagator,''
Phys.\ Lett.\ B {\bf 115} (1982) 468.
%%CITATION = PHLTA,B115,468;%%

%\cite{Ellwanger:2001yy}
\bibitem{Ellwanger:2001yy}
U.~Ellwanger and N.~Wschebor,
%``Solvable confining gauge theories at large N,''
Phys.\ Lett.\ B {\bf 517} (2001) 462
[hep-th/0107093];
%%CITATION = HEP-TH 0107093;%%
%\cite{Ellwanger:2001yr}
%\bibitem{Ellwanger:2001yr}
%U.~Ellwanger and N.~Wschebor,
%``Confinement with Kalb-Ramond fields,''
JHEP {\bf 0110} (2001) 023
[hep-th/0107196].
%%CITATION = HEP-TH 0107196;%%

%\cite{Polonyi:1995jz}
\bibitem{Polonyi:1995jz}
J.~Polonyi,
``Confinement and renormalization'', hep-ph/9511243;
%%CITATION = HEP-PH 9511243;%%
%\cite{Polonyi:2000mc}
%\bibitem{Polonyi:2000mc}
%J.~Polonyi,
``Confinement as crossover'', hep-ph/0012265.
%%CITATION = HEP-PH 0012265;%%
\end{thebibliography}