\section{Introduction}

Four-dimensional $SU(N)$ gauge theory at zero temperature is known to be
in a confining phase for all values of the bare coupling.
A very large amount of work has been performed over the last decade
in an effort to isolate the types of configurations in the
functional measure responsible
for maintaining one confining phase for arbitrarily weak coupling
\cite{Rev}, \cite{LAT}.
Nevertheless, a direct derivation
of this unique
feature of $SU(N)$ theories (shared only by
non-abelian ferromagnetic spin systems in $2$ dimensions)
has remained elusive.

The origin of the difficulty is clear. It is the multi-scale nature of
the problem: passage from a short distance ordered regime, where weak
coupling perturbation theory is applicable,
to a long distance strongly coupled disordered regime, where
confinement and other collective phenomena emerge.
Systems involving such dramatic change in physical
behavior over different scales are hard to treat. Hydrodynamic turbulence,
involving passage from laminar to turbulent flow, is another
well-known example,
which, in fact, shares some striking qualitative features with the
confining QCD vacuum.

The natural framework for addressing the problem from first principles
is a Wilsonian renormalization group (RG) block-spinning procedure bridging
short to long scales. The use of lattice regularization, i.e. the framework
of lattice gauge theory (LGT) \cite{W}, is virtually mandatory in this
context. There is no other known usable
non-perturbative formulation
of gauge theory that gives the path integral in closed form
preserving non-perturbative gauge invariance and positivity of the
transfer matrix (reflection positivity).
Attempts at exact blocking constructions towards the
`perfect action' along the Wilsonian renormalized trajectory \cite{H},
however, turn out, not surprisingly, to be exceedingly complicated.

There are, nonetheless, approximate RG decimation procedures that
can provide bounds on judicially chosen quantities.
The basic idea in this paper is to
obtain both upper and lower bounds for the
partition function and the partition function in the presence of
external center flux. The bounds are obtained by employing
approximate decimations of the `potential moving' type \cite{M},
\cite{K}, which
can be explicitly computed to any accuracy by simple algebraic operations.
This leads to a rather simple construction constraining the behavior of the
exact partition functions in the presence and in
the absence of center flux; and, through them, the exact vortex free energy
order parameter. The latter is the ratio of these two partition functions.
It is thus shown to exhibit
confining behavior for all values
$0 < \beta < \infty$,  of the inverse coupling $\beta=4/g^2$
defined at lattice spacing $a$ (UV cutoff). An earlier outline of
the argument was given in \cite{T1}.

As it will become clear in the following, there are two main ingredients
here that allow this type of result to
be obtained. The first is the use of approximate decimations that are
easily explicitly computable at every step, while correctly reflecting the
nature of RG flow in the exact theory. The second is to consider
only partition functions, or (differences of) free energies, rather than
the RG evolution of a
full effective action that would allow computation of any observable at
different scales. This more narrowly focused approach results into
tremendous simplification compared to a general RG blocking
construction.

The presentation is for the most part quite explicit. Some simple
propositions, mostly containing basic bounds, serve as
building blocks of the argument. They are enumerated by roman numerals in
the text below. Most proofs have been relegated to a
series of appendices so as not to clutter what is essentially a
simple construction. Only the case of gauge group $SU(2)$ is considered
explicitly here. The same development, however, can be applied to other
groups, and, most particularly, to $SU(3)$ which exhibits identical
behavior under the approximate decimations.

It will be helpful at this point to provide an outline of the
steps in the argument developed in the rest of the paper.
In section \ref{DEC}, starting with the pure $SU(2)$  LGT with
partition function defined on a lattice of spacing $a$, we
define a class of approximate decimation transformations to a
coarser lattice of spacing $ba$.
In section \ref{Z} the resulting partition function on this decimated lattice
is shown to be an upper bound on the partition function on
the original lattice. A similar rule can be devised for obtaining a
partition function on the decimated lattice which gives a lower bound
on the original partition function.
One then interpolates between these bounds.
For some appropriate value of the interpolating parameter, one thus
obtains an exact
integral representation of the original partition function. This
representation is in terms of
an effective action defined on the decimated lattice of spacing $ab$
plus a bulk free energy contribution resulting from the blocking $a \to ab$.
Now, any such interpolation is not unique,
and it is indeed  expedient to consider different interpolation
parametrizations.
The resulting partition function representation is then invariant under
such parametrization variations in its effective action.
The other important ingredient is that the effective action in this
representation is constrained between the effective actions
corresponding to the upper and lower bound partition functions.
Iterating
this procedure in successive decimations, a representation of the
partition function is  obtained on progressively coarser lattices of
spacing $a \to ab \to ab^2 \to \cdots \to ab^n$.

In section \ref{TZ} we consider the partition function in the
presence of external
center flux. This is the flux of a center vortex, introduced by a
$Z(2)$ `twist' in the action, and rendered topologically stable by winding
around the lattice torus. The decimation-interpolation  procedure
just outlined for the partition function can be applied also in the
presence of the external flux. A  representation of the twisted
partition function on progressively coarser lattices can then be
obtained in a completely analogous manner.

The ratio of the twisted to the untwisted partition function is the
vortex free energy order parameter. Its behavior as a function
of the size of the system characterizes the system's possible phases.
By known correlation inequalities it can, furthermore, be related to the
Wilson and t'Hooft order parameters. Our representations
of the twisted and untwisted partition functions may now be used
to represent the ratio (section \ref{Z-/Z}).
One may exploit the parametrization invariance of these representations
to ensure that the bulk free energy contributions resulting in each
decimation step $ab^{m-1} \to ab^m$  explicitly cancel between numerator and
denominator in the ratio. One is then left with a representation
of the vortex free energy solely in terms of an effective action
defined on a lattice of spacing $ab^n$.

Now this effective action is constrained by
the effective actions corresponding to the upper and lower bounds.
The latter are easily explicitly computable by straightforward iteration
of the potential-moving decimation rules. Under successive
transformations they flow, for space-time dimension $d\leq 4$ and
any original coupling $g$ defined at
spacing $a$, to the strong coupling regime.  This is the regime where the
coefficients in the character expansion of the exponential of the action
become sufficiently small for the strong coupling cluster expansion to
converge. Confining behavior is the immediate result for the
vortex free energy, and, hence, `area law' behavior for the Wilson loop
(section \ref{CONf}).

As it is well-known the theory contains only one free parameter, a
physical scale which is conventionally taken to be (some multiple of)
the string tension. This fact comes out in a natural way in the
context of RG decimations, as we will see in the following.
Fixing this scale then determines the dependence $g(a)$.
The fact that $g(a)\to 0$ as $a\to 0$ is an essentially
qualitative consequence of the flow exhibited by the decimations.

Some concluding remarks are given in section \ref{SUM}.

\section{Decimations} \label{DEC}
\setcounter{equation}{0}
\setcounter{Roman}{0}

We work on a hypercubic lattice $\Lambda \subset {\rm\bf Z}^d$ of
length $L_\mu$ in the $x^\mu$-direction, $\mu=1,\ldots ,d$,
in units of the lattice spacing $a$. Individual bonds, plaquettes,
3-cubes, etc are generically denoted by $b$, $p$, $c$, etc. More
specific notations such as $b_\mu$ or $p_{\mu\nu}$  are
used to indicate elementary $m$-cells of particular orientation.
We use the standard framework and common notations of LGT with
gauge group $G$. Group elements are generically denoted by $U$,
and the bond variables by $U_b \in G$. In this paper we take
$G=SU(2)$.

We start with some appropriate plaquette action $A_p$ defined
on $\Lambda$, which, for definiteness, is taken to be
the Wilson action
\begin{equation}
A_p(U_p,\beta) ={\beta\over 2}\;{\rm Re}\,\chi_{1/2}(U_p) \;, \qquad
U_p=\prod_{b\in \partial p} U_b \;,\label{Wilson}
\end{equation}
with $\beta=4/g^2$ defining the lattice coupling $g$.
The character expansion
of the exponential of the plaquette action function is given by
\begin{equation}
\exp \left(A_p(U,\beta)\right)
   = \sum_j\;d_j\, F_j(\beta)\,\chi_j(U) \label{exp}
\end{equation}
with Fourier coefficients:
\begin{equation}
 F_j(\beta) = \int\,dU\;
\exp \left(A_p(U,\beta)\right) \,{1\over d_j}\,\chi_j(U)\;.\label{Fourier}
\end{equation}
Here $dU$ denotes Haar measure on $G$, and $\chi_j$ the
character of the $j$-th representation of dimension $d_j$.
So, for SU(2), the only case considered explicitly here, all
characters are real, $j=0, {1\over 2}, 1,
{3\over 2}, \ldots$, and $d_j=(2j+1)$. (\ref{Fourier}) implies that
$F_0\geq F_j$, all $j\not=0$. Explicitly, one finds
\begin{equation}
F_j(\beta) = {2\over \beta}\, I_{d_j}(\beta) \,
\end{equation}
in terms of the modified Bessel function $I_\nu$.

It will be convenient to work in terms of normalized coefficients:
\begin{equation}
c_j(\beta) = {\displaystyle F_j(\beta) \over \displaystyle F_0(\beta)} \;, \label{ncoeffs}
\end{equation}
so that
\begin{eqnarray}
\exp \left(A_p(U,\beta)\right)  &=&  F_0\,\Big[\, 1 + \sum_{j\not= 0}
d_j\,c_j(\beta)\,\chi_j(U)\,
 \Big] \nonumber \\
    & \equiv & F_0\;f_p(U,\beta)\,.  \label{nexp}
\end{eqnarray}
The (normalized) partition function on lattice $\Lambda$ is then
\begin{equation}
Z_\Lambda(\beta) =
\int dU_\Lambda\;\prod_{p\in \Lambda}\,f_p(U_p,\beta)\equiv
\int\,d\mu_\Lambda^0
\;,\label{PF1}
\end{equation}
where $dU_\Lambda\equiv \prod_{b\in \Lambda}dU_b$, and expectations
are computed with the measure $d\mu_\Lambda =
d\mu_\Lambda^0 / Z_\Lambda(\beta)$.

The action (\ref{Wilson}) is  such that
\begin{equation}
F_j (\beta)\geq 0\;, \qquad \mbox{hence}\quad 1\geq c_j(\beta)
\geq 0\qquad\quad \mbox{all}\quad j \;,
\end{equation}
which implies that the measure defined by (\ref{PF1}) satisfies reflection
positivity (RP) both in planes without sites and in planes with sites.
Note that $\lim_{\beta\to \infty}c_j(\beta)=1$.

Let $\Lambda^{(n)}$ be the hypercubic lattice of spacing $b^na$,
with integer $b\geq 2$, and
$Z_{\Lambda^{(n)}}(\{c_j(n)\})$ denote a  partition
function of the form (\ref{PF1}) defined on
$\Lambda^{(n)}$ in terms of
some given set of coefficients $\{c_j(n)\}$:
\begin{eqnarray}
Z_{\Lambda^{(n)}}(\{c_j(n)\}) & = &
  \int dU_{\Lambda^{(n)}} \prod_{p\in \Lambda^{(n)}}
\Big[\, 1 + \sum_{j\not= 0} d_j \,
c_j(n)\chi_j(U_p)\,\Big] \nonumber \\
& \equiv & \int dU_{\Lambda^{(n)}} \prod_{p\in \Lambda^{(n)}}\,f_p(U_p,n)
\equiv \int\,d\mu_{\Lambda^{(n)}}^0
\,,\label{PF2}
\end{eqnarray}
where $dU_{\Lambda^{(n)}}\equiv \prod_{b\in \Lambda^{(n)}}dU_b$.
We also employ the notations
\begin{equation}
g_p(U,n) \equiv f_p(U,n) -1 = \sum_{j\not= 0} d_j\, c_j(n)
\, \chi_j(U) \;,\label{g}
\end{equation}
and $\| \cdot\|$ for the $\|\cdot\|_\infty$-norm:
\begin{equation}
\|g(n)\| = \sum_{j\not=0} d_j^2\,  c_j(n) \,.\label{gnorm}
\end{equation}

One has the simple but basic result:

\prop{For $Z_{\Lambda^{(n)}}(\{c_j(n)\})$ given by (\ref{PF2})
with $c_j(n) \geq 0$ for all $j$, and periodic boundary conditions,

(i) $\dZ{n}(\{c_j(n)\})$ is an increasing function of each
$c_j(n)$:
\begin{equation}
\partial \dZ{n}(\{c_i(n)\}) / \partial c_j(n) \geq 0 \; ;\label{PFder0}
\end{equation}

(ii)
\begin{equation}
Z_{\Lambda^{(n)}}(\{c_j(n)\}) \geq \Big[\,1 + \sum_{j\not=0} d_j^2\,
c_j(n)^6 \,\Big]^{|\Lambda^{(n)}|} \; .\label{PFlowerb1}
\end{equation}  }
(\ref{PFder0}) is an immediate consequence of RP in planes without sites.
The proof of (\ref{PFlowerb1}), also based on RP, is given in Appendix A.
Strict inequality in fact holds in (\ref{PFder0}) and (\ref{PFlowerb1}),
with equality only in the trivial case where all $c_j(n)$'s vanish.
In particular, one has
\begin{equation}
Z_{\Lambda^{(n)}}(\{c_j(n)\}) \; > \; 1 \,.\label{PFlowerb2}
\end{equation}
Simple as (\ref{PFlowerb2}) is, it is not trivial, as it requires
non-negativity of $c_j(n)$'s, and will be useful in the following.

\subsection{Construction of decimation transformations} \label{DEC1}

To perform an RG transformation $a \to ba$,
the lattice is partitioned into $d$-dimensional
decimation cells of side length $ba$. Various approximate
decimation transformations may be devised involving the
`weakening', i.e. decreasing the $c_j$'s  of interior plaquettes, while
compensating by `strengthening', i.e. increasing $c_j$'s  of
boundary plaquettes of each cell. The simplest such scheme \cite{M},
which is  adopted in the following,  implements
complete removal of interior plaquettes. This may be pictured \cite{K}
as moving the potentials due to interior plaquette interactions
to the boundary.

This `potential moving' may be performed  as the composition of
elementary steps.
The elementary potential moving  step is defined in terms of
a $3$-dimensional cell of side length $ba$ in a given
decimation direction, say the $x^\kappa$-direction,
and length $a$ in the other two directions $\mu$, $\nu$.
Two such $3$-cells adjacent along the
$\kappa$-direction are shown in Figure~\ref{Dec1fig}. The $(b -1)$
interior plaquettes in each cell perpendicular
to $x^\kappa$ (shaded) are removed, i.e.
\begin{equation}
A_p(U_p) \to  0 \label{potmove1}
\end{equation}
for the action at their original location, and displaced (arrows) in the
positive $x^\kappa$ direction to the position of the corresponding
plaquette (bold) on the cell boundary. There the displaced interior
plaquettes are combined with the boundary plaquette
into one plaquette $p$ with action
`renormalized' by some appropriate  amount\footnote{One
may take this renormalization factor to depend on the move direction,
but we need not consider these more general transformations here.}
$\zeta_0$:
\begin{equation}
A_p(U) \to  \zeta_0\,A_p(U) \;.\label{potmove2}
\end{equation}
\begin{figure}[ht]
\resizebox{15cm}{!}{\input{Dec1.pstex_t}}
\caption{Basic plaquette moving operation, $b=2$ \label{Dec1fig}}
\end{figure}
A complete transformation consists of performing this elementary operation
successively in every lattice direction $\kappa=1, \ldots, d$
in such a way that eventually one is left only with plaquette interactions
on a lattice of spacing $ba$.
In practice, there is no reason for a choice
other than $b=2$, but, for clarity, we keep general (integer) $b$.
The result of a complete transformation
is given by equations (\ref{RG1})-(\ref{RG5}) below, to which a reader may
turn directly.

To describe this process in more detail, let  the lattice be
partitioned into $d$-dimensional hypercubic decimation cells $\sigma^d$
of side length $b a$ in each lattice direction. Plaquettes interior
to a $\sigma^d$ are defined as those  not wholly contained in its
$(d-1)$-dimensional boundary $\partial \sigma^d$.
Consider the effect
of successive application of the elementary moving operation to plaquettes
of fixed orientation, say $[\mu\nu]$.
There are $(d-2)$ normal directions $\kappa_i \not= \mu,\nu$,
$i=1, \cdots, d-2$,
in which a plaquette $p_{\mu\nu}$ can be moved.
Interior $p_{\mu\nu}$'s in each $\sigma^d$ are first moved
to the cell boundary $\partial \sigma$
in groups of $(b-1)$ parallel plaquettes,
along , say, the positive $\kappa_1$-direction (as in Figure 1). They end up
in the face $\sigma^{(d-1)}_{\kappa_1} \subset \partial \sigma^d$
perpendicular to the $\kappa_1$-axis.
There each group is identified with the plaquette present at that
location and merged in one plaquette $p_{\mu\nu} \in
\sigma^{(d-1)}_{\kappa_1}$
with a `renormalized' action (\ref{potmove2})).
Similarly, $p_{\mu\nu}$ plaquettes in each face
$\sigma^{(d-1)}_{\kappa_i}\subset \partial \sigma^d$,
with $i\not= 1, \mu,\nu$ are moved along the $\kappa_1$-axis in
groups of $(b-1)$  to the face $\sigma^{(d-2)}_{\kappa_1\kappa_i}
\subset \partial \sigma^{(d-1)}_{\kappa_i}$
normal to the $\kappa_1$ and $\kappa_i$ directions, where they are
merged and renormalized.

There are now $(d-3)$ directions inside the $(d-1)$-dimensional face
$\sigma^{(d-1)}_{\kappa_1}$ in
which a $[\mu\nu]$-plaquette can move.
Thus in proceeding to apply  the elementary moving operation successively
in all directions, the once-moved-renormalized $p_{\mu\nu}$'s
in $\sigma^{(d-1)}_{\kappa_1}$ are next
moved, in groups of $(b-1)$ plaquettes in the positive $\kappa_2$-direction,
to the face $\sigma^{(d-2)}_{\kappa_1\kappa_2} \subset \partial
\sigma^{(d-1)}_{\kappa_1}$. Similarly,
the once-moved-renormalized $p_{\mu\nu}$'s inside a face
$\sigma^{(d-2)}_{\kappa_1\kappa_i}$ are moved provided $\kappa_2$ is among the
$(d-4)$ available directions normal to a $[\mu\nu]$-plaquette inside
$\sigma^{(d-2)}_{\kappa_1\kappa_i}$.

Continuing this process in the remaining directions
$\kappa_i$, $i=3,\ldots,(d-2)$, the set of
$[\mu\nu]$-plaquettes on the initial lattice ends up in the
$2$-dimensional faces
$\sigma^2_{\kappa_1\kappa_2\ldots\kappa_{(d-2)}} \subset \partial
\sigma^3_{\kappa_1\kappa_2\ldots\kappa_{(d-3)}}
\linebreak
\subset \cdots \subset
\partial \sigma^{(d-1)}_{\kappa_1}$.

The above process, described for plaquettes of one
fixed orientation $[\mu\nu]$, is carried out for each of the
$d(d-1)/2$ possible choices of plaquette orientation \cite{K}.

The end result of the process is then a lattice having
elementary $2$-faces of side length $b a$, each tiled by
$b^2$ plaquettes of side length $a$.
The action of each of these $b^2$ plaquettes has been
renormalized according to (\ref{potmove2}) by a total factor of
\begin{equation}
\zeta_0^{(d-2)} \equiv \zeta
\;. \label{totalren}
\end{equation}
This is expressed by (\ref{RG5}) below.

The integrations over the bonds interior to each
$2$-face of side length $ba$ are now carried out. This merges the
$b^2$ tiling plaquettes into a single plaquette of side length $ba$.
These integrations are exact
and do not change the value of the partition
function that resulted after the completion of the plaquette moving
operations. We, however, allow further renormalizing the result of these
integrations by introducing, in addition to $\zeta_0$, another parameter,
$r$ (cf. (\ref{RG2}) below).
This completes the decimation transformation to a hypercubic
lattice of spacing spacing $ba$.

The important feature of this decimation transformation is
that it preserves the original one-plaquette form of the action,
so the result can again be represented in the form (\ref{nexp}).
The transformation rule for successive decimations
\begin{eqnarray}
& & a\, \to b\, a \,\to\, b^2 a \to\, \cdots \to\,b^{n-1} a
 \to \,b^n a \to \cdots\nonumber \\
 & & \Lambda \to \Lambda^{(1)} \to \Lambda^{(2)} \to \cdots \to
\Lambda^{(n-1)} \to \Lambda^{(n)} \to \cdots\;,\nonumber
\end{eqnarray}
is then:
\begin{equation}
f_p(U,n-1)\to F_0(n)\,f_p(U,n) = F_0(n)\,\Big[ 1 + \sum_{j\not= 0}
d_j\,c_j(n)\,\chi_j(U) \Big]  \,.\label{RG1}
\end{equation}
The $n$-th step coefficients $F_0(n)$, $c_j(n)$ are obtained from
the coefficients $c_j(n-1)$ of the previous step by
\begin{equation}
c_j(n) = \hat{c}_j(n)^{b^2 r}\; , \label{RG2}
\end{equation}
\begin{equation}
F_0(n) = \hat{F}_0(n)^{b^2} \label{RG3}
\end{equation}
where
\begin{equation}
\hat{c}_j(n)\equiv \hat{F}_j(n)/\hat{F}_0(n) \leq 1  \;, \qquad j\not= 0\;,
\label{RG4}
\end{equation}
and
\begin{equation}
\hat{F}_j(n)= \int\,dU\;\Big[\,f(U,n-1)\,\Big]^\zeta\,
{1\over d_j}\,\chi_j(U)
\; .  \label{RG5}
\end{equation}
The $n=0$ coefficients are the coefficients
$c_j(\beta)$ on the original lattice $\Lambda$.
(\ref{RG5}) encodes the end result of the plaquette moving - renormalization
operations described above, with $\zeta$ of the form (\ref{totalren});
and (\ref{RG2}), (\ref{RG3}) that of the
subsequent 2-dimensional integrations, and further renormalization by
the parameter $r$.

It is easily seen that $f_p(U, n) \;>\; 0$ given that this holds for $n=0$
(cf. (\ref{exp}), (\ref{nexp})).
The effective plaquette action
on lattice $\Lambda^{(n)}$ of spacing $b^n a$
is then
\begin{eqnarray}
f_p(U, n) & = & \Big[\, 1 + \sum_{j\not= 0} d_j\,c_j(n)\, \chi_j(U)\,
 \Big]  \label{nexp1} \\
& \equiv & \exp\Big(\, A_p(U, n)\,\Big)   \;, \label{actdef}
\end{eqnarray}
with effective couplings defined by the character expansion
\begin{equation}
A_p(U, n) = \beta_0(n) + \sum_{i\not= 0} \beta_i(n)\,d_i\,\chi_i(U) \;.
\label{effact}
\end{equation}

A point on notation. In the above we used the notations $F_0(n)$,
$c_j(n)$, $\beta_j(n)$, etc, which do not display the full set of
explicit or implicit dependences of these quantities. Thus, a more complete
notation is:
\begin{eqnarray}
c_j(n) &=& c_j(\,n,b,\zeta,r,\{c_j(n-1)\}\,)   \nonumber \\
F_0(n) &=& F_0(\,n,b,\zeta,\{c_j(n-1)\}\,) \label{short-hand}
\end{eqnarray}
Dependence on the original coupling $\beta$ comes, of course,
iteratively through the coefficients $\{c_j(n-1)\}$ of the preceding step.
Because of the iterative nature of many of the arguments in this paper
several  explicit and implicit dependences propagate to most of the
quantities used in the following. To prevent notation from getting out of
hand we generally employ short-hand notations such as those on the l.h.s.
of (\ref{short-hand}), unless specific reference to particular
dependences is required.

The resulting partition function after $n$ such
decimation steps is:
\begin{equation}
Z_\Lambda(\beta, n) =
\prod_{m=1}^n  F_0(m)^{|\Lambda|/b^{md}}\; Z_{\Lambda^{(n)}}(\{c_j(n)\})
\; ,\label{PF2a}
\end{equation}
with  $Z_{\Lambda^{(n)}}(\{c_j(n)\})$ of the form (\ref{PF2}) and
coefficients (\ref{short-hand}) resulting after $n$ steps according to
(\ref{RG2}) - (\ref{RG5}).
The bulk free energy density resulting
from decimating from scale $a$ to $b^n a$ is then
$\sum_{m=1}^n \ln F_0(m) /b^{md}\,$,
each term in this sum representing the contribution from
$b^{(m-1)}a \to b^m a$ as specified by (\ref{RG1}).
The partition function (\ref{PF2a}) is, of course, not equal to
the original partition function $Z_\Lambda(\beta)$ of (\ref{PF1})
since the decimation transformation is not exact. How they are related
will be addressed below.

\subsection{Some properties of the decimation transformations}\label{DEC2}

The transformation rule specified by (\ref{RG1})-(\ref {RG5}) is
meaningful for real positive $\zeta$. Here, however, a basic
distinction can be made.
As it is clear from (\ref{RG5}),
for {\it integer} $\zeta$ the important property of positivity of the Fourier
coefficients in (\ref{RG1}) is maintained at each decimation step:
\begin{equation}
F_0(n)\geq 1\;, \qquad 1\geq c_j(n)\geq 0 \qquad \qquad
(\mbox{integer} \ \zeta) \;. \label{+c}
\end{equation}
This means that reflection positivity
is maintained at each decimation step.
This clearly is not  guaranteed to be
the case for non-integer $\zeta$.
Thus non-integer $\zeta$ results in transformations that, in general,
violate the reflection positivity of the theory (assuming a
reflection positive original action).

It is important in this connection that,
after each decimation
step, the resulting action retains the original one-plaquette form,
but will generally contain {\it all} representations in (\ref{effact}).
Furthermore, among the effective couplings
$\beta_j(m)$ negative ones will occur.
These features are present in general, even after a single decimation
step $a\to ba$ starting, as we did, with the single (fundamental)
representation Wilson action (\ref{Wilson}).  For integer $\zeta$, however,
the resulting effective action (\ref{effact}), even in the presence of some
negative couplings, still defines a reflection positive measure,
since, as just noted, the expansion of its exponential (\ref{nexp1}) gives
positive coefficients (\ref{+c}).

It is also worth noting that, given a set of initial coefficients
(\ref{ncoeffs}), the transformation
rule (\ref{RG2}) - (\ref{RG5}) with integer $\zeta$ can be explicitly
evaluated, to any desired accuracy, by purely algebraic operations, namely
repeated application of the KG reduction rule
\begin{equation}
\chi_i\,\chi_j =\sum_{k=|i-j|}^{i+j} \chi_k \label{KG}
\end{equation}
in (\ref{RG5}) and character orthogonality -- no actual integrations
need be carried out.

The choice (cf. (\ref{totalren}))
\begin{equation}
\zeta_0 = 1 +(b -1 ) \qquad \Longrightarrow \qquad
\zeta=b^{d-2}
\label{MKz}
\end{equation}
is special. It increases the couplings of
receiving plaquettes, at each basic moving step,
by an amount exactly equal to that of the
corresponding displaced plaquettes. This, together with $r=1$, is
essentially the original
choice in \cite{M} as reformulated in \cite{K}, and will be referred to as
MK decimation. It will be important in the following.\footnote{It is worth
noting in this context that in numerical investigations
of the standard MK recursions in gauge theories \cite{NT-BGZ}
fractional $b$, ($1< b <2$), which by (\ref{MKz}) corresponds to
non-integer $\zeta$, has often been used.}

There are various other interesting properties of the decimations
that can be derived from (\ref{RG2}) - (\ref{RG5}).
The following one is particularly important. The norm  (\ref{gnorm})
of the coefficients obtained by application of (\ref{RG2}) - (\ref{RG5})
with integer $\zeta\geq 1$ and $r=1$ satisfies (Appendix D):
\begin{equation}
||g(n+1)|| \leq \Big[\,\zeta\,||g(n)||\,\Big]^{b^2}
\Big[\, 1+ ||g(n)||\,\Big]^{(\zeta-1)b^2} \,.\label{gnormrecur}
\end{equation}

Assume now that
\begin{equation}
||g(n)|| \leq \exp (- C_n)\,. \qquad \quad C_n > 0 \,, \label{gnormU1}
\end{equation}
for some $n$. Then
\begin{equation}
||g(n+1)|| \leq \Big[\,\zeta\,||g(n)||\,\Big]^{b^2} \exp \left(\,
(\zeta-1)\, b^2\,\right)
\leq \exp \Big[ -(\, C_n - k\,) b^2 \Big] \,, \label{gnormrecurU1}
\end{equation}
where $k= \ln \zeta + (\zeta-1)$. The recursion
\begin{equation}
C_{n+1} = C_n b^2 - k b^2  \label{gnormrecurU2}
\end{equation}
gives
\begin{equation}
C_{n+m} = \Big[\, C_n - {b^2 k\over b^2-1}\,\Big] b^{2m} + {b^2 k\over b^2-1}
\,.\label{gnormsoln}
\end{equation}
\prop{If for some $n$ the norm of coefficients (\ref{gnorm}) obeys
(\ref{gnormU1}) with
\begin{equation}
C_n > {b^2 k\over b^2-1}\, , \label{gnormU2}
\end{equation}
then, under iteration of the decimation transformation
(\ref{RG2}) - (\ref{RG5}), $||g(n+m)||\to 0$ as
$m\to \infty$ according to (\ref{gnormsoln}). }\\
This fall-off behavior is immediately
recognizable as ``area-law''.

If one assumes that $c_j(n)$ are small enough so that the theory
is within the strong coupling regime, this behavior can be immediately
deduced for the leading coefficient $c_{1/2}(n)$ directly from
(\ref{RG2}) - (\ref{RG5}):
\begin{equation}
c_{1/2}(n+1) = c_{1/2}(n)^{b^2}
\exp \Big(\,[\,\ln \zeta + O(c_{1/2}(n))\,]\,b^2\, \Big)\,. \label{RGstrong}
\end{equation}
The result (\ref{gnormrecurU1}) gives then
an estimate of the corrections due to all higher representations.
What is noteworthy here, however, is that the condition (\ref{gnormU2})
is rather weaker than the commonly stated conditions  for being inside the
convergence radius of the strong coupling cluster expansion (cf.
section \ref{CONf}).

We note two further properties of the decimation
transformations (\ref{RG1}) - (\ref{RG5}). The first is that with $r=1$
they become exact in space-time dimension $d=2$ since then, from
(\ref{totalren}),  $\zeta=1$.
The second is that, with $\zeta=b^{(d-2)}$, vanishing coupling
$g=0$ is a fixed point in any $d$, i.e. MK decimation is
exact at zero coupling. This follows simply from the fact that
\[ \lim_{\beta\to \infty} \Big[\int d\nu(x)\;  e^{\beta f(x)} \Big]^{1/\beta}
=\mbox{ess. sup}\   e^{f(x)} \equiv \| e^f\|  \]
for any normalized measure $d\nu(x)$. Applying this to the  result of
performing the plaquette moving operation starting from
(\ref{nexp}), and with $p^\prime\in \Lambda$ labeling the plaquettes
tiling the plaquettes $p\in \Lambda^{(1)}$, one has
\begin{eqnarray}
& & \lim_{\beta\to \infty} \Bigg[\int dU_\Lambda \,
\prod_{p\in \Lambda^{(1)}}
\prod_{p^\prime \in p}\exp\Big(\beta b^{(d-2)}\,{1\over 2}
\chi_{1/2}(U_{p^\prime})\Big) \Bigg]^{1/\beta}  \nonumber \\
& = &
\prod_{p\in\Lambda^{(1)}} \Big\|\exp \Big(b^{(d-2)}{1\over2} \chi_{1/2}\Big)
\Big\|^{b^2} = e^{|\Lambda|} =
\lim_{\beta\to \infty} \Bigg[\int dU_\Lambda\,\prod_{p\in \Lambda}
\exp\Big(\beta \,{1\over 2}
\chi_{1/2}(U_p)\Big) \Bigg]^{1/\beta} \;.\qquad
\end{eqnarray}
This clearly holds also for $r\not= 1$, as is evident from
the fact that $\lim_{\beta\to \infty} c_j(\beta)=1$. This fixed point is
easily seen to be unstable.

\section{Partition function} \label{Z}
\setcounter{equation}{0}
\setcounter{Roman}{0}

Since our decimations are not exact RG transformations, the partition
function does not in general remain invariant under them.
The subsequent development hinges on the following two basic propositions
that relate partition functions under such a decimation.

\subsection{Upper and lower bounds}\label{u-lPF}
Consider a partition $Z_{\Lambda^{(n-1)}}$ on lattice $\Lambda^{(n-1)}$ of the
form (\ref{PF2}) given in terms of some set of coefficients $\{c_j(n-1)\}$.
Apply a decimation transformation (\ref{RG1}) - (\ref{RG5}) performed with
$\zeta=b^{(d-2)}$. Denote the resulting coefficients by $c_j^U$, $F_0^U$,
i.e.
\begin{eqnarray}
c_j^U(n,r) & \equiv & c_j(\, n,b,\,\zeta=b^{(d-2)}, r, \{c_j(n-1)\}\,)
\label{upperc}\\
F_0^U(n) & \equiv & F_0( \, n,b,\, \zeta=b^{(d-2)},  \{c_j(n-1)\}\,)
\label{upperF} \;.
\end{eqnarray}
Note that
\begin{equation}
c_j^U(n,r) = c_j^U(n,1)^r \, .
\label{upperc1}
\end{equation}

\prop{
For $Z_{\Lambda^{(n-1)}}$ of the form (\ref{PF2}),
a decimation transformation (\ref{RG1}) -
(\ref{RG5}) with $\zeta=b^{d-2}$ and $0 < r\leq 1$  results in an upper
bound on $Z_{\Lambda^{(n-1)}}$:
\begin{equation}
Z_{\Lambda^{(n-1)}}(\{c_j(n-1)\})\,  \leq \,  F_0^U(n)^{|\Lambda^{(n)}|}\,
Z_{\Lambda^{(n)}}(\{c_j^U(n,r)\})\;.\label{U}
\end{equation}

The r.h.s. in (\ref{U}) is a monotonically
decreasing function of $r$ on $0 < r\leq 1$.
}

Given partition function $Z_{\Lambda^{(n-1)}}$ on lattice
$\Lambda^{(n-1)}$ of the form (\ref{PF2}) in terms of some set of
coefficients $\{c_j(n-1)\}$, let
\begin{eqnarray}
c_j^L(n) & \equiv & c_j(n-1)^6 \label{lowerc1}\\
F_0^L(n) & \equiv & 1 \;.\label{lowerF1}
\end{eqnarray}

\prop{
For $\dZ{(n-1)}$, $\dZ{n}$ of the form (\ref{PF2}):
\begin{equation}
Z_{\Lambda^{(n)}}(\{c_j^L(n)\}) \, \leq \, Z_{\Lambda^{(n-1}}(\{
c_j(n-1)\}) \;.  \label{L}
\end{equation}
}

The proof of III.1 is given in Appendix A,
where somewhat stronger results than (\ref{U})
are actually obtained. III.2 is a corollary of
(\ref{PFlowerb1}) (Appendix A). For the argument in the rest of this paper,
the precise form of the lower bound is in fact not important.
By II.1(i) a further lower bound is   obtained by replacing
$c_j^L(n)$ in III.2 by, for example,
\begin{equation}
c_j^L(n) \equiv  c_j(n-1)^6 \,c_j^U(n,r) \label{lowerc2}
\end{equation}
since $0\leq c_j^U(n,r)\leq 1$. Another choice is to
simply set
\begin{equation}
c_j^L(n)=0 \,,\label{lowerc3}
\end{equation}
which is a restatement of (\ref{PFlowerb2}).

A related lower bound, which, in analogy to the upper bound in III.1,
can be formulated directly in terms of the
transformations (\ref{RG1}) - (\ref{RG5}), is obtained by taking
$c_j^L(n)$ in III.2 to be given by:
\begin{eqnarray}
c_j^L(n) & \equiv & c_j(\, n,b,\,\zeta=1, r=1, \{c_j(n-1)\}\,) \nonumber\\
           & = & c_j(n-1)^{b^2} \,,
\label{lowerc4}\\
F_0^L(n) & \equiv & F_0( \, n,b,\, \zeta=1,  \{c_j(n-1)\}\,) \nonumber \\
           & = & 1 \; .\label{lowerF2}
\end{eqnarray}
With this choice of $c_j^L(n)$, note that III.1 - III.2  imply the
fact that the decimations (\ref{RG1}) - (\ref{RG4})
become exact for $d=2$ and $r=1$.
III.1 says that, after removal of interior plaquettes,  modifying the
couplings of the remaining plaquettes
by taking $\zeta=b^{d-2}$   (and $r\leq 1$)
results into overcompensation.
III.2 says that decimating plaquettes while leaving the couplings of the
remaining plaquettes unaffected ($\zeta=1$, $r=1$) results in
undercompensation.
The proof of III.2 for $c_j^L(n)$ given by (\ref{lowerc4})
is similar to that of II.1,
but need not be given here, since the weaker
bounds above will suffice.

In the following it will in fact be more convenient to take (\ref{lowerc2}) or
(\ref{lowerc3}) for the definition of the lower bound coefficients
$c_j^L(m)$.  Use of the stronger lower bounds
above may be preferable
for numerical investigations, but does not contribute anything further
to the argument in this paper.

III.1 and III.2 give  upper and lower bounds on the partition
function after a decimation step.
It is then natural to interpolate between these bounds.

\subsection{Interpolation between upper and lower bounds}\label{interbounds}

Introducing a parameter $\alpha \in [0,1]$, we define
coefficients $\tilde{c}_j(m,\alpha,r)$
interpolating between $c_j^L$ at $\alpha=0$ and $c_j^U$
(\ref{upperc}) at $\alpha=1$:
\begin{equation}
\tilde{c}_j(m,\alpha, r) = (1-w(\alpha))\, c_j^L(m) +
w(\alpha)\, c_j^U(m,r) \;, \quad \qquad 0 < r \leq 1.
\label{interc1}
\end{equation}
with
\begin{equation}
w(0)=0\;, \qquad \quad w(1)=1\;,  \quad \qquad w^\prime(\alpha) >  0
\;. \label{interc2}
\end{equation}
For example,
\begin{equation}
w(\alpha) = {e^\alpha-1\over e-1}  \label{w}
\end{equation}
There is clearly a variety of other choices than (\ref{interc1}) for these
interpolating coefficients.
We always require that
\begin{equation}
\partial\, \tilde{c}_j(m,\alpha,r) /\partial \,\alpha > 0  \;,
\label{interc3}
\end{equation}
which is satisfied by (\ref{interc1}) - (\ref{interc2}).

Similarly, we define coefficients
interpolating between (\ref{lowerF1}) and (\ref{upperF}). For our purposes
it will be convenient to take
\begin{equation}
\tilde{F}_0(m,h, \alpha,t) = F_0^U(m)^{h_t(\alpha)} \label{interF1} \;,
\end{equation}
where $h_t$ denote a family of monotonically increasing smooth functions
of $\alpha$, labeled by a parameter
$t \in [t_a,t_b]$, and such that
\begin{equation}
h_t(0)=0\;, \qquad h_t(1)=1 \,. \label{hlimits}
\end{equation}
We write $h_t(\alpha) \equiv h(\alpha,t)$. Examples
are\footnote{Supplementing
these definitions at $\alpha=0$ as needed is understood. Thus, $h(\alpha,t)=
0$ on $\alpha \leq 0$ in the first example in (\ref{h}); and
standard smoothing in the second example: replace $\alpha$ in $h$ by
$g_\epsilon(\alpha)=\int \rho_\epsilon(\alpha-x) g(x)dx$,  where
$g(x)=x$ for $x>0$, $g(x) =0$ for $x\leq 0$, and $\rho_\epsilon(x)$
is $C^\infty$, has support inside $|x|^2\leq \epsilon^2$ and
satisfies $\rho_\epsilon \geq 0$ and $\int \rho_\epsilon =1$. }
\begin{eqnarray}
h(\alpha,t) & = & \exp \left( -\, \sigma(t)\,{1-\alpha\over \alpha}\right)\;,
\qquad h(\alpha,t) =\alpha^{\sigma(t)}\, , \qquad
h(\alpha,t) = \tanh\,({\ \ \alpha \over \sigma(t)\,(1-\alpha)})\,,
\nonumber \\
& & \qquad \qquad \qquad \hspace{2cm}  0 <  \alpha \leq 1,\qquad
0 < t_a \leq t \leq t_b  < \infty \,, \label{h}
\end{eqnarray}
where $\sigma(t)$ is a smooth monotonically increasing positive function on
$[t_a,t_b]$, e.g. $\sigma(t) = t$.

The interpolating partition function on $\Lambda^{(m)}$ constructed
from $\tilde{c_j}$ and $\tilde{F}_0$ is now defined by
\begin{equation}
\tilde{Z}_{\Lambda^{(m)}}(\beta,h,\alpha,t,r)
= \tilde{F}_0(m,h,\alpha,t)^{|\Lambda^{(m)}|}\,
Z_{\Lambda^{(m)}}(\{\tilde{c}_j(m,\alpha,r)\}) \label{interPF1}
\end{equation}
where
\begin{eqnarray}
Z_{\Lambda^{(m)}}(\{\tilde{c}_j(m,\alpha,r)\})
   & = &
\int dU_{\Lambda^{(m)}}\;\prod_{p\in \Lambda^{(m)}}\,\Big[ 1
+ \sum_{j\not= 0} d_j\, \tilde{c}_j(m,\alpha, r)
\, \chi_j(U_p) \Big]  \nonumber \\
 & \equiv & \int dU_{\Lambda^{(m)}}\;\prod_{p\in \Lambda^{(m)}}\,
f_p(U_p,m,\alpha,r) \,. \label{interPF2}
\end{eqnarray}
Combining II.1, (\ref{interc3}) and the fact that
$\tilde{F}_0$ is, by definition, also an increasing function of
$\alpha$ one has \\
\prop{
The interpolating free energies
$\ln Z_{\Lambda^{(m)}}(\{\tilde{c}_j(m,\alpha,
r)\})$ and $\ln \tilde{Z}_{\Lambda^{(m)}}(\beta,h,
\alpha,t,r)$ are increasing functions of $\alpha$:
\begin{equation}
\partial \ln Z_{\Lambda^{(m)}}\Big(\{\tilde{c}_j(m,\alpha,r)
\}\Big) /\partial \alpha \,>\,  0
\,.  \label{interPFder1}
\end{equation}
\\
}
Equality in (\ref{interPFder1}) applies only in the
trivial case were all the coefficients $\tilde{c}_j$'s vanish.

In terms of (\ref{interPF1}), III.1 and III.2 give
\begin{equation}
\tilde{Z}_{\Lambda^{(m)}}(\beta,h,0,t,r)
\leq  Z_{\Lambda^{(m-1)}} \leq
\tilde{Z}_{\Lambda^{(m)}}(\beta,h,1,t,r) \,. \label{interI1}
\end{equation}
Now $\tilde{Z}_{\Lambda^{(m)}}(\beta,h,
\alpha,t,r)$ is continuous in $\alpha$. It follows from
(\ref{interI1}) that there exist a value of
$\alpha$ in $(0,1)$:
\begin{equation}
\alpha(m, h,t,r,\{c_j(m-1)\},b,\Lambda)\equiv
\alpha_{\Lambda,h}^{(m)}(t,r)   \label{interI2}
\end{equation}
such that
\begin{equation}
\tilde{Z}_{\Lambda^{(m)}}(\beta,h,\alpha_{\Lambda,h}^{(m)}(t,r),
t,r) =  Z_{\Lambda^{(m-1)}}  \,.\label{interI3}
\end{equation}
In other words, at each given value of $t$, $r$, there exist
a value of $\alpha$ at which the partition
function on $\Lambda^{(m)}$, resulting from a decimation transformation
$\Lambda^{(m-1)} \to \Lambda^{(m)}$, equals the partition function on
$\Lambda^{(m-1)}$. This value is unique by III.3.
By construction, $\alpha_{\Lambda,h}^{(m)}(t,r)$ is such
that (\ref{interI3}) remains invariant under variation of
$t$, $r$ in their domain of definition, i.e.
$\alpha_{\Lambda,h}^{(m)}(t,r)$ represents the
level surface of the function $\tilde{Z}_{\Lambda^{(m)}}(\beta,h,\alpha,
t,r)$ fixed by the value $Z_{\Lambda^{(m-1)}}$.
The parametrization invariance under varying $t$ will be important
later.

We now examine the dependence
on $t$, $r$  in (\ref{interI2})
more closely. Given $Z_{\Lambda^{(m-1)}}$ and
some interpolation $h$, assume that (\ref{interI3}) is satisfied
at the point $(t_0, r_0, \alpha=\alpha_{\Lambda,h}^{(m)})$. Then, by the
implicit function theorem, applicable by III.3, there is a
function $\alpha_{\Lambda,h}^{(m)}(t,r)$ with
continuous derivatives such that
$\alpha_{\Lambda,h}^{(m)}(t_0,r_0)=\alpha_{\Lambda,h}^{(m)}$,
and uniquely satisfies (\ref{interI3}) in  a sufficiently small neighborhood
of $ (t_0, r_0, \alpha_{\Lambda,h}^{(m)})$. But since a
solution to (\ref{interI3}) exists for each choice of $t,r$ in their
domain of definition, this neighborhood
can be extended by a standard
continuity argument to all points of this domain.
$\alpha_{\Lambda,h}^{(m)}(t,r)$ then represents the regular
level surface of the function (\ref{interPF1}) fixed  by (\ref{interI3}).
Furthermore,
\begin{equation}
{\partial \alpha_{\Lambda,h}^{(m)}(t,r)\over \partial t} =
v(\alpha_{\Lambda,h}^{(m)}(t,r), t, r) \;,
\label{alphtder1}
\end{equation}
where
\begin{equation}
v(\alpha, t, r) \equiv
- { \displaystyle {\partial h(\alpha,t) / \partial t}
\over{\displaystyle  {\partial h(\alpha,t)\over \partial \alpha} +
A_{\Lambda^{(m)}}(\alpha, r)} }
\;,\label{alphtder2}
\end{equation}
with
\begin{equation}
A_{\Lambda^{(m)}}(\alpha, r) \equiv  {1\over \ln F_0^{U}(m) }\,
{1\over |\Lambda^{(m)}| }\,
{\partial \over \partial\alpha }\ln Z_{\Lambda^{(m)}}\,\Big(\{\tilde{c}_j( m,
\alpha, r)\}\Big) > 0\;. \label{alphtder3}
\end{equation}
We will always assume that $h$ is chosen such that
$\partial h/\partial t$ is negative.  This is the case with the examples
(\ref{h}). Then, from (\ref{alphtder2}), $v>0$
on $0 <\alpha < 1$, with $v=0$ at $\alpha=0$ and $\alpha=1$.

It is also useful to equivalently view $\alpha_{\Lambda,h}^{(m)}(t,r)$ as
the solution to the ODE
\begin{eqnarray}
d\alpha/dt & =& v(\alpha, t, r) \,,
\qquad \alpha\in (0,1)\;,\label{ODE}\\
\alpha(t_0) & =&  \alpha_{\Lambda,h}^{(m)} > 0\;,  \qquad t_0 \in [t_a,t_b]\;.
\nonumber
\end{eqnarray}
Then standard results of ODE theory imply the existence of a
unique solution in a neighborhood of $\alpha_{\Lambda,h}^{(m)}>0$,
which can in fact be extended indefinitely forward for all
$t\geq t_0$.\footnote{Indeed, v is differentiable on
$\alpha_{\Lambda,h}^{(m)}\leq \alpha\leq 1$ and
vanishes at $\alpha=1$.}

A short computation using (\ref{alphtder1}) gives
\begin{equation}
{d h(\alpha_{\Lambda,h}^{(m)}(t,r),t)\over dt} =
 - {\partial \alpha_{\Lambda,h}^{(m)}(t,r)\over \partial t} \,
A_{\Lambda^{(m)}}(\alpha_{\Lambda,h}^{(m)}(t,r), r)
\,, \label{htder}
\end{equation}
as it should for consistency with (\ref{interI3}).
(\ref{htder}) and (\ref{alphtder1}) make apparent what the
effect of  a parametrization change due
to a shift in $t$ is.
Increasing (decreasing) $t$ increases (decreases) the  contribution of
$\ln Z_{\Lambda^{(m)}}(\{\tilde{c}_j(m,
\alpha_{\Lambda,h}^{(m)}(t,r),r)\})$ while
decreasing (increasing) by an equal amount the contribution
from $\ln F_0^{U}(m)\,h\big(\alpha_{\Lambda,h}^{(m)}(t,r), t
\big)\,|\Lambda^{(m)}|$, so that the sum stays
constant and equal to $\ln Z_{\Lambda^{(m-1)}}$ in accordance with
(\ref{interI3}).

The derivative w.r.t. $r$ is similarly given by
(\ref{alphrder1}) - (\ref{alphrder2}) in Appendix B.
Now, by (III.1), the upper bound in (\ref{interI1})
is optimized for $r=1$, which would appear to make consideration
of other $r$ values unnecessary.
The reason one may want, however, to vary
$r$ away from unity is the following.

The values $\alpha_{\Lambda,h}^{(m)}(t,r)$ lie in the
interval $(0,1)$.
Consider the possibility that one finds that
$\alpha_{\Lambda,h}^{(m)}(t_m,1)$ differs from $1$ only by terms that
vanish as the lattice size grows. This means that, since
$v \geq 0$ in (\ref{alphtder1}),
$\alpha_{\Lambda,h}^{(m)}(t,1)$
is, to within such terms, a constant function of $t$ for all $t\geq t_m$.
For the purposes of the
argument in the following sections we want to exclude
this possibility, and ensure that, at least in some neighborhood of
a chosen $t$ value, the derivative (\ref{alphtder1}) is non-vanishing
by an amount independent of lattice size.

We require that
\begin{equation}
\delta^\prime  < \alpha_{\Lambda,h}^{(m)}(t,r)< 1-\delta \;,
\label{collar}
\end{equation}
with  $\delta > 0$, $\delta^\prime > 0$ independent of the lattice size
$|\Lambda^{(m)}|$. The lower bound requirement is easily shown (Appendix B)
to be automatically satisfied by combining II.1 and (\ref{interI3}).
As it is also shown in Appendix B, one may always ensure that the upper
bound requirement in (\ref{collar}) holds
by choosing  the decimation parameter $r$ to
vary, if necessary, away from unity in the domain
\begin{equation}
1 \geq r \geq 1-\epsilon  \;, \label{rdomain}
\end{equation}
where $0 < \epsilon \ll 1$  with $\epsilon$ independent of $|\Lambda^{(m)}|$.

With (\ref{collar}) in place, (\ref{alphtder1}) and (\ref{htder})
imply (Appendix B) that
\begin{equation}
{\displaystyle \partial \alpha_{\Lambda,h}^{(m)}\over \partial t} (t,r)
\geq  \eta_1(\delta) > 0 \, , \qquad \qquad
- {\displaystyle d h\over \displaystyle dt}(\alpha_{\Lambda,h}^{(m)}(t,r),t)\geq
\eta_2(\delta)>0  \, , \label{dercollar}
\end{equation}
where $\eta_1$, $\eta_2$ are lattice-size independent.
Furthermore, if (\ref{collar}) already holds for $r=1$,
it also holds for any $r$ in (\ref{rdomain}).
We may as well then simplify matters in the following by setting   the
parameter $r$ to the value $r=1-\epsilon$
with some fixed small $\epsilon$. This $\epsilon$ may eventually be taken
as small as one pleases after
a sufficiently large number of decimations have been performed.
This has an obvious meaning in the context of iterating the
decimation transformation as pointed out in subsection \ref{disc1} below.
We accordingly simplify notation by dropping explicit reference to
$r$, except on occasions when a statement is made for general
$r$ values. Thus we write $\alpha_{\Lambda,\,h}^{(m)} (t) \equiv
\alpha_{\Lambda,\,h}^{(m)}(t,1-\epsilon)$, $c^U_j(m)
\equiv c^U_j(m, 1-\epsilon))$, etc.

\subsection{Representation of the partition function on decimated lattices}
\label{repZ}
So, starting on the original lattice spacing $a$, with partition function
given in terms of coefficients $\{c_j(\beta)\}$, one may iterate the
procedure represented by (\ref{interI1}) - (\ref{interI3}).
Taking the same interpolation family $h$ in every cycle,
an iteration cycle consists of the following steps.
\begin{enumerate}
\item[(i)] A decimation transformation $\Lambda^{(m-1)}\to \Lambda^{(m)}$
given by the rules (\ref{RG1}) -
(\ref{RG5}) applied to the coefficients in $Z_{\Lambda^{(m-1)}}$, and
resulting into the upper bound coefficients on $\Lambda^{(m)}$
according to (\ref{upperc}) - (\ref{upperF}) and (\ref{U}). Similarly,
a lower bound
on $\Lambda^{(m)}$ is obtained according to (\ref{L})  with lower bound
coefficients given by (\ref{lowerF1}) and (\ref{lowerc2}) or (\ref{lowerc3}).
\item[(ii)] Interpolation  between the resulting
upper and lower bound partition functions on $\Lambda^{(m)}$
according to (\ref{interc1}), (\ref{interF1}), and
(\ref{interPF1}), (\ref{interPF2}).
\item[(iii)] Fixing the value
$0 < \alpha_{\Lambda,h}^{(m)}(t) < 1$, eq. (\ref{interI2}),
so that the $(m-1)$-th step partition function $Z_{\Lambda^{(m-1)}}$ is
preserved, eq. (\ref{interI3}).
\item[(iv)] Picking a value of the  parameter $t=t_m$, to fix
the coefficients
$\{\tilde{c}_j(m,\alpha_\Lambda^{(m)}(t_m))\}$ of the resulting
partition function $Z_{\Lambda^{(m)}}$, and return to step (i).
\end{enumerate}

This scheme for the coefficients in $Z_{\Lambda^{(m)}}$ may be
depicted as follows:
\begin{equation}
\begin{array}{c}
\begin{array}{ccc}
\hfill &  c_j(\beta) &  \hfill\\
&  \begin{picture}(60,15)
\put(60,15){\vector(-4,-1){80}}
\end{picture}
\begin{picture}(38,8)
\put(20,6){\vector(0,-2){20}}
\end{picture}
\begin{picture}(60,15)
\put(1,15){\vector(4,-1){80}}
\end{picture}  & \\
 & & \\
\end{array} \\
\begin{array}{ccccc}
\hfill \{c_j^L(1)\} & \leq & \{\tilde{c}_j(1,\alpha_{\Lambda,\,h}^{(1)}
(t_1))\} &\leq & \{c_j^U(1)\}\hfill\\
&  \begin{picture}(30,10)
\put(30,10){\vector(-3,-1){50}}
\end{picture}    &
\begin{picture}(38,8)
\put(20,6){\vector(0,-2){20}}
\end{picture}     &
\begin{picture}(30,10)
\put(1,10){\vector(3,-1){50}}
\end{picture}  &  \\
  &  &    &  &   \\
\hfill \{c_j^L(2)\} & \leq & \{\tilde{c}_j(2,\alpha_{\Lambda,\,h}^{(2)}
(t_2))\} &\leq & \{c_j^U(2)\}\hfill\\
&  \begin{picture}(30,10)
\put(30,10){\vector(-3,-1){50}}
\end{picture}    &   \begin{picture}(38,8)
\put(20,6){\vector(0,-2){20}}
\end{picture}                 &
\begin{picture}(30,10)
\put(1,10){\vector(3,-1){50}}
\end{picture}  &  \\
&  &   &   & \\
\vdots & & \vdots & & \vdots
\end{array}\\
\end{array}    \label{S1}
\end{equation}

The result after $n$ iterations is then:
\begin{eqnarray}
Z_\Lambda(\beta) &= & \int dU_\Lambda\;\prod_{p\in \Lambda}\,f_p(U,\beta)
\label{O}\\
  & =& \left[\,\prod_{m=1}^n \tilde{F}_0(m,h,\alpha_{\Lambda,\,h}^{(m)}(t_m),
t_m)^{|\Lambda|/ b^{md}}\,\right]\;
\; Z_{\Lambda^{(n)}}\,\Big(\{\tilde{c}_j( n, \alpha_{\Lambda,\,h}^{(n)}(t_n))
\}\Big) \,. \label{A}
\end{eqnarray}
(\ref{A}) is an {\it exact integral representation} on the
decimated lattice $\Lambda^{(n)}$ of the
partition function $Z_\Lambda$ originally defined on the undecimated
lattice $\Lambda$ by the integral representation
(\ref{PF1}) or (\ref{O}).

III.3 allows the iterative procedure leading to (\ref{A}) to be
implemented in a slightly  different manner, one that turns out later to
be more convenient for our purposes. Since by III.3
\begin{equation}
Z_{\Lambda^{(m)}}\,\Big(\{\tilde{c}_j(m, \alpha_{\Lambda,\,h}^{(m)}(t_m))\}
\Big) \leq
Z_{\Lambda^{(m)}}\,\Big(\{\tilde{c}_j(m, 1)\}\Big) =
Z_{\Lambda^{(m)}}\,\Big(\{c_j^U(m)\}\Big) \,,\label{UPF}
\end{equation}
an upper bound for each successive iteration step is also obtained by applying
III.1 to the r.h.s. rather than the l.h.s. of the inequality sign in
(\ref{UPF}). The only resulting modification  in the above procedure
is in step (i): the upper bound coefficients
$c^U_j(m)$ and $F_0^U(m)$ on $\Lambda^{(m)}$
are computed according to (\ref{upperc}) and (\ref{upperF}) but now
using the set $\{c_j^U(m-1)\}$ rather than the set
$\{\tilde{c}_j(m-1, \alpha_{\Lambda,\,h}^{(m-1)}(t_{m-1}))\}$ as the
coefficient set of the previous step.

The same alternative can be applied to the lower bounds in (\ref{S1}).
Since, again by III.3, one has
\begin{equation}
Z_{\Lambda^{(m)}}\,\Big(\{c_j^L(m)\}\Big) =
Z_{\Lambda^{(m)}}\,\Big(\{\tilde{c}_j(m, 0)\}\Big) \leq
Z_{\Lambda^{(m)}}\,\Big(\{\tilde{c}_j(m, \alpha_{\Lambda,\,h}^{(m)}(t_m))\}
\Big) \,, \label{LPF}
\end{equation}
a lower bound for each successive iteration step is also obtained by applying
III.2 to the l.h.s. rather than the r.h.s. of the inequality sign
in (\ref{LPF}). If one adopts (\ref{lowerc3}), this makes no
difference since the lower bound coefficients equal zero at every step.
If one uses (\ref{lowerc2}), the resulting modification to (\ref{S1}) is
that in step (i) the lower bound coefficients
$c^L_j(m)$ on $\Lambda^{(m)}$
are now computed
using the set $\{c^L_j(m-1)\}$ rather than
$\{\tilde{c}_j(m-1, \alpha_{\Lambda,\,h}^{(m-1)}(t_{m-1}))\}$ as the
coefficient set of the previous step.

One may adopt either or both modifications following from (\ref{UPF}) or
(\ref{LPF}). Adopting both, the iterative scheme for the coefficients in
$Z_{\Lambda^{(m)}}$ replacing (\ref{S1}) is:
\begin{equation}
\begin{array}{c}
\begin{array}{ccc}
\hfill &  c_j(\beta) &  \hfill\\
&  \begin{picture}(60,15)
\put(60,15){\vector(-4,-1){80}}
\end{picture}
\begin{picture}(20,10)
\put(12,8){\vector(0,-2){20}}
\end{picture}
\begin{picture}(60,15)
\put(1,15){\vector(4,-1){80}}
\end{picture}  &  \\
  &  &  \\
\end{array} \\
\begin{array}{ccccc}
\hfill \{ c_j^L(1) \} \qquad & \leq &
\{\tilde{c}_j(1,\alpha_\Lambda^{(1)}(t_1))\}
&\leq &  \qquad \{c_j^U(1)\}\hfill\\
 &  &    &  &   \\
\begin{picture}(30,10)
\put(6,10){\vector(0,-2){20}}
\end{picture} &   &
\begin{picture}(30,10)
\put(18,10){\vector(0,-2){20}}
\end{picture}                &  &
\qquad \begin{picture}(20,10)
\put(1,10){\vector(0,-2){20}}
\end{picture}  \\
 &  &      &  &    \\
\hfill \{ c_j^L(2) \} \qquad & \leq &
\{\tilde{c}_j(2,\alpha_\Lambda^{(2)}(t_2))\}
 &\leq & \qquad \{c_j^U(2)\}\hfill  \\
  &   &        &   &     \\
\begin{picture}(30,10)
\put(6,10){\vector(0,-2){20}}
\end{picture}  &    &
\begin{picture}(30,10)
\put(18,10){\vector(0,-2){20}}
\end{picture}                         &  &
\qquad \begin{picture}(20,10)
\put(1,10){\vector(0,-2){20}}
\end{picture} \\
  &  &       &  &   \\
\vdots\quad\;  & & \  \vdots & & \ \vdots
\end{array}\\
\end{array}    \label{S2}
\end{equation}

This again leads, after $n$ iterations, to the representation (\ref{A}).
Note, however, that the actual numerical value of
$\alpha_{\Lambda,\,h}^{(m)}(t_m)$ in (\ref{A}), fixed at each step
by requiring (\ref{interI3}), will, in general, be different depending on
whether scheme (\ref{S1}) or (\ref{S2}) is used for the iteration.
Also note that the upper bounds $c^U_j(m)$ in (\ref{S2}) are not
optimal compared to those in (\ref{S1}). The scheme (\ref{S2}), however,
turns out to be more convenient for our purposes in the following.

\subsection{Discussion of the representation (\ref{A})}\label{disc1}
As indicated by the notation, on any finite lattice, the
$\alpha_{\Lambda,\,h}^{(m)}$ values possess
a lattice size dependence. This weak dependence enters as a
correction that vanishes  inversely with lattice size.
Indeed, by the standard results on the existence of
the thermodynamic limit of lattice systems, for a partition function
$Z_{\Lambda^{(m)}}(\{c_j\})$ of the form (\ref{PF2})
on lattice $\Lambda^{(m)}$
with torus topology (periodic boundary conditions):
\begin{equation}
\ln Z_{\Lambda^{(m)}}(\{c_j\})= |\Lambda^{(m)}|\,\varphi(\{c_j\}) +
\delta\varphi_{\Lambda^{(m)}}(\{c_j\}) \, ,
\end{equation}
$\varphi(\{c_j\})$ being the free energy per unit volume in the
infinite volume limit, and  $\delta\varphi_{\Lambda^{(m)}}(\{c_j\})\leq
O(\mbox{constant})$.\footnote{That is, there are no
`surface terms' for torus topology. In fact surface terms arising with
other, e.g. free, boundary conditions can be precisely defined as the
difference in the free energies computed with periodic versus such other
boundary conditions \cite{Fi}.}
From this and (\ref{interI3}) it is straightforward to show that
\begin{equation}
\alpha_{\Lambda,h}^{(m)}(t,r) =
 \alpha_h^{(m)}(t,r) + \delta
\alpha_{\Lambda,h}^{(m)}(t,r)  \label{alphsplit1}
\end{equation}
with $\delta \alpha_{\Lambda,h}^{(m)}(t,r) \to 0$ as some inverse power
of lattice size in the large volume limit.
In fact, we have already established the presence of a lattice-size
independent contribution in $\alpha_{\Lambda,h}^{(m)}(t,r)$ in an
alternative manner through (\ref{collar}),
i.e. the fact that  in (\ref{alphsplit1}) one must have
\begin{equation}
\alpha_h^{(m)}(t,r) > \delta^\prime \;.\label{alphsplit2}
\end{equation}
An explicit expression for $\delta^\prime$ is given by (\ref{alphlowerb1}),
(\ref{alphlowerb2}).

At weak and strong coupling the $\alpha_h^{(m)}(t,r)$
values may be
estimated analytically by comparison with the weak and
strong coupling expansions, respectively.
In general, starting from (\ref{interI1}), the location of
$\alpha_{\Lambda,\,h}^{(m)}$ satisfying (\ref{interI3}) may be formulated
as the fixed point of a contraction mapping.
This allows in principle its numerical determination, for given values of
all other parameters, to any desired accuracy.

For our purposes here, however, the actual numerical values
of the $\alpha_{\Lambda,\,h}^{(m)}$'s, beyond the fact that they are fixed
between $0$ and $1$, will not be directly relevant.
The main application of the representation (\ref{A}) in this paper will
be to relate the behavior of the exact theory to
that of the easily computable approximate decimations bounding it without
explicit  knowledge of the actual  $\alpha_{\Lambda,\,h}^{(m)}$ values.

It is important to be clear about the meaning of (\ref{A}).
The partition function $Z_\Lambda(\beta)$ is originally
given by its integral representation (\ref{O}) on
lattice $\Lambda$ of spacing $a$. (\ref{A}) then gives another integral
representation of $Z_\Lambda(\beta)$ in terms of an integrand defined
on the coarser lattice $\Lambda^{(n)}$ of spacing $b^na$ plus
a total bulk free energy contribution resulting from
decimating between scales $a$ and $b^na$. The action
$A_p(U,n,\alpha_{\Lambda,h}^{(n)})$ in $\dZ{n}(\{\tilde{c}_j(n,
\alpha_{\Lambda,h}^{(n)}\})$ is constructed to reproduce
this one physical quantity, i.e. the free energy $\ln Z_\Lambda(\beta)$,
nothing more and nothing less. In particular, it is {\it not} implied
that this action on $\Lambda^{(n)}$ can also be used to exactly
compute any other observable. For that one would need to attempt
the previous development from scratch with the corresponding
operator inserted in the integrand.

Recall that, by (\ref{interc3}), the coefficients $\tilde{c}_j(n,\alpha,r)$'s
are increasing in $\alpha$, and $\tilde{c}_j(n,1,r) =c_j^U(n,r)$,
$\;\tilde{c}_j(n,0,r) =c_j^L(n)$:
\begin{equation}
c_j^L(n) <  \tilde{c}_j(\,n,\alpha_{\Lambda,h}^{(n)}(t)\,) <
c_j^U(n)\; , \qquad \quad
\quad 0 < \alpha_{\Lambda,h}^{(n)(t)}(t) <  1 \;.\label{cineq5}
\end{equation}
Thus,
the coefficients $\tilde{c}_j(\,n,\alpha_{\Lambda,h}^{(n)}(t)\,)$
in the representation (\ref{A}) are bounded from above by
$c_j^U(n)$ no matter what the actual values of
$\alpha_{\Lambda,h}^{(n)}(t)$ are.

When considering the implications of this bound under
successive decimations the advantage of employing scheme (\ref{S2}),
rather than (\ref{S1}), becomes clear.
The coefficients $c_j^U(n)$ on the r.h.s. column
in (\ref{S2}) are obtained by straightforward iteration of the decimation
rules (\ref{RG2})-(\ref{RG5}) with $\zeta=b^{d-2}$; i.e. only
knowledge of the $c_j^U(n-1)$, not of the $\tilde{c}_j(n-1,
\alpha_{\Lambda,h}^{(n-1)}(t_{n-1}))$, is
required to obtain  the $c_j^U(n)$ at the $n$-th step.
The flow of these $c_j^U(n)$ coefficients then constrains the flow of the
exact representation coefficients  $\tilde{c}_j(n,
\alpha_{\Lambda,h}^{(n)}(t_n))$
according to (\ref{cineq5}) from above.
In particular,
{\it if the $c_j^U(n)$'s on the r.h.s. column in (\ref{S2})
approach the strong coupling  fixed point, i.e.
\begin{equation}
F_0^U(n)\to 1, \quad \quad
c_j^U(n) \to 0, \qquad  \mbox{as}\quad  n\to \infty \,,\label{scfp}
\end{equation}
so must
the $\tilde{c}_j(n,\alpha_{\Lambda,h}^{(n)})$'s in the representation
(\ref{A}).}\footnote{
To strictly draw the same conclusion from the alternative scheme
(\ref{S1}) requires an
additional step, such as showing that the $c_J^U(n)$'s computed
according to the scheme (\ref{S1}) flow to the strong coupling regime
if those computed according to (\ref{S2}) do.}

Now the coefficients $c_j^U(n,r)$ at $r=1$ are the MK decimation
coefficients (cf. section \ref{DEC2}).
As it is well-known, the MK decimations
for $SU(2)$ (and also $SU(3)$) are found by explicit evaluation to
indeed flow to the strong coupling fixed point (\ref{scfp})
for all starting $\beta<\infty$ and $d\leq 4$. Above the critical dimension
$d=4$, the decimations result in free spin wave behavior ($c_j^U(n,1)
\to 1$ as $n\to \infty$) starting from any
$\beta > \beta_0$, where $\beta_0 =O(1)$.

Here, for reasons discussed at the end of section \ref{interbounds},
we take $r$ in the range (\ref{rdomain}). This may be viewed as
fixing the direction from
which the point $\zeta=b^{(d-2)}$, $r=1$ in the parameter space of the
iteration (\ref{RG2}) - (\ref{RG5}) is approached.
This is actually irrelevant for the flow behavior of the $
c_j^U(n,1-\epsilon)\equiv c_j^U(n)$ since, in the case of $SU(2)$ considered
here, this point  is a structurally stable point of the
iteration.\footnote{It is,
however, very much relevant in cases where this point
is not structurally stable, e.g. in $U(1)$.}

Note that  zero lattice coupling, $g=0$, is a fixed
point as it is
for the MK decimations. This is also
evident from $\lim_{\beta\to\infty}c_j(\beta)=1$ and III.2.

What does (\ref{scfp}) combined with (\ref{cineq5})
imply about the question of confinement in the exact theory?
The
fact that the long distance part, $\dZ{n}(\{\tilde{c}_j(n,
\alpha_{\Lambda,h}^{(n)})\})$, in (\ref{A}) flows in
the strong coupling regime does not suffice to answer the question.
It is the combined contributions
from all scales between $a$ and $b^na$ in (\ref{A}) that add up to give
the exact free energy $\ln Z_\Lambda(\beta)$. Indeed, recall that,
by a parametrization change by shifts in $t$ at each decimation step,
one can shift the relative amounts assigned to these
various contributions keeping the total sum fixed (cf. remarks immediately
following (\ref{htder}). This parametrization freedom will in fact
be important in the following. On the other hand, the fact that
by (\ref{cineq5})
the flow of $\tilde{c}_j(n,\alpha_{\Lambda,h}^{(n)}(t_n))$ to the strong
coupling regime is independent of such parametrization changes
is strongly suggestive. At any rate,
to unambiguously determine the long distance behavior of the theory
one needs to consider appropriate long distance order parameters.

\section{`Twisted' partition function} \label{TZ}
\setcounter{equation}{0}
\setcounter{Roman}{0}
The above derivation leading to the representation (\ref{A}) for
the partition function  cannot be applied in the presence of
observables without modification. Thus, in the presence of
operators involving external sources, such as the Wilson or
't Hooft loop, translation invariance is lost. Reflection
positivity is also reduced to hold only in the plane bisecting
a rectangular loop.
Fortunately, there are other order parameters that can
characterize the possible phases of the theory while
avoiding most of these complications.
They are the well-known vortex
free energy, and its transform with respect to the center of
the gauge group (electric flux free energy).
They are in fact the natural order parameters in the present context
since they are constructed out of partition functions, i.e.
partition functions in the presence of external fluxes.

Let $Z_\Lambda(\tau_{\mu\nu}, \beta)$ denote the partition function
with action modified by the `twist' $\tau_{\mu\nu}$, i.e. an element of the
group center, for every plaquette on a
coclosed set of plaquettes ${\cal V}_{\mu\nu}$ winding through the
periodic lattice in the $(d-2)$ directions perpendicular to the
$\mu$, and $\nu$-directions, i.e. winding through every
$[\mu\nu]$-plane for fixed $\mu, \nu$:
\begin{equation}
A_p(U_p) \to A_p(\tau_{\mu\nu} U_p) \;, \qquad \mbox{if} \quad
p\in {\cal V}_{\mu\nu}\;. \label{twist1}
\end{equation}
A nontrivial twist  ($\tau_{\mu\nu}\not=1$) represents a discontinuous
gauge transformation on the set ${\cal V}_{\mu\nu}$ with
multivaluedness in the group center. Thus, for group $SU(N)$, it introduces
vortex flux characterized by elements of $\pi_1(SU(N)/Z(N))=Z(N)$.
The vortex is rendered topologically
stable by being wrapped around the lattice torus.

In the case of $SU(2)$ explicitly  considered here, there is only one
nontrivial element, $\tau_{\mu\nu}=-1$.
As indicated by the notation
$Z_\Lambda(\tau_{\mu\nu},\beta)$, the twisted partition function  depends only
on the directions in which ${\cal V}_{\mu\nu}$
winds through the lattice, not
the exact shape or location of ${\cal V}_{\mu\nu}$.
This expresses the mod 2 conservation of flux.
Indeed, a  twist $\tau_{\mu\nu}=-1$ on the plaquettes forming
a  coclosed set ${\cal V}_{\mu\nu}$ can be moved to the plaquettes forming any
other homologous
coclosed set ${\cal V}^{\ \prime}$ by the change of variables
$U_b \to -U_b$ for each bond $b$
in a set of bonds cobounded by ${\cal V} \cup {\cal V}^{\ \prime}$,
leaving  $Z_\Lambda(\tau_{\mu\nu},\beta)$ invariant. By the same token,
$Z_\Lambda(\tau_{\mu\nu},\beta)$ is invariant under changes
mod 2 in the number of
homologous coclosed sets in $\Lambda$ carrying a twist.
In the following, for definiteness, we fix, say, $\mu=1$, $\nu=2$, and drop
further explicit reference to the $\mu$, $\nu$ indices. Also,
we write $Z_\Lambda(-1,\beta) \equiv Z_\Lambda^{(-)}(\beta)$.

(\ref{twist1}) implies that $Z_\Lambda^{(-)}$ is obtained from
$Z_\Lambda$ by the replacement
\begin{equation}
f_p(U_p,a) \to f_p(-U_p,a)=
\Big[\, 1 + \sum_{j\not= 0} (-1)^{2j}\,d_j\,c_j(\beta)\,
\chi_j(U_p)\,\Big]  \label{twist2}\,, \qquad \mbox{for each} \quad p\in {\cal V} \,,
\end{equation}
in (\ref{PF1}), (\ref{nexp}), i.e. only half-integer representations on
plaquettes in ${\cal V}$ are
affected. In general then, the twisted version of the partition function
(\ref{PF2}) on $\Lambda^{(n)}$ is
\begin{equation}
Z^{(-)}_{\Lambda^{(n)}}(\{c_j(n)\}) =
\int dU_{\Lambda^{(n)}}\;
\prod_{p\in \Lambda^{(n)}}\,f^{(-)}_p(U_p,n)  \; ,
\label{PF1atwist}
\end{equation}
with
\begin{equation}
f^{(-)}_p(U_p,n) =
\Big[\, 1 + \sum_{j\not= 0} (-1)^{2j\,S_p[{\cal V}]}\,d_j\,c_j(n)\,
\chi_j(U_p)\,\Big]\;.\label{PF1btwist}
\end{equation}
$S_p[{\cal V}]$ denotes the characteristic function of the plaquette set ${\cal V}$,
i.e. $S_p[{\cal V}]=1$ if $p\in {\cal V}$, and $S_p[{\cal V}]=0$ otherwise.
A simple result (Appendix A) of obvious physical significance is:

\prop{ With $c_j(n) \geq 0$, all $j$,
\begin{equation}
Z^{(-)}_{\Lambda^{(n)}}(\{c_j(n)\}) \leq Z_{\Lambda^{(n)}}(\{c_j(n)\}) \,.
\label{Z>Z-}
\end{equation} }
Strict inequality holds in fact in
(\ref{Z>Z-}) for any nonvanishing $\beta$ on any finite lattice.

Application of the decimation operation
defined in section \ref{DEC}  on some given
$Z^{(-)}_{\Lambda^{(m-1)}}$ of the form (\ref{PF1atwist}) results in the rule
\begin{equation}
f^{(-)}_p(U, m-1) \to F_0(m)\,f^{(-)}_p(U,m) = F_0(m) \,
\Big[ 1 + \sum_{j\not= 0} (-1)^{2j\,S_p[{\cal V}]}\,d_j\,c_j(m)\,\chi_j(U) \Big]
\, ,\label{RG1twist}
\end{equation}
with coefficients $F_0(m)$, $c_j(m)$ computed according to the rules
(\ref{RG2}) - (\ref{RG5}). Starting on lattice $\Lambda$, the twisted
partition function resulting after $n$ such steps is
\begin{equation}
Z_\Lambda^{(-)}(\beta, n) = \prod_{m=1}^n
F_0(m)^{|\Lambda|/b^{md}}\; Z_{\Lambda^{(n)}}^{(-)}(\{c_j(n)\})
\; . \label{PF2twist}
\end{equation}
Note that the flux is carried entirely in
$Z_{\Lambda^{(n)}}^{(-)}$. Indeed, bulk free energy contributions
from each $\Lambda^{(m-1)} \to \Lambda^{(m)}$ decimation step arise from local
moving-integration operations within cells of side
length $b$ on $\Lambda^{(m-1)}$,
i.e. topologically trivial subsets, and are thus insensitive to the flux
presence.
The evolution with $n$ of the effective action in $Z_{\Lambda^{(n)}}^{(-)}$
then determines the manner in which flux spreads,
which is characteristic of the phase the system is in.

\subsection{Upper and lower bounds}
In the presence of the flux, the measure in (\ref{PF1atwist}) possesses
the property of reflection positivity only in $(d-1)$-dimensional
planes perpendicular to any one of
the directions $\rho \not= 1,2$ in which ${\cal V}$ winds around the lattice.
One way of dealing with this is to simply consider the quantity
\begin{equation}
Z^+_{\Lambda^{(n)}}(\{c_j(n)\})\equiv  {1\over 2}
\Big(Z_{\Lambda^{(n)}}(\{c_j(n)\}) +
Z_{\Lambda^{(n)}}^{(-)}(\{c_j(n)\})\Big) \label{Zplus}
\end{equation}
instead of $Z_{\Lambda^{(n)}}^{(-)}$.
It is indeed easily checked that reflection positivity holds for
the measure in $Z^+_{\Lambda^{(n)}}$ in all planes.
A direct consequence of this (Appendix A) is then the analog of II.1:

\prop{For $\dZ{n}^+(\{c_j(n)\})$ given by (\ref{Zplus})
with $c_j(n) \geq 0$ for all $j$, and periodic boundary conditions,

(i) $\dZ{n}^+(\{c_j(n)\})$ is an increasing function of each
$c_j(m)$:
\begin{equation}
\partial \dZ{n}^+(\{c_i(n)\}) / \partial c_j(n) \geq 0 \; ;\label{PFplusder0}
\end{equation}

(ii)
\begin{equation}
\dZ{n}^+(\{c_j(n)\}) \geq \Big[\,1 + \sum_{j\not=0} d_j^2\,
c_j(n)^6 \,\Big]^{|\Lambda^{(n)}|} \; .\label{PFpluslowerb1}
\end{equation}
}
Again, in these bounds equality holds only in the trivial case where
all $c_j(n)$'s vanish. In particular, one has
\begin{equation}
\dZ{n}^+(\{c_j(n)\}) \; > \; 1 \,.\label{PFpluslowerb2}
\end{equation}
Note that these bounds are identical to
those in II.1. This signifies the obvious fact that they bound from below by
underestimating the bulk free energies proportional to the lattice volume,
whereas the lattice size dependence of the free energy discrepancy between
$\dZ{n}(\{c_j(n)\})$ and $\dZ{n}^{(-)}(\{c_j(n)\})$ is much weaker.

Upper and lower bound statements analogous to III.1 and III.2 can be
obtained for $Z^+_{\Lambda^{(n)}}$. One has:

\prop{
For $Z_{\Lambda^{(n-1)}}^+$ of the form
(\ref{Zplus}),
a decimation transformation (\ref{RG1twist}), (\ref{RG2}) -
(\ref{RG5}) with $\zeta=b^{d-2}$ and $0 < r\leq 1$  results in an upper
bound on $Z_{\Lambda^{(n-1)}}^+$:
\begin{equation}
Z^+_{\Lambda^{(n-1)}}(\{c_j(n-1)\})\,  \leq \,
F_0^U(n)^{|\Lambda^{(n)}|}\,
Z^+_{\Lambda^{(n)}}(\{c_j^U(n,r)\})\;. \label{Uplus}
\end{equation}
The r.h.s. in (\ref{Uplus}) is a monotonically
decreasing function of $r$ on $0 < r\leq 1$.
}

\prop{
For $Z_{\Lambda^{(n-1)}}^+$ of the form (\ref{Zplus}):
\begin{equation}
Z^+_{\Lambda^{(n)}}(\{c_j^L(n)\}) \, \leq \, Z^+_{\Lambda^{(n-1}}(\{
c_j(n-1)\}) \;, \label{Lplus}
\end{equation}
where the coefficients $c_j^L(n)$ are given by (\ref{lowerc1}).
}

The proof of IV.3, as well as that of IV.4, an easy corollary
of IV.2, are given in Appendix A. It then follows from (\ref{PFplusder0}))
that (\ref{Lplus}) holds also with coefficients $c_j^L(n)$ given by
(\ref{lowerc2}) or (\ref{lowerc3}).
Again, in analogy to III.2, IV.4 also holds
with $c_j^L$ given by (\ref{lowerc4}), but this form will not be used here.

\subsection{Representation of $Z_\Lambda + Z_\Lambda^{(-)}$
on decimated lattices}

The procedure of section \ref{Z}
leading to the representation (\ref{A}) for $Z_\Lambda$
can now be applied to $Z^+_\Lambda= (Z_\Lambda + Z_\Lambda^{(-)})/2$.
One introduces the
interpolating coefficients $\tilde{c}_j(m,\alpha,r)$
given by eq. (\ref{interc1}), and
$\tilde{F}_0(m,h,\alpha,t)$ given by eq. (\ref{interF1}) for some
choice of interpolation function $h$ such as given by the examples
(\ref{h}). The quantity corresponding to
(\ref{interPF1}) is then given by
\begin{equation}
\tilde{Z}^+_{\Lambda^{(m)}}(\beta,h,\alpha,t,r)
= \tilde{F}_0(m,h,\alpha,t)^{|\Lambda^{(m)}|}\,
Z^+_{\Lambda^{(m)}}(\{\tilde{c}_j(m,\alpha,r)\}) \label{interPF1plus}
\end{equation}
where
\begin{equation}
Z^+_{\Lambda^{(m)}}(\{\tilde{c}_j(m,\alpha,r)\})
= {1\over2} \Big( Z_{\Lambda^{(m)}}(\{\tilde{c}_j(m,\alpha,r)\}) +
Z^{(-)}_{\Lambda^{(m)}}(\{\tilde{c}_j(m,\alpha,r)\}) \Big)
\label{interPF2plus}
\end{equation}
with $Z_{\Lambda^{(m)}}(\{\tilde{c}_j(m,\alpha,r)\})$ given by
(\ref{interPF2}) and $Z^{(-)}_{\Lambda^{(m)}}(\{\tilde{c}_j(m,\alpha,r)\})$
given by (\ref{PF1atwist}) - (\ref{PF1btwist}) with
coefficients $\tilde{c}_j(m,\alpha,r)$.
We then have the analog of III.3:

\prop{
The interpolating free energies
$\ln Z^+_{\Lambda^{(m)}}(\{\tilde{c}_j(m,\alpha,
r)\})$ and
$\ln \tilde{Z}^+_{\Lambda^{(m)}}(\beta,h,
\alpha,t,r)$ are
increasing functions of $\alpha$:
\begin{equation}
\partial \ln Z^+_{\Lambda^{(m)}}\Big(\{\tilde{c}_j(m,\alpha, r)\} \Big)
/\partial \alpha \, > \,  0
\,.  \label{interPFder1plus}
\end{equation}
}

In terms of (\ref{interPF1plus}), IV.3 and IV.4 give
\begin{equation}
\tilde{Z}^+_{\Lambda^{(m)}}(\beta,h,0,t,r)
\leq Z^+_{\Lambda^{(m-1)}} \leq
\tilde{Z}^+_{\Lambda^{(m)}}(\beta,h,1,t,r) \, . \label{interI1plus}
\end{equation}
which implies that there exist a value of
$\alpha$ in $(0,1)$:
\begin{equation}
\alpha^+(m, h,t,r,\{c_j(m-1)\},b,\Lambda)\equiv
\alpha_{\Lambda,\, h}^{+(m)}(t,r)   \label{interI2plus}
\end{equation}
such that
\begin{equation}
\tilde{Z}^+_{\Lambda^{(m)}}(\beta,h,\alpha_{\Lambda,\, h}^{+(m)}(t,r),
t,r) =  Z^+_{\Lambda^{(m-1)}}  \,.\label{interI3plus}
\end{equation}
This value is unique, for given values of $t, r$, by IV.5.
$\alpha_{\Lambda,\, h}^{+(m)}(t,r)$ gives the regular level surface
of the function $\tilde{Z}^+_{\Lambda^{(m)}}(\beta,h,\alpha,t,r)$
fixed by the value $Z^+_{\Lambda^{(m-1)}}$.

All the considerations concerning the dependence on the parameters
$t, r$ in the previous section carry over directly to
$\alpha_{\Lambda,\, h}^{+(m)}(t, r)$.
In particular, one has
\begin{equation}
{\partial \alpha_{\Lambda,\, h}^{+(m)}(t,r)\over \partial t} =
v^+(\alpha_{\Lambda,\,h}^{+(m)}(t,r), t, r) \;,
\label{alphplustder1}
\end{equation}
where
\begin{equation}
v^+(\alpha, t, r) \equiv
- { \displaystyle {\partial h(\alpha,t) / \partial t}
\over{\displaystyle  {\partial h(\alpha,t)\over \partial \alpha} +
A^+_{\Lambda^{(m)}}(\alpha, r)} }
\;,\label{alphplustder2}
\end{equation}
with
\begin{equation}
A^+_{\Lambda^{(m)}}(\alpha, r) \equiv  {1\over \ln F_0^{U}(m) }\,
{1\over |\Lambda^{(m)}| }\,
{\partial \over \partial\alpha }
\ln Z^+_{\Lambda^{(m)}}\,\Big(\{\tilde{c}_j( n,
\alpha, r)\}\Big) > 0\;. \label{alphplustder3}
\end{equation}
Again, we always assume that $h$ is chosen such that
$\partial h/\partial t$ is negative. Then, from (\ref{alphplustder2}), $v^+>0$
on $0 <\alpha < 1$, with $v^+=0$ at $\alpha=0$ and $\alpha=1$.  Also
\begin{equation}
{d h(\alpha_{\Lambda,h}^{+(m)}(t,r),t)\over dt}
= - {\partial \alpha_{\Lambda,h}^{+(m)}(t,r)\over \partial t} \,
A^+_{\Lambda^{(m)}}(\alpha_{\Lambda,h}^{+(m)}(t,r), r)
\,. \label{hplustder}
\end{equation}
The derivative w.r.t. $r$ is similarly given by
(\ref{alphplusrder1}).

The values (\ref{interI2plus}) obey
\begin{equation}
\delta^{+\,\prime} < \alpha_{\Lambda,\,h}^{+(m)}(t,r)<
1-\delta^+  \label{collarplus}
\end{equation}
with lattice-size independent, positive $\delta^+$ and $\delta^{+\,\prime}$.
Again, the lower bound is automatically satisfied, whereas the upper
bound is ensured by letting the parameter $r$ vary, if necessary,
in (\ref{rdomain}) (cf. Appendix B).
From this it follows that the analog of (\ref{dercollar}):
\begin{equation}
{\displaystyle \partial \alpha_{\Lambda,\,h}^{+(m)}\over \partial t} (t,r)
\geq  \eta^+_1(\delta^+) > 0 \, , \qquad \qquad
- {\displaystyle d h\over \displaystyle dt}(\alpha_{\Lambda,\,h}^{+(m)}(t,r),t) \geq
\eta^+_2(\delta^+)>0  \, , \label{dercollarplus}
\end{equation}
holds for some lattice-size independent $\eta^+_1$, $\eta^+_2$.
Since, furthermore, (\ref{collarplus}) holds for any $r$ if it already
holds for $r=1$, we may again set
$r = 1-\epsilon$, and, according to the convention introduced in the
previous section, write $\alpha_{\Lambda,\,h}^{+(m)}(t) \equiv
\alpha_{\Lambda,\,h}^{+(m)}(t, 1 -\epsilon)$,
etc.

As in the last section, one may iterate this procedure of
performing a decimation transformation to produce upper and lower bounds
according to (\ref{interI1plus}), and then fixing the value
(\ref{interI2plus}) of the interpolating parameter $\alpha$ according to
(\ref{interI3plus}).
Assume that we choose the same interpolation family $h$ at every step.
Then starting from the original lattice, after $n$ iterations one obtains
\begin{eqnarray}
Z^+_\Lambda(\beta) &= & {1\over 2} \Big(
Z_\Lambda(\beta) + Z^{(-)}_\Lambda(\beta)\Big)
\nonumber \\
  & =& \left[\,\prod_{m=1}^n \tilde{F}_0(m, h,\alpha_{\Lambda,\,
h}^{+(m)}(t_m), t_m)^{|\Lambda|/ b^{md}}\,\right]\;
\; Z_{\Lambda^{(n)}}^+\,\Big(\{\tilde{c}_j( n, \alpha_{\Lambda,\, h}^{+(n)}
(t_n))\}\Big) \,. \label{B}
\end{eqnarray}

The discussion in subsection \ref{disc1} concerning the representation
(\ref{A}) of $Z_\Lambda$ applies equally well to (\ref{B}). In particular,
note that again the existence of the large volume limit implies that
\begin{equation}
\alpha_{\Lambda,h}^{+(m)}(t,r) =
 \alpha_h^{+(m)}(t,r) + \delta
\alpha_{\Lambda,h}^{+(m)}(t,r)  \label{alphplussplit1}
\end{equation}
with $\delta \alpha_{\Lambda,h}^{+(m)}(t,r) \to 0$ as some inverse power of
lattice size in the  $|\Lambda^{(m)}|\to \infty$ limit.
Alternatively, (\ref{collarplus}) already implies that one must have
a lattice-size independent contribution $\alpha_h^{+(m)}(t,r) \,>\,
\delta^{+\,\prime}\,>\,0$
in (\ref{alphplussplit1}) (cf. Appendix B).

Again, either scheme (\ref{S1}) or (\ref{S2}) may be used to
obtain (\ref{B}). For the reasons already noted, however,
the latter scheme is more convenient for our considerations.
Note, furthermore, that the bounding coefficients
$c^U_j(m)$ and $c_j^L(m)$ in this scheme are
the same for $Z_\Lambda$ and $Z^+_\Lambda$ since they do not depend
on $\alpha_{\Lambda,\,h}^{(m-1)}$ or $\alpha_{\Lambda,\,h}^{+(m-1)}$.
We, therefore, adopt it in what follows as the common iteration scheme
for  $Z_\Lambda$ and $Z^+_\Lambda$:
\begin{equation}
\begin{array}{c}
\begin{array}{ccc}
 & \quad   c_j(\beta) \qquad &  \\
&  \begin{picture}(60,20)
\put(45,20){\vector(-4,-1){100}}
\end{picture}
\begin{picture}(30,20)\put(8,20){\vector(0,-2){20}}\end{picture}
\begin{picture}(60,20)
\put(1,20){\vector(4,-1){100}}
\end{picture}  &
\end{array} \\
\begin{array}{ccccc}
\hfill \{c_j^L(1)\} \quad & \leq &
\{\tilde{c}_j(1,\alpha_{\Lambda,\,h}^{(1)}(t_1))\}, \;
\{\tilde{c}_j(1,\alpha_{\Lambda,\,h^+}^{+(1)}(t^+_1))\}
&\leq & \quad \{c_j^U(1)\}\hfill\\
\begin{picture}(30,20)
\put(8,20){\vector(0,-2){20}}
\end{picture} &   &
\begin{picture}(30,20)\put(10,20){\vector(0,-2){20}}\end{picture}
          &  &
\qquad \begin{picture}(30,20)
\put(1,20){\vector(0,-2){20}}
\end{picture}  \\
\hfill \{c_j^L(2)\} \quad & \leq &
\{\tilde{c}_j(2,\alpha_{\Lambda,\,h}^{(2)}(t_2))\}, \;
\{\tilde{c}_j(2,\alpha_{\Lambda,\,h^+}^{+(2)}(t^+_2))\}
&\leq & \quad \{c_j^U(2)\}\hfill  \\
\begin{picture}(30,20)
\put(8,20){\vector(0,-2){20}}
\end{picture}  &    &
\begin{picture}(30,20)\put(10,20){\vector(0,-2){20}}\end{picture}
              &  &
\qquad \begin{picture}(30,20)
\put(1,20){\vector(0,-2){20}}
\end{picture} \\
\vdots \; \quad & & \vdots & & \vdots \;
\end{array}\\
\end{array}    \label{S3}
\end{equation}
In (\ref{S3}) and in the following, the more detailed notation $h^+$ and
$t^+$ is used for the choice of interpolation and
$t$-parameter values occurring in (\ref{B}) whenever they  need be
distinguished from those used in the representation (\ref{A}) for
$Z_\Lambda$, which can, of course, be chosen independently.

As indicated by the notation, even for common choice of interpolation
$h=h^+$ and of all other parameters, the values of
$\alpha_{\Lambda,\,h}^{+(m)}(t ,r)$
fixed by the requirement (\ref{interI3plus}) are a priori distinct
from those of $\alpha_{\Lambda,\,h}^{(m)}(t,r)$ fixed by
(\ref{interI3}). It is easily seen, however, that for sufficiently
large lattice volume they must nearly coincide. We examine this
difference more precisely below.

\section{The ratio $Z_\Lambda^{(-)}/Z_\Lambda$}\label{Z-/Z}
\setcounter{equation}{0}
\setcounter{Roman}{0}

We may now compare $Z_\Lambda$ and $Z_\Lambda + Z^{(-)}_\Lambda$ by
means of their representations (\ref{A}) and (\ref{B}) on
successively decimated lattices.
Consider then the ratio of $Z_\Lambda + Z_\Lambda^{(-)}$ and $Z_\Lambda$ as
given by (\ref{B}) and (\ref{A}) with common choice
of interpolation $h=h^+$ after one decimation:
\begin{eqnarray}
\left(\,1+ {Z_\Lambda^{(-)} \over Z_\Lambda }\,\right) & = &
{ 2 \tilde{Z}^+_{\Lambda^{(1)}}\,(\beta, h,
\alpha_{\Lambda,\, h}^{+(1)}(t^+),\, t^+) \over
\tilde{Z}_{\Lambda^{(1)}}\,(\beta, h,
\alpha_{\Lambda,\,h}^{(1)}(t),\, t) } \label{ratio1a} \\
& = & \left(\,{ \tilde{Z}_{\Lambda^{(1)}}\,(\beta, h,
\alpha_{\Lambda,\, h}^{+(1)}(t^+),\, t^+) \over
\tilde{Z}_{\Lambda^{(1)}}\,(\beta, h,
\alpha_{\Lambda,\,h}^{(1)}(t),\, t) }\,\right) \left(\,1+
{ Z_{\Lambda^{(1)}}^{(-)}\,\Big(\{\tilde{c}_j(1,\alpha_{\Lambda,\,h}^{+(1)}
(t^+))\}\Big)
\over Z_{\Lambda^{(1)}}\,\Big(\{\tilde{c}_j(1,\alpha_{\Lambda,\,h}^{+(1)}
(t^+))\}\Big) }
\,\right)
\label{ratio1}
\end{eqnarray}
By construction, the r.h.s. is invariant under independent variations
of $t$ and $t^+$.
Now since by IV.1
\begin{equation}
 1< \left(\,1+ {Z_\Lambda^{(-)} \over Z_\Lambda }\,\right) < 2 \; \qquad
\mbox{and} \qquad
1 < \left(\,1+
{ Z_{\Lambda^{(1)}}^{(-)}\,\Big(\{\tilde{c}_j(1,\alpha_{\Lambda,\,h}^{+(1)}
(t^+))\}\Big)
\over Z_{\Lambda^{(1)}}\,\Big(\{\tilde{c}_j(1,\alpha_{\Lambda,\,h}^{+(1)}(t^+))
\}\Big) }
\,\right) <2 \;,\label{ratiobounds}
\end{equation}
it follows that\footnote{(\ref{ratioconstr}) clearly
holds for general values of the parameter $r$, not just for the values
(\ref{rdomain}) used in (\ref{ratio1}).}
\begin{equation}
{1\over 2} < { \tilde{Z}_{\Lambda^{(1)}}\,(\beta, h,
\alpha_{\Lambda,\, h}^{+(1)}(t^+),\, t^+) \over
\tilde{Z}_{\Lambda^{(1)}}\,(\beta, h,
\alpha_{\Lambda,\,h}^{(1)}(t),\,  t) }
 < 2 \;. \label{ratioconstr}
\end{equation}
Though the bounds (\ref{ratiobounds}) are rather crude, the resulting
constraint (\ref{ratioconstr}) is quite informative.
First, it says that if in the equality (\ref{interI3}), i.e.
\[ \tilde{Z}_{\Lambda^{(1)}}\,(\beta, h,
\alpha_{\Lambda,\,h}^{(1)}(t), t) = Z_\Lambda \]
one substitutes for $\alpha_{\Lambda,\,h}^{(1)}(t)$ the wrong level
surface $\alpha_{\Lambda,\, h}^{+(1)}(t)$,
the resulting discrepancy in the free energy per unit volume
is at most $O(1/|\Lambda^{(1)}|)$. Furthermore, (\ref{ratioconstr})
constrains by how much $\alpha_{\Lambda,h}^{+(1)}(t)$
can differ from $\alpha_{\Lambda,h}^{+(1)}(t^+)$ at $t=t^+$.
From the definition (\ref{interPF1}) and III.3, the change
in $\tilde{Z}_{\Lambda^{(1)}}\,
(\beta,h, \alpha, t, r)$
under a shift $\delta \alpha$ in $\alpha$ satisfies
\begin{equation}
|\,\delta \ln \tilde{Z}_{\Lambda^{(1)}}\,
(\beta, h, \alpha, t, r)| >
|\,\delta \alpha |\,|\Lambda^{(1)}| \ln F_0^U(1)\,
{\partial h(\alpha,t)\over \partial \alpha} \;. \label{varcon}
\end{equation}
When combined
with (\ref{varcon}), the constraint (\ref{ratioconstr}), taken at general $r$,
implies that one
must have
\begin{equation}
| \alpha_{\Lambda,\,h}^{+(1)}(t,r) -
\alpha_{\Lambda,\,h}^{(1)}(t,r) | \leq
O({1\over |\Lambda^{(1)}|}) \;. \label{alphdiff}
\end{equation}
This implies that in (\ref{alphsplit1}), (\ref{alphplussplit1}) one
has $\alpha_h^{(1)}(t)=\alpha_h^{+(1)}(t)$, i.e. any difference occurs only in
the parts $\delta \alpha_{\Lambda,\,h}^{(1)}$, $\delta \alpha_{\Lambda,\,h}^{
+(1)}$ that vary inversely with lattice size. Thus, in the large volume
limit, this difference becomes unimportant if one is
interested only in the computation of partition functions, or
bulk free energies. This, however, is
not the case for free energy differences such as the ratio (\ref{ratio1a}).
Indeed, any discrepancy of the size (\ref{alphdiff}) means that
the first factor in (\ref{ratio1}) can contribute as much as the second
factor in round brackets. Thus the expression  for the ratio
of the twisted to the untwisted partition function given by (\ref{ratio1}),
though exact, is not immediately useful for extracting  this
ratio on the coarser lattice.

To address this issue one may make use of the $t$-parametrization invariance
of (\ref{ratio1}).
First  the cancellation of
the bulk energies generated in the integration from
scale $a$ to $ba$ is made explicit as follows.
For any given $t^+_1$, choose $t_1$ in  $\tilde{Z}_{\Lambda^{(1)}}
\,(\beta,  \alpha_{\Lambda,h}^{(1)}(t_1), t_1)$ so
that
\begin{equation}
h(\alpha_{\Lambda,\,h}^{(1)}(t_1), t_1) =
h(\alpha_{\Lambda,\,h}^{+(1)}(t^+_1), t^+_1) \;.\label{hequal1}
\end{equation}
This is clearly always possible by
(\ref{dercollar}) and (\ref{dercollarplus}), and by (\ref{alphdiff});
in fact,  $t_1-t^+_1 = O(1/|\Lambda^{(1)}|)$.
Then (\ref{ratio1a}) assumes the form
\begin{equation}
\left(\,1+ {Z_\Lambda^{(-)} \over Z_\Lambda }\,\right) =
{ 2 Z^+_{\Lambda^{(1)}}\,\Big(\{\tilde{c}_j(1,\alpha^+_{\Lambda,\,h}(t^+_1))
\}\Big)
\over Z_{\Lambda^{(1)}}\,\Big(\{\tilde{c}_j(1,\alpha_{\Lambda,\,h}(t_1))
\}\Big) } \;. \label{ratio2}
\end{equation}

We may now iterate this  procedure  performing $(n-1)$ decimation steps
according to the scheme (\ref{S3}), at each step
choosing  $t_m$, $t^+_m$ such that
\begin{equation}
h(\,\alpha_{\Lambda,\,h}^{(m)}(t_m), t_m) =
h(\,\alpha_{\Lambda,\,h}^{+(m)}(t^+_m), t^+_m) \;, \qquad
m=1,\ldots (n-1) \;.  \label{hequal2}
\end{equation}

Carrying out a final $n$-th decimation step one obtains
\begin{eqnarray}
\left(\,1+ {Z_\Lambda^{(-)} \over Z_\Lambda }\,\right) & = &
{ 2\, \tilde{Z}^+_{\Lambda^{(n)}}\,(\beta, h,
\alpha_{\Lambda,\, h}^{+(n)}(t^+),\, t^+) \over
\tilde{Z}_{\Lambda^{(n)}}\,(\beta, h,
\alpha_{\Lambda,\,h}^{(n)}(t),\, t) } \label{ratio3} \\
 & = & { \tilde{Z}_{\Lambda^{(n)}}\,(\beta, h,
\alpha_{\Lambda,\,h}^{+(n)}(t^+),\, t^+) \over \tilde{Z}_{\Lambda^{(n)}}\,
(\beta, h,  \alpha_{\Lambda,\,h}^{(n)}(t),\, t)  } \,\left(\,1+
{ Z_{\Lambda^{(n)}}^{(-)}\,\Big(\{\,\tilde{c}_j(n,\alpha_{\Lambda,\,h}^{+(n)}
(t^+))\,\}\Big)
\over Z_{\Lambda^{(n)}}\,\Big(\{\,\tilde{c}_j(n,\alpha_{\Lambda,\,h}^{+(n)}
(t^+))\,\}\Big) } \right)
\label{ratio4}
\end{eqnarray}
The argument for $n=1$ (eq. (\ref{ratio1})) above may now be applied to
(\ref{ratio4}) to conclude
\begin{equation}
| \alpha_{\Lambda,h}^{+(n)}(t,r) -
\alpha_{\Lambda,h}^{(n)}(t,r) | \leq
O({1\over |\Lambda^{(n)}|}) \;. \label{alphdiffN}
\end{equation}
Any such discrepancy between $\alpha_{\Lambda,h}^{+(n)}(t)$ and
$\alpha_{\Lambda,h}^{(n)}(t)$ in (\ref{ratio4}) presents the same problem
for extracting the ratio at scale $b^n a$ as at scale $ba$.
In this  sense  (\ref{ratio4}) is not qualitatively different from
the $n=1$ case (\ref{ratio1}). Transferring the discrepancy to
large $n$, however, allows a technical simplification as we see below.

Next, consider (\ref{ratio3}) rewritten as
\begin{equation}
\left(\,1+ {Z_\Lambda^{(-)} \over Z_\Lambda }\,\right) =
\left( { Z^+_{\Lambda^{(n-1)}} \over \tilde{Z}^+_{\Lambda^{(n)}}\,(\beta,  h,
\alpha_{\Lambda,\,h}^{(n)}(t),\, t) } \right)\,\left(\,1+
{ Z_{\Lambda^{(n)}}^{(-)}\,\Big(\{\,\tilde{c}_j(n,\alpha_{\Lambda,\,h}^{(n)}
(t))\,\}\Big)
\over Z_{\Lambda^{(n)}}\,\Big(\{\, \tilde{c}_j(n,\alpha_{\Lambda,\,h}^{(n)}
(t))\,\}\Big) }
\,\right)
\label{ratio5}
\end{equation}
by use of (\ref{interI3plus}). By construction (cf. (\ref{interI3})),
$\alpha_{\Lambda,\,h}^{(n)}(t)$ is such
that the r.h.s. in (\ref{ratio3}), hence in (\ref{ratio5}),
is invariant under changes in the
parameter $t$; but note
that the two $\alpha_{\Lambda,\,h}^{(n)}$-dependent factors in
round brackets on the r.h.s. in (\ref{ratio5}) are {\it not}
separately invariant.
If, for some given $t$, $\alpha_{\Lambda,\,h}^{(n)}(t)$ is larger
(smaller) than $\alpha_{\Lambda,\,h}^{+(n)}(t)$, then,
by IV.5, $\tilde{Z}^+_\Lambda\,(\beta, h,
\alpha_{\Lambda,\,h}^{(n)}(t),\,t)$ is larger (smaller) than
$\tilde{Z}^+_{\Lambda^{(n)}}\,(\beta,h, \alpha_{\Lambda,\,h}^{+(n)}(t),
\, t)= Z^+_{\Lambda^{(n-1)}}$,
and the second factor in round brackets on the r.h.s. of (\ref{ratio5})
overestimates (underestimates) the ratio $Z_\Lambda^{(-)}/ Z_\Lambda $.
It is then natural to ask whether there exist a value $t=
t_{\Lambda,h}^{(n)}$ such that
\begin{equation}
\tilde{Z}^+_{\Lambda^{(n)}}\,\Big(\beta, h,\alpha_{\Lambda, h}^{(n)}(
t_{\Lambda,h}^{(n)}),\, t_{\Lambda,h}^{(n)} \Big)
= Z^+_{\Lambda^{(n-1)}} \;.\label{interIfixplus}
\end{equation}
Note that
the graphs of $\alpha_{\Lambda, h}^{(n)}(t)$ and
$\alpha_{\Lambda, h}^{+(n)}(t)$ must intersect at $t_{\Lambda,\,h}^{(n)}$.

A unique solution to (\ref{interIfixplus}) indeed exists as shown in
Appendix C provided
\begin{equation}
A_{\Lambda^{(n)}}(\alpha, r)  \geq
A^+_{\Lambda^{(n)}}(\alpha, r)   \label{A>A+}
\end{equation}
with $r$ in (\ref{rdomain}).  An equivalent statement to
(\ref{A>A+}) is
\begin{equation}
A_{\Lambda^{(n)}}(\alpha, r)  \geq
A^{(-)}_{\Lambda^{(n)}}(\alpha, r) \, ,  \label{A>A-}
\end{equation}
where $A^{(-)}_{\Lambda^{(m)}}(\alpha, r)$ is defined
by (\ref{alphplustder3}) but with $Z^+_{\Lambda^{(n)}}$ replaced by
$Z_{\Lambda^{(n)}}^{(-)}$.
Assume now that under successive decimations the coefficients
$c^U_j(m)$ in (\ref{S3}) evolve within the convergence radius of the
strong coupling cluster expansion. Taking then $n$  in
(\ref{ratio3}) sufficiently
large, we need establish inequality (\ref{A>A+})\footnote{For Abelian
systems, comparison inequalities of
the type (\ref{A>A+}) either follow from Griffith's inequalities, or can be
approached by the same methods. All such known
methods fail in the non-Abelian case.} only at
strong coupling. Within this expansion it is a straightforward exercise
to establish the validity of (\ref{A>A+}), with strict inequality
on any finite lattice.

We summarize the above development in the following:\\
\prop{
Consider $n$ successive decimation steps performed according to the
scheme (\ref{S3}).
Assume that there is an $n_0$ such that the upper bound coefficients
$c^U_j(n)$ become sufficiently small for $n\geq n_0$.

Then the ratio of the twisted to the untwisted partition
function on lattice $\Lambda$, of spacing $a$, has a representation on lattice
$\Lambda^{(n)}$, of spacing $b^na$ and $n \geq n_0$, given by:
\begin{equation}
{Z_\Lambda^{(-)}(\beta) \over Z_\Lambda(\beta) } =
{ Z_{\Lambda^{(n)}}^{(-)}\,\Big(\{\,\tilde{c}_j(n,\alpha_\Lambda^{*\,(n)})\,\}
\Big) \over Z_{\Lambda^{(n)}}\,\Big(\{\, \tilde{c}_j(n,\alpha_\Lambda^{*\,(n)})
\,\}\Big) } \;,
\label{ratio6}
\end{equation}
where
\begin{equation}
\alpha_\Lambda^{*\,(n)}\equiv \alpha_{\Lambda, h}^{(n)}(t_{\Lambda,\,h}^{(n)})
 \;.\label{alphstar}
\end{equation}
Here, the function $\alpha_{\Lambda, h}^{(n)}(t)$ is defined
by (\ref{interI3}), i.e. is the solution for $\alpha$ to
\begin{equation}
\tilde{Z}_{\Lambda^{(n)}}\,(\beta, h, \alpha,\, t) =
Z_{\Lambda^{(n-1)}}\;, \label{interI3A}
\end{equation}
and $t_{\Lambda,\,h}^{(n)}$ is defined by
(\ref{interIfixplus}), i.e. is the solution for $t$ to the equation
\begin{equation}
\tilde{Z}^+_{\Lambda^{(n)}}\,(\beta, h,\alpha_{\Lambda, h}^{(n)}(t),\,t)
= Z^+_{\Lambda^{(n-1)}} \;.\label{interIfixplusA}
\end{equation}
}

As indicated by the notation in (\ref{alphstar}),
any dependence on $h$ must cancel in $\alpha_\Lambda^{*\,(n)}$.
Indeed, Cauchy's form of the intermediate  value theorem gives
\begin{equation}
{ \ln Z_{\Lambda^{(n)}}^{(-)}\,\Big(\{\,\tilde{c}_j(n,\alpha)\,\}\Big) -
\ln Z_{\Lambda^{(n)}}^{(-)}\,\Big(\{\,\tilde{c}_j(n,\alpha_\Lambda^{*\,(n)})
\,\}\Big)
\over \ln Z_{\Lambda^{(n)}}\,\Big(\{\,\tilde{c}_j(n,\alpha)\,\}\Big) -
\ln Z_{\Lambda^{(n)}}\,\Big(\{\,\tilde{c}_j(n,\alpha_\Lambda^{*\,(n)})\,\}
\Big) }
= { A_{\Lambda^{(n)}}^{(-)}(\xi) \over A_{\Lambda^{(n)}}(\xi) }
\leq 1 \;, \label{Cauchy}
\end{equation}
for some $\xi$ between $\alpha_\Lambda^{*\,(n)}$ and $\alpha$,
and use of (\ref{A>A-}) was made to obtain the last  inequality.
Setting $\alpha$ equal to $1$ in (\ref{Cauchy}), combining with
(\ref{ratio6}), and using III.3, IV.5, gives
\begin{equation}
{Z_\Lambda^{(-)} \over Z_\Lambda } \geq
{ Z_{\Lambda^{(n)}}^{(-)}\,\Big(\{\,c^U_j(n)\,\}\Big)
\over Z_{\Lambda^{(n)}}\,\Big(\{\,c^U_j(n)\,\}\Big) } \; .
\label{ratiolower}
\end{equation}
The upper bound coefficients in (\ref{S3}) then, which correspond to
upper bounds for the partition functions $Z_\Lambda$ and $Z_\Lambda^{(-)}$,
give a lower bound for the ratio
$Z_\Lambda^{(-)}/Z_\Lambda$.\footnote{This result was first
stated a long time ago in \cite{T}.}
Setting $\alpha=0$ in (\ref{Cauchy}), similarly yields an upper bound.
Thus:\\
\prop{
With  the same conditions as in V.1 the
ratio of the twisted to the untwisted partition
function on lattice $\Lambda$ of spacing $a$ is bounded on lattice
$\Lambda^{(n)}$ of spacing $b^na$ by:
\begin{equation}
{ Z_{\Lambda^{(n)}}^{(-)}\,\Big(\{\,c^L_j(n)\,\}\Big)
\over Z_{\Lambda^{(n)}}\,\Big(\{\,c^L_j(n)\,\}\Big) }
\geq
{Z_\Lambda^{(-)} \over Z_\Lambda }
\geq
{ Z_{\Lambda^{(n)}}^{(-)}\,\Big(\{\,c^U_j(n)\,\}\Big)
\over Z_{\Lambda^{(n)}}\,\Big(\{\,c^U_j(n)\,\}\Big) } \; .
\label{ratiolowerupper}
\end{equation}
}

Now, the ratio of the interpolating
partition functions (\ref{interPF2plus})
and (\ref{interPF2}) interpolates monotonically between
the upper and lower bounds in (\ref{ratiolowerupper}) since
\begin{equation}
{d\over d\alpha }\; { Z^{(-)}_{\Lambda^{(n)}}(\{\tilde{c}_j(n,\alpha\})
\over
Z_{\Lambda^{(n)}}(\{\tilde{c}_j(n,\alpha)\}) } < 0 \label{ratioder}
\end{equation}
by (\ref{A>A-}). It follows that there exist a unique value
$\alpha^{*\,(n)}_\Lambda$ of $\alpha$ at which this ratio
of the interpolating partition functions equals
$Z_\Lambda^{(-)} / Z_\Lambda $.
This is a restatement of (\ref{ratio6}), but makes explicit the fact that
this value is independent of $h$. In fact, it shows that all dependence
on parametrization choices, i.e. the choice of parameters $t_m$ made
in successive decimations, eventually cancels in
$\alpha^{*\,(n)}_\Lambda$. Indeed, the latter can depend only on the number
of decimations $n$ and the initial coupling $\beta$, since this is all the
upper and lower bounds in (\ref{ratiolowerupper}) depend on.
This, in retrospect, is as expected, since all bulk free-energy
contributions depending on such choices were canceled in finally
arriving at (\ref{ratio6}), but V.2 makes it manifest.

(\ref{ratiolowerupper}) was obtained as a corollary of (\ref{ratio6}).
An alternative approach would be to proceed in
the reverse direction, i.e.
establish (\ref{ratiolowerupper})
directly, from which (\ref{ratio6}) would follow by interpolation
between the upper and lower bounds as in the previous paragraph.
In other words, follow also in the case of the
ratio of the partition functions the approach followed separately for
the untwisted and twisted partition functions in the previous sections.
This is further discussed in Appendix C.

\section{Confinement}\label{CONf}
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\subsection{Order parameters}\label{CONFOP}
The vortex free energy $F_\Lambda^{(-)}$ is defined by  the
ratio of partition functions considered in the previous section:
\begin{equation}
\exp(-F_\Lambda^{(-)}(\beta)) = {Z_\Lambda^{(-)}(\beta)
\over Z_\Lambda(\beta)} \;.
\label{vfe}
\end{equation}
It represents the free energy cost for adding a vortex to the vacuum,
the $Z(2)$ flux of the inserted vortex being rendered stable by
wrapping  around the toroidal lattice. As has been discussed in the
literature, all possible phases of gauge theory (Higgs, Coulomb, or
confinement) can be characterized by the behavior of (\ref{vfe})
as one lets the lattice become large. In particular, having
taken the vortex to wind through the
lattice in the directions $\kappa=3, \ldots, d$,
a confining phase is signaled by the asymptotic behavior
\begin{equation}
F_\Lambda^{(-)}(\beta) \sim L\,
\exp(\,-\hat{\sigma}(\beta) |A|\,) \;, \label{vfeconf}
\end{equation}
where $L\equiv \prod_{\kappa\not= 1,2}\,L_\kappa$, and
$A\equiv L_1 L_2$. (\ref{vfeconf}) represents exponential spreading
of the flux introduced by the twist on the set ${\cal V}$ in the
transverse directions (creation of mass gap), with $ \hat{\sigma}(\beta)$
giving the exact string tension.
Note that, according to (\ref{vfeconf}), $F_\Lambda^{(-)}(\beta))\to 0$
as $|\Lambda|\to \infty $ in any power-law fashion, i.e. one
has `condensation' of the vortex flux.
The behavior (\ref{vfeconf}) is dictated by physical reasoning \cite{tH},
\cite{MP},
and explicitly realized within the strong coupling expansion.
As such free energies differences are generally notoriously difficult to
measure  accurately, demonstration of the behavior (\ref{vfeconf})
by numerical simulations at large $\beta$'s has been achieved only
relatively recently \cite{KT}, \cite{Fetal}.

The $Z(2)$ Fourier transform of (\ref{vfe})
\begin{equation}
\exp(-F_\Lambda^{\rm el}(\beta)) = {1\over 2}\Big(
\,1 - {Z_\Lambda^{(-)}(\beta)
\over Z_\Lambda(\beta)} \,\Big)
\label{efe}
\end{equation}
gives the corresponding dual (w.r.t. the gauge group center) order
parameter, the color electric free energy.  (\ref{vfe}) and (\ref{efe})
are ideal pure long-range order parameters. They do not suffer from
the physically irrelevant but technically quite bothersome
complications, such as loss of translational
invariance, or mass renormalization and other short
range contributions, that arise from the explicit introduction of
external sources. Such external current sources are introduced
in the definition of the Wilson and t'Hooft loops. Furthermore, the
behavior of the latter can be bounded by that of
(\ref{vfe}) and (\ref{efe}) \cite{TY}.
In particular, the  following relation holds. Let $C$ be a rectangular loop
of minimal area $S$ lying in a $2$-dimensional $[12]$-plane. Then
\cite{TY}:
\begin{equation}
\vev{W[C]}_\Lambda \leq \left[ \exp(-F_\Lambda^{\rm el}\right]^{S/A} \,,
\label{W-vfebound}
\end{equation}
where $W[C]=\chi_{_{1/2}}\,\Big(\prod_{b\in C} U_b\Big)$ is the usual
Wilson loop observable. It follows from (\ref{W-vfebound}) that
confining behavior (\ref{vfeconf}) of the vortex free energy
implies confining behavior (`area-law') for the Wilson loop.

\subsection{Strong coupling cluster expansion and confinement}

We now return to our considerations at the end of section \ref{Z}
regarding the flow of the coefficients  $\tilde{c}_j(n,
\alpha_{\Lambda,h}^{(n)}(t))$ in our partition function
representations (\ref{A}) and (\ref{B}).
This flow is bounded from above by that of
the MK coefficients $c_j^U(m)$ regardless of the specific
value assumed by the $\alpha_{\Lambda,h}^{(m)}(t_m)$'s at each decimation step
(cf (\ref{cineq5})). Furthermore, by explicit evaluation under
the iteration rules  (\ref{RG2}) - (\ref{RG5}), one finds that
$c_j^U(n) \to 0$
as $n\to \infty$ for any initial $\beta$, provided $d\leq 4$.
Thus, given any initial $\beta$, one may always take the number of iterations
$n$ large enough
so that the coefficients $c_j^U(n)$ become small
enough to be within the region of convergence of the
strong coupling expansion. Then by V.1:
\begin{equation}
\exp(-F_\Lambda^{(-)}(\beta)) =
{ Z_{\Lambda^{(n)}}^{(-)}\,\Big(\{\,\tilde{c}_j(n,\alpha_\Lambda^{*\,(n)})\,\}
\Big) \over Z_{\Lambda^{(n)}}\,\Big(\{\, \tilde{c}_j(n,\alpha_\Lambda^{*\,(n)})
\,\}\Big) } \;. \label{vfeA}
\end{equation}
The vortex free energy may then be evaluated in terms of the coefficients
$\tilde{c}_j(n,\alpha_\Lambda^{*\,(n)})$ directly on lattice $\Lambda^{(n)}$
of spacing $b^n a$ within a convergent strong coupling polymer
expansion.

Recall that, in the pure lattice gauge theory context, a polymer is
a set $Y$ of connected plaquettes containing no `free' bond, i.e. no bond
belonging to only one plaquette in $Y$ (see e.g. \cite{Mu}).
The activity of a polymer $Y$ is defined by
\begin{equation}
z(Y)= \int\;\prod_{b\in Y} dU_b\;\prod_{p\in Y} g_p(U,n) \;, \label{zY}
\end{equation}
where
\begin{equation}
g_p(U,n) = \sum_{j\not= 0} d_j\, \tilde{c}_j(n,\alpha_\Lambda^{*\,(n)})
\, \chi_j(U_p) \;.\label{g1}
\end{equation}
The polymer expansion is then
\begin{equation}
\ln\dZ{n}
= \sum_{X \subset \Lambda^{(n)}} \,a(X)\;\prod_{Y_i \in X} \;
z(Y_i)^{n_i} \;, \label{clusterexp1}
\end{equation}
where the sum is over all linked clusters of polymers in $\Lambda^{(n)}$,
each cluster $X$ consisting of a connected set of polymers
$Y_i$, $i=1,\ldots,k_X$ with multiplicities $n_i$.  The
combinatorial factor $a(X)$ is given by
\begin{equation}
a(X)=\sum_{G(X)} (-1)^{l(G)} \, , \label{combfactor}
\end{equation}
where the sum is over all connected graphs on $X$ (full set (including
multiplicities)
$\{Y_i\}$ as vertices with a line connecting overlapping polymers)
and $l(G)$ is the number of lines in the graph.

In the case of $\dZ{n}^{(-)}$, the presence of the
flux enters the activities
(\ref{zY}) through the replacement (\ref{twist2}). We denote the resulting
activities by $z^{(-)}(Y)$. This replacement
does not affect polymers that are wholly contained in a simply
connected part of $\Lambda^{(n)}$, since, in this case, the flux
can be removed by a change of variables in the integrals in (\ref{zY}).
Only clusters that contain at least one non-simply connected
polymer forming a topologically non-trivially closed surface
can be affected. Thus, one has
\begin{equation}
\ln\dZ{n}^{(-)} - \ln \dZ{n}
= \sum_{X \subset \Lambda^{(n)}} \,a(X)\;\left(\,\prod_{Y_i \in X} \;
z^{(-)}(Y_i)^{n_i} - \,\prod_{Y_i \in X} \;
z(Y_i)^{n_i}\right)\;, \label{clusterexp2}
\end{equation}
where the sum is only over all such topologically nontrivial
linked clusters, the contribution of all other clusters
canceling in the difference. The minimal cluster of this type consists
of a single polymer which is a
2-dimensional plane $\Pi: x_\mu=$const., $\mu=3,\ldots,d$ on $\Lambda^{(n)}$,
thus of size $A^{(n)}=L_1^{(n)}L_2^{(n)}$, and activity
\begin{equation}
z(\Pi)=  \sum_{{\rm half-int.}\atop j\geq 1/2}
\tilde{c}_j(n,\alpha_\Lambda^{*(n)})^{A^{(n)}}
= \tilde{c}_{1/2}(n,\alpha_\Lambda^{*(n)})^{\,A^{(n)}} \,[\, 1+
\sum_{{\rm half-int.}\atop j\geq 3/2}
\left({\tilde{c}_j(n,\alpha^{*(n)}) \over \tilde{c}_{1/2}(n,\alpha^{*(n)})}
\right)^{A^{(n)}}\,] \;.\label{lead}
\end{equation}
(Note that the terms from the higher representations in (\ref{lead}) become
utterly negligible in the large volume limit.)
There are $L^{(n)}=\prod_{\kappa\not=1,2}L_\kappa^{(n)}$ such minimal
clusters giving the leading contribution
in (\ref{clusterexp2}). This leading contribution is thus
seen to give the confining behavior (\ref{vfeconf}).
Nonleading contributions come from nonminimal clusters
consisting of $\Pi$ with or without `decorations', and  additional polymers
touching $\Pi$. Such corrections have been evaluated
in terms of the character expansion coefficients
(the $\tilde{c}_j(n,\alpha_\Lambda^{*(n)})$'s in our case) to quite high order
\cite{Mu}. They can be shown to exponentiate, so that
\begin{equation}
{1\over L}\,F_\Lambda^{(-)}(\beta) = \exp(- \hat{\sigma}_\Lambda\,A)
\end{equation}
with
\begin{eqnarray}
\hat{\sigma}_\Lambda
& = & {1\over b^{2n}}\,\kappa_\Lambda(n,\alpha_\Lambda^{*(n)}) \nonumber\\
& = & {1\over b^{2n}}\, \Big[\, \kappa(n,\alpha_\Lambda^{*(n)})
+ O\left( (\tilde{c}_{j+1/2}/\tilde{c}_{1/2})^{L_\mu^{(n)}}\right)
+ O(n/A^{(n)}) \,\Big] \;, \label{sigma1}
\end{eqnarray}
where \cite{Mu}
\begin{equation}
\kappa(n,\alpha_\Lambda^{*(n)}) =
\Big[-\ln \tilde{c}_{1/2}(n,\alpha_\Lambda^{*(n)})
- 4\,\tilde{c}_{1/2}(n,\alpha_\Lambda^{*(n)})^4 +8\,\tilde{c}_{1/2}(n,
\alpha_\Lambda^{*(n)})^6 + \ldots \Big] \,. \label{sigma2}
\end{equation}
By the convergence of the expansion \cite{Ca}, the large volume limit
exists and is given given by $\hat{\sigma}=\kappa(n,\alpha^{*(n)})/b^{2n}$,
where $\alpha^{*(n)}$ is the lattice independent part of
$\alpha_\Lambda^{*(n)}$ (cf. (\ref{alphsplit1})).

The number of iterations $n$ in the above expressions is taken large enough
so that, given some initial $\beta$ on $\Lambda$, the resulting
$c_j^U(n)$ are within the
expansion convergence regime, and one can write the representation
(\ref{vfeA}) by V.1. This implies the existence of a scale, a point to
which we return below. Otherwise, $n$ is arbitrary.
By construction, our procedure is such that the ratio (\ref{vfe})
is reproduced under successive decimations.
Thus, given (\ref{vfeA}) at some $n$, suppose one performs one more
decimation to lattice $\Lambda^{(n+1)}$. The condition that
determines $\alpha_\Lambda^{*(n+1)}$ such that (\ref{vfeA}) is
preserved is
\begin{equation}
\kappa_\Lambda(n,\alpha_\Lambda^{*(n)}) = {1\over b^2}\,\kappa_\Lambda(n+1,
\alpha_\Lambda^{*(n+1)}) \;,\label{kappaI}
\end{equation}
which then results in constant string tension $\hat{\sigma}_\Lambda$
under successive decimations.
Using (\ref{sigma2}) with (\ref{interc1}) and (\ref{RGstrong}),
it is an easy exercise to solve (\ref{kappaI}), at least
to leading approximation, for $\alpha^{*(n+1)}$.
The $t$-parameter value $t_{h}^{(n+1)}$ this $\alpha^{*(n+1)}$
corresponds to can then also be easily obtained, if desired, from
\[ h(\alpha^{*(n)},t)\ln F_0^U(n+1)\,|\Lambda^{(n+1)}| +
\ln \dZ{n+1}\,\Big(\{\, \tilde{c}_j(n+1,\alpha^{*\,(n+1)})
= \ln \dZ{n}\,\Big(\{\, \tilde{c}_j(n,\alpha^{*\,(n)})
\Big) \,,  \]
with $\ln\dZ{n}$, $\ln\dZ{n+1}$ given by (\ref{clusterexp1}) - in fact, to
leading approximation, $\ln\dZ{n+1}$ can be ignored.
Note that this amounts to replacing
the set of the two equations (\ref{interI3A}) -
(\ref{interIfixplusA}) in V.1 by their ratio and
one of them. This is indeed the most convenient procedure once
(\ref{vfeA}) has been achieved.

\subsection{String tension and asymptotic freedom}

$\kappa(n,\alpha^{*(n)})$ is the string tension in lattice units of
lattice $\Lambda^{(n)}$. It is a complicated,
but well-defined function  of the original coupling $\beta= 4/g^2$ defined on
lattice $\Lambda$,  eq. (\ref{Wilson}) (cf. remarks following
(\ref{ratioder})). We write
\begin{equation}
\kappa(n,\alpha^{*(n)}) \equiv \hat{\sigma}(n,g) \,.\label{sigma3}
\end{equation}
In dimensional units the asymptotic string tension in
$d=4$ (\ref{sigma1}) - (\ref{sigma2}) is then
\begin{eqnarray}
\sigma  & = & {1\over a^2}
{1\over b^{2n}}\, \hat{\sigma}(n,g)
\label{sigma4a} \\
& = & {1\over a^2} \, \hat{\sigma}(g) \,. \label{sigma4b}
\end{eqnarray}
Here, as remarked above, $n$ is assumed greater than some required smallest
$n(g)$.
This (dynamically generated) physical scale, or some chosen multiple of
it, is the only parameter in
the theory. Fixing it specifies how the coupling $g$ must vary
with changes of the (unphysical) lattice spacing $a$.

It is convenient, and customary, to introduce a fixed scale $\Lambda_0$
serving as an arbitrary unit of physical scales. Setting
\begin{equation}
\Lambda_0^{\:-1} = a b^n \, ,\label{length}
\end{equation}
determines the lattice spacing $a$ such that it takes $n$ steps to
reach length scale  $1/\Lambda_0$:
\begin{equation}
n= {1\over \ln b} \ln {1\over a\, \Lambda_0} \,. \label{n-a}
\end{equation}
Fixing the string tension, given in units of $\Lambda_0$:
\begin{equation}
\sigma = k \Lambda_0^2 \;,\label{sigma5}
\end{equation}
implies
\begin{equation}
\hat{\sigma}(n,g) = k \label{sigmaI}
\end{equation}
for some constant $k$.
(\ref{sigmaI}) specifies the dependence of the bare coupling
$g$ on $n$, hence, through (\ref{n-a}), the dependence on the
lattice spacing $a$. It gives then the value $g(a)$ specified by the
value of the string tension. (This is, of course, equivalent to fixing
(\ref{sigma4b}) directly.)

Since
\[ \hat{\sigma}(n+1,g+\Delta g)^{1/2} - \hat{\sigma}(n,g)^{1/2} =
b\, [\,\hat{\sigma}(n,g+\Delta g)^{1/2} - \hat{\sigma}(n,g)^{1/2}\,] +
(b-1) \hat{\sigma}(n,g)^{1/2} \]
and  $\Delta a = -(b-1)a/b$ for $\Delta n=1$, one has from
(\ref{sigmaI}):
\begin{equation}
{\Delta \sqrt{\hat{\sigma}} \over \Delta g } (a {\Delta g\over \Delta a})
= \sqrt{\hat{\sigma}}  \label{sigmaIdiff}
\end{equation}
If $(a{\Delta g /\Delta a})\equiv\beta(g)$, the `beta-function', is known,
(\ref{sigmaIdiff}) can be integrated directly for $\hat{\sigma}(g)$.
(This introduces a dimensional integration constant which can serve as
the scale $\Lambda_0$).
This is in fact the familiar
textbook argument were one {\it assumes} the existence of a
string tension so as to get (\ref{sigmaIdiff}), in which the standard weak
coupling perturbative expression for the beta function is then used.

For us, however, the existence of a non-zero string tension is the
outcome of the process of successive decimations to coarser scales as
developed above. This process embodies all relevant information in the theory.
In particular, it also supplies the specification of the function $g(a)$.

One can indeed construct the function $g(a)$ directly as follows:\\
(i) Starting with some initial value of $\beta=4/g^2$ perform
successive decimations following the flow into the strong coupling
regime with resulting string tension $\kappa(n,\alpha^{*(n))})$,
eq. (\ref{sigma2}), at some $n=n_0$. Let $k$ denote the value of this
string tension. The corresponding
value of the lattice spacing $a_0$ is given by (\ref{n-a}), and $g=g(a_0)$. \\
(ii) Fix the string tension as in (\ref{sigmaI}). This is then satisfied at
$n_0$, $g(a_0)$. \\
(iii) Vary $g$ away from $g(a_0)$ to determine $g$ such that, under
successive decimations following the flow into the strong coupling
regime, the resulting string tension satisfies (\ref{sigmaI}) for
$n=n_0+1$.\\
(iv) Repeat (iii) for $n=n_0+2, n_0+3,\cdots, n_0-1, \cdots$. \\
This provides the functional relation $g(a)$. In particular, for
$b=2$, it gives the sequence of values $g(a_0/2^l)$, $l=1,2,\ldots$,
starting from some value $g(a_0)$.\footnote{This is the analog in the
present context of the `staircase' procedure in  \cite{Cr}.}

Note that, according to (i) above,
the number of decimations $n_0$ at which one chooses to apply
V.1 to obtain (\ref{vfeA}), (\ref{sigma2}) amounts to fixing the string
tension. This is the only physical parameter in the theory.
A specification of $\Lambda_0$ is a specification of the value
$g(a_0)$ at spacing $a_0$, which is a
convention of no physical import.

One then has in principle a constructive method for obtaining
$g(a)$ by a sequence of simple algebraic operations.
This is the coupling $g(a)$ as defined in the physical
non-perturbative renormalization scheme
specified by keeping the string tension fixed.

A straightforward illustration of the method is provided by setting all
$\alpha_\Lambda^{(n)}=1$, i.e. apply it to the flow according to the
upper bound coefficients $c_j^U$ in (\ref{S1}). This yields
$g(a)$ as given by MK decimations.
We cannot apply it explicitly to the
case of interest, i.e. the flow following the middle column
coefficients in (\ref{S2}),
since we do not determine them explicitly in this paper.
The qualitative features at strong and weak coupling, however, are readily
discernible.

At strong coupling, i.e. small initial $\beta$, the number
of decimations needed to reach a given string tension is of
order unity, i.e. the lattice spacing $a$ is large: $a = O(\Lambda_0^{\,-1})$,
and one is very far from any continuum limit.
Successive decimations, by  construction, reproduce the behavior seen
within the strong coupling expansion, and
the familiar strong coupling variation given by $\beta(g) \sim g \ln g$ is
the result, as can be checked by a short computation.

The opposite limit of large initial $\beta$ corresponds to large
number of decimations, hence $a \ll \Lambda_0^{\,-1}$. Indeed, recall that
$g=0$ is a fixed point of the decimations. Hence, for $\beta\to \infty$,
one necessarily has $n\to \infty$ in order for, say, the leading upper bound
coefficient $c_{1/2}^U(n)$ to reach any prescribed value $ < 1$.
Thus, $a\Lambda_0 \to 0$. Note that this limit is well-defined by construction
since everything is bounded and continuous under successive
decimations.
Asymptotic freedom, i.e. the statement
that $g(a)\to 0$ as $a\to 0$, is then a direct qualitative
consequence of the flow produced by the decimations.

It is instructive to examine the actual manner in which $g(a)\to 0$
under the upper bound decimations, i.e. the
$c_j^U(m)$'s in (\ref{S2}). Comparing two $g$ values that differ by
one decimation step ($b=2$), one finds
\begin{equation}
{1\over g^2(a)} ={1\over g^2(2a)} + 2b_0 \,\ln2 + O(g^2)
\end{equation}
for sufficiently small $g(a)$. The constant $b_0=(1-1/b^2)/(24\ln b)$
underestimates
the value $11/24\pi^2$ obtained in a continuum perturbative calculation
by only about $3\%$.

The actual flow (middle column in (\ref{S2})) is faster,
corresponding to somewhat larger $b_0$. According to RG lore,
a beta-function defined by other means, such
as fixing some renormalized coupling within weak coupling perturbation
theory, should coincide, in its universal
first two terms, with that defined by the above physical non-perturbative
scheme. This, however, is outside the scope of, and
not of direct relevance for the main argument in this paper.

To reiterate, the above procedure
completely specifies the dependence $g(a)$ in the physical
renormalization scheme defined by keeping the
string tension fixed, and this dependence is necessarily such
that $g(a)\to 0$ as $a\to 0$.

\section{Concluding remarks}\label{SUM}

In summary, we obtained a representation of the vortex free energy, originally
defined on a lattice of spacing $a$,
in terms of partition functions on a lattice of spacing $ab^n$.
The effective action in this representation
is bounded by the corresponding effective action resulting
from potential moving decimations (MK decimations) from spacing $a$ to
spacing $ab^n$. The latter are explicitly computable. Confining behavior
is the result, starting from any initial coupling $g$ on spacing $a$,
by taking the number of decimation $n$ large enough.

It is worth remarking again that in an
approach based on RG decimations the fact that the only
parameter in the theory is a physical scale emerges in a natural way.
Picking a number of decimations can be related to
fixing the string tension. That this can
be done only after flowing into the strong coupling regime
reflects the fact that this dynamically generated scale is an `IR effect'.
The coupling $g(a)$ is completely determined in its dependence on $a$
once the string tension is fixed.
In particular, $g(a) \to $ as $a\to 0$.
Note that this implies that there
is no physically meaningful or unambiguous way of non-perturbatively viewing
the short distance regime independently of the long distance regime.
Computation of all physical observable quantities
in the theory must then give a multiple of the string tension or
a pure number. In the absence of other interactions, this scale provides
the unit of length; there are in fact no free
parameters.\footnote{This is part of the meaning of the common
saying ``QCD is the perfect theory''.}

There is a variety of other results related to the approach in this
paper that could not be included here. We note, in particular,
that the same procedure  can be immediately transcribed to the
Heisenberg $SU(2)$ spin model.
Also, apart from analytical results, the considerations in this
paper may be combined with Monte Carlo RG techniques to
constrain the numerical construction of improved actions at different
scales, a subject of perennial interest to the practicing lattice
gauge theorist. We hope to report on these matters elsewhere.

This research was partially supported by
NSF grant NSF-PHY-0555693.

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