% Corrections by DCB Dec 18 1996
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% Date: Wed, 18 Dec 1996 11:56:43 -0500
% From: Tom Hurd
% To: db5d@faraday.clas.Virginia.EDU
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\begin{document}
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\title{A non-Gaussian fixed point for $\phi^4$ in $4-\ep$ dimensions - I.}
\author{D. Brydges\thanks{Research supported by NSF Grant DMS 9401028 }
\\Dept. of Mathematics\\University of Virginia\\
Charlottesville, VA 22903 \and
J. Dimock\thanks{Research supported by NSF Grant PHY9400626}\\
Dept. of Mathematics \\
SUNY at Buffalo \\
Buffalo, NY 14214 \and T. R. Hurd\thanks{Research supported by the
Natural Sciences and Engineering Research Council of Canada.}\\Dept. of
Mathematics and Statistics\\McMaster University\\Hamilton,
Ontario\\L8S 4K1}
\maketitle
\begin{abstract}
We consider the $\f ^4$ quantum field theory in four dimensions.
The Gaussian part of the measure is modified to simulate $4-\ep$
dimensions where $\ep$ is small and positive. We give a renormalization group
analysis for the infrared behavior of the resulting model.
We find that the Gaussian fixed point is unstable but that there is
a hyperbolic non-Gaussian fixed point a distance $\cO(\ep)$ away. In a
neighborhood of this fixed point we construct the stable manifold.
\end{abstract}
\newpage
\tableofcontents
\section{Introduction}
We consider a Euclidean $\phi^4$ quantum field theory in $d$ dimensions as
given by functional integrals of the form
\be {\int [...] e^{-V(\phi)} d \mu_v (\phi) \over \int e^{-V(\phi)} d \mu_v ( \phi )
} \ee
The integral is over some collection of functions $\phi $ on $\bf R^d$
and $\mu_v$ is a Gaussian measure with covariance $v = (-\De
)^{-1}$. Up to terms that can be absorbed by adjusting $v$ the
potential $V$ has the form \be \label{potential} V(\phi)
= \la \int :\phi^4:_v + \mu \int :\phi^2:_v \ee where $\la$ is a
coupling constant and $\mu $ is a mass. It is a basic problem of
quantum field theory to establish the existence of such integrals and
study their properties. The problem is also relevant to the study of
critical properties in statistical mechanical systems, in which case
$\phi$ is interpreted as an order parameter.
We are interested in the case when $\la$ and $\mu$ are small.
Furthermore we focus on the IR (infrared, long distance) behavior of
the model and so insert an UV (ultraviolet, short distance)
regularization, imposed as a regularization of the covariance
$v$. With this modification the model is well-defined in finite volume
and one seeks to take the infinite volume limit.
We attack the problem by the renormalization group (RG) method. One
successively integrates out short distance modes and then rescales to
obtain a new effective potential. The new potential is a lot more
complicated. However the leading terms are of the form
(\ref{potential}) with new coupling constants $(\la', \mu' )$. One
repeats the transformation and studies the flow of the theory. An
attracting fixed point is supposed to encapsulate the long distance
behavior of all models that flow to it.
The massless Gaussian measure ($\la =0, \mu = 0$) is a fixed point
for all $d$. Suppose $d > 4$. In a space of general potentials one
finds that the linearization around this fixed point has $\mu $
growing ( ``relevant'') and has $\la $ shrinking ( ``irrelevant''). One
has a hyperbolic fixed point as in figure 1. If one selects $\mu=
\mu(\la)$ to lie on the stable manifold then the fixed point is
strictly attracting. These are the critical field theories.
\begin{center}
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\begin{center}
{\sc Figure 1: $RG$ flow for $d = 4$}
\end{center}
If $d=4$ then the $\la$ term is stable (``marginal''). However this
only refers to the linearization, and an analysis of higher order
terms shows that $\la$ still flows to zero, so the qualitative picture
is the same as for $d>4$. A rigorous treatment for $d=4$ (in a lattice
model) has been given by Gawedzki and Kupiainen \cite{GaKu85a},
\cite{GaKu86}. Other treatments can be found in
\cite{FMRS87},\cite{MaPo85},\cite{MaPo89}.
For $d<4$ one finds that $\la$ becomes a second relevant variable and
the analysis becomes cloudy.
To gain insight into this question one can study the flow equations in
$4-\ep$ dimensions not just for integer $\ep$, but also for $\ep$
small and positive. Expanding in powers of $\ep$ and keeping the
lowest orders one predicts that there is a second fixed point which
lies at $\la = \cO(\ep)$ and hence is non-Gaussian. (see figure 2.)
The stable manifold around this fixed point is again supposed to
correspond to critical theories. The predictions one gets for these
theories (e.g. for critical exponents) turn out to be pretty good not
just for small $\ep$ but also for $\ep$ integer. This was one of the
early successes of the RG approach (see \cite{WiFi72},\cite{WiKo74} ).
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\begin{center}
{\sc Figure 2: $RG$ flow for $d = 4-\epsilon$}
\end{center}
Our goal in this paper is to give a rigorous version of this. We
start with the theory in $d=4$ but modify the Gaussian measure so that
its covariance has scaling behavior appropriate for $4-\ep$
dimensions. This is taken as the definition of the theory in $4- \ep$
dimensions. Assuming $\ep$ is small we analyze the complete RG flow
including all the remainder terms. We prove the existence of the
second fixed point and the associated stable manifold corresponding
to critical field theories.
The analysis in the present paper is carried out at the level of
polymer activities, and in infinite volume. In a subsequent paper
\cite{BrDiHu97} we make the connection with finite volume. We also
analyze the long distance behavior of the two point function for the
critical theories and find that the non-Gaussian fixed point gives
rise to an anomalous decay rate.
Our analysis uses RG tools created for other models
(\cite{BrYa90}, \cite{DiHu91}, \cite{DiHu93}, \cite{BrDiHu94a},
\cite{BrDiHu94b}, \cite{BrDiHu95} ). We particularly rely on
a technical companion paper \cite{BrDiHu96a} which contains a number
of innovations and has fresh proofs of all the basic theorems.
The model is admittedly a bit artificial. However we hope that these
techniques can be adapted to study a more realistic model such as
$N$-component $\phi^4$ in $d=3$ with $1/N$ as a small parameter. This
is a favorite subject for more heuristic treatments. We are also
confident that our methods can be adapted rather easily to give a
fresh treatment of the $d=4$ results.
We note that there is previous work on the existence of non-Gaussian
fixed points. For hierarchical $\phi^4$ models these include a
$d=4-\ep$ version and a $d=3, N-$component version \cite{GaKu83a}, a
computer assisted proof of existence for d=3, N=1
\cite{KoWi86},\cite{KoWi91}, and a proof for the infinitesimal RG
\cite{Fel87}. The analysis by Felder \cite{Fel85} of planar graph
$\phi ^{4}$ in $4 + \epsilon $ dimensions is particularly close
in spirit to this paper.
In the rest of the introduction we define the model, define the RG
transformations, and preview our results.
\subsection{The model}
We shall formulate the model both on $\bR^4$ and on tori
$\La_N=\bR^4/(L^N\bZ)^4$ of side $L^N$. Working for the moment on $\bR^4$,
the Gaussian measure $\mu_v$ is taken to have mean zero and covariance
\bea
v(x-y)&=& \int ^\infty_1\ d\alpha \ \alpha^{\ep/2-2}\
e^{-|x-y|^2/4\alpha}\nn\\ &=& \int_{\bR^4} dp \ p^{-2-\ep}\ \G(1+\ep/2,p^2)\
e^{ip\cdot(x-y)}
\label{}
\eea
where the incomplete Gamma function is
$ \G(a,x)=\int^\infty_x\ e^{-t}\ t^{a-1}\ dt $.
One
can see from this that $v$ is the kernel of an ultra--violet regularization of the
operator $(-\De)^{-1-\ep/2}$. This means it scales like $|x|^{-2+\ep}$ for large
values of
$|x|$, and thus can be taken as a definition of a (regularized) inverse Laplacian
in $4-\ep$ dimensions.
The analysis we present actually holds for a wider class of covariances, with
$e^{-|x-y|^2/4\alpha}$ replaced by more general functions $f(|x-y|/\sqrt \alpha)$.
For discrete RG problems such as ours, the covariance is written as a sum
\be\label{second} v(x-y)=\sum_{j=0}^\infty L^{-(2-\ep)j} C(L^{-j}(x-y)) \ee
where each term is a rescaling with scale factor $L>1$ of the ``fluctuation''
covariance
\be
C(x)= \int ^{L^2}_1\ d\alpha \ \alpha^{\ep/2-2}\ e^{-x^2/4\alpha}
\ee
When we work on a torus $\La_N$, we define $v_N$ by summing over periods
$z\in(L^N\bZ)^4$, and truncating the sum over scales. This yields
\be v_N(x-y)= \sum_{j=0}^N L^{-(2-\ep)j} C_j(L^{-j}(x-y)) \label{fourth} \ee
where
\[ C_j(x)=\sum_{z\in (L^{N-j}\bZ)^4} C(x+z)
\]
Note that $ C_j$ is almost independent of $j$ when $N-j$ is large.
Let $\mu = \mu_{v_N}$ be a Gaussian measure with mean zero and
covariance $v_N$. Since $v_N$ is smooth this measure is realized
on a Sobolev space of $\cC ^{3}$ functions on $\La_N$.
The model is now the study of the measure
\be e^{-V(\f)} d\mu_v(\f) =
\exp(-V(\La_N, \f , v; \la , \z, \mu)) d \mu_{v_N} (\f) \ee
where the potential is
\be V(\La,\f, v; \la, \z, \mu) =
\la \int_{\La} :\phi^4:_v + \z \int_{\La} :(\pa \f)^2:_v +
\mu \int_{\La} :\phi^2:_v \label{originalcc} \ee
Here we have allowed for an adjustment in the field strength by
including the $\z$ term.
The measure is well defined for
$N < \infty$, but our interest is in the limit(s) as $N \to \infty$.
We also need the potential localized in subsets
$X$ of $\La_N$. In this case we find it convenient to allow two
versions of the field strength and for $\z = (\z_1,\z_2)$ define
\bea V(X,\f, v; \la, \z, \mu) &=& \nn \\
&&\hspace{-.5in}\la \int_X :\phi^4:_v + \z_1 \int_X :(\pa \f)^2:_v +
\z_2 \int_X : \f (-\De)\f:_v +
\mu \int_X :\phi^2:_v
\label{cc} \eea
When $X = \La_N$ we can integrate by parts and recover
the previous version with field strength $\z_1 + \z_2$.
\subsection{RG transformations}
Let $\cZ(\f) = e^{-V(\f)}$.
We want to consider integrals of the form $\int \cZ(\f) d \mu_v(\f)$
or more generally convolutions
\be (\mu_v *\cZ)(\f)=\int d\mu_v(\z) \ \cZ(\f+\z) \ee
Now from (\ref{second}) or (\ref{fourth}) the covariance
$v=v_N$ has the decomposition
\be \label{fifth} v_N(x-y) = \sum_{k=0}^{N} \hat C_k(x-y) \ee
where $\hat C_k(x)
= L^{-(2-\ep)k} C_k(L^{-k}x)$
and $C_k =C$ in infinite volume.
The RG idea is that one can then write $\mu_v$
as an iterated convolution
\be \mu_v *\cZ =\mu_{\hat C_{N}}*\mu_{\hat C_{N-1}}*\dots
*\mu_{\hat C_1}*\mu_{\hat C_0}*\cZ \ee
Or if we define
\be \hat \cZ_j = \mu_{\hat C_{j-1}}*\dots *\mu_{\hat C_1}*
\mu_{\hat C_0}*\cZ \ee
then we have a sequence on densities $\tilde \cZ_0 = \cZ,
\hat \cZ_1,\hat \cZ_2,\dots$ related by
\be \tilde \cZ_{j+1} = \mu_{ \hat C_j} * \hat \cZ_j \ee
We want to study $\hat \cZ_N$ as $N \to \infty$.
We scale the density $\hat \cZ_j$ on $\La_N$ to
$ \cZ_j$ on $\La_{N-j}$ by
\be \cZ_j(\f) = \hat \cZ_j(\f_{ L^{-j}}) \ee
where for any $r>0$ the rescaled field $\f_r(x)$ is defined to be
\be \f_r(x)=r^{1-\ep/2}\f(rx) \label{rescal}
\ee
Then the densities are related by
\be \cZ_{j+1}(\f) = (\mu_{C_j} * \cZ_j)(\f_{L^{-1}}) \ee
This is the basic RG transformation.
Each fluctuation convolution is well defined and controllable because of the
smoothness and exponential decay of the covariances $C_j$. Each step is almost
independent of $j$ for a large torus, and is actually independent of $j$ for
infinite volume.
The result of all this is that the study of the original
integral $\mu_v * \cZ$ is replaced by the study of an (almost) autonomous
discrete flow in the space of densities. The problem is to follow the flow.
\subsection{The RG flow}
We now sketch our treatment of the RG flow, referring to subsequent sections for
precise statements and proofs.
To control the flow it is important to keep track of the local
structure of the densities $\cZ_j$. To begin with we have the
exponential of a local functional. This form is not preserved but
something like it is true. We find that we can always write the
densities in the form
\be
\cZ_j(\f)
=
(\cE xp \ A_j)(\La_{N-j}, \f)
\equiv
\sum_{\{ X_i \} } \prod_i A_j(X_i,\phi)
\ee
Here the $\cE$xponential is defined by the sum which is over all
partitions $\{X_i\}$ of the volume $\La$ into unions of unit blocks
called polymers. The functionals $A_j(X,\f)$ are called polymer
activities, have tree decay in $X$, and only depend on the restriction
of $\f$ to $X$. The expansion is called a polymer expansion. Our
formalism using these polymer expansions as a tool for controlling the
RG was initiated by Brydges and Yau \cite{BrYa90}. Similar ideas but
adapted towards numerical computations were developed by Mack and
Pordt \cite{MaPo89}. We review the results we need in section 2.
The RG mapping $\cZ_j \to \cZ_{j+1}$ of densities is now replaced by a
mapping $A_j \to A_{j+1}$ of polymer activities. At
the level of polymer activities the RG transformation also makes sense
in infinite volume, and is independent of $j$. In this paper we only
study this infinite volume transformation. The problem of
whether this infinite volume transformation correctly represents the
infinite volume limit of the model is deferred to a paper in
preparation \cite{BrDiHu97}.
In sections 3,4 we will show that the activities $A_j$ can be written
in the form \be A_j(X, \f) = B_j(X,\f,v;\la_j, \z_j, \mu_j,w_j) +
R_j(X, \f) \ee Here $B_j(X,\f)$ is an explicit functional of certain
effective coupling constants $(\la_j, \z_j, \mu_j)$ and a function
$w_j$ on $\La_{N-j} \times \La_{N-j}$. The leading contribution comes
when $X= \De=$ a unit block. We have
\be
B_j(\De, \f,v;\la_j, \z_j, \mu_j, w_j)
=
\exp \bigl( -V(\De, \f, v; \la_j, \z_j, \mu_j)\bigr ) + ...
\ee
with further contributions computed in second order perturbation
theory. The term $R_j$ is a remainder term incorporating all higher
orders of perturbation theory.
The polymer activities $A_j$ are thus parametrized by variables
$(\la_j, \z_j, \mu_j,R_j, w_j)$. These variables transform nicely
under the RG. Assuming roughly that
\[ \la_j= \cO (\ep),
(\z_1)_j= \cO(\ep), (\z_2)_j= \cO(\ep^2),
\mu_j= \cO (\ep^2), R_j = \cO (\ep^3) \]
and that $\ep$ is
sufficiently small we find a flow equation of the form:
\bea \label{flow}
\la_{j+1}
&=&
L^{2\ep}\left(\la_j - a(w_j)\la_j^2 + \cO (\ep^3)\right) \nn \\[.2in]
(\z_i)_{j+1}
&=&
L^{\ep}\left( (\z_i)_j -b_i(w_j)\la_j^2 + \cO (\ep^3)\right) \nn
\\[.2in]
\mu_{j+1}
&=&
L^{2+\ep}\left(\mu_j - c(w_j)\la_j^2 - d(w_j)\la_j (\z_{1,j} +\z_{2,j})
+\cO(\ep^3)\right)\nn \\[.2in]
R_{j+1}
&=&
\cO(\ep^3) \nn \\[.2in]
w_{j+1}(x)
&=&
L^{2-\ep} \left(w_j(Lx) + C(Lx) \right)
\eea
with $b_2 =0$. The initial
conditions should have $R_0 =0, w_0=0$. The
function $w_j$ can be explicitly computed and converges to a limit
$w_{\infty}$ as $j \to \infty$. We have nevertheless included it
as a dynamical variable to make the flow autonomous in infinite
volume.
The powers of $L$ in the first three equations are bigger by a factor
of $L^{\ep}$ than we would expect if we were really in $4- \ep$
dimensions. This is due to the fact that our spatial integrals are in
4 rather than $4-\ep$ dimensions.
With generic initial data we cannot iterate the RG mapping
indefinitely for we have three expanding variables and would soon
leave the region of definition of the mapping. Some further analysis
is needed.
The origin is an unstable fixed point. To find the second fixed point
replace $a(w_j)$ by its limiting value $a(w_{\infty})$ and ignore the
higher order terms in the $\la$ equation. One finds an approximate
fixed point at
\be
\bar\la={L^{2\ep}-1 \over L^{2\ep}a(w_{\infty})} = \cO(\ep)
\ee
Our further analysis is carried out in a neighborhood of this approximate
fixed point. The deviation $\tilde \la = \la - \bar \la$ replaces
$\la$ and satisfies \be \tilde \la_{j+1} = (2- L^{\ep}) \tilde \la_j +
\cO(\ep^3) \label{newflow} \ee Note that $(2-L^{\ep}) <1$ so that
$\tilde \la_j$ is a contracting variable.
In the new variables $(\tilde \la, \z,\mu,R, w)$ we have a mapping on
a neighborhood of the origin in a Banach space for which the
linearization has two expanding directions $(\z, \mu)$ and the rest
contracting $(\tilde \la , R, w)$, and for which the nonlinear part is
very small. By a version of the stable manifold theorem there is a
hyperbolic fixed point in this neighborhood, and associated with this
fixed point is a stable manifold of dimension 2 and an unstable
manifold of codimension 2. The stable manifold is given as the graph
of a $C^{\infty}$ function $\z= \z(\tilde \la , R,w), \mu = \mu(\tilde
\la , R,w)$, or specializing to the initial values $R=0,w=0$ it has
the form $\z= \z(\tilde \la), \mu = \mu(\tilde \la )$. Densities
corresponding to points on this manifold flow to the density of the
fixed point. These are the critical field theories. The details of
this argument are given in section 5.
\section{Review}\label{sec-review}
In order to make the present paper reasonably self--contained, we include
here a concise review of the definitions and results for a single RG
transformation, adapted to the problem at hand. The reader wishing all the details in
a more general setting is directed to the original paper \cite{BrDiHu96a}.
\subsection{Polymer expansions}
At the $j$th RG step, we have a {\it base space} $\La= \La_{N-j}$ which is a
4--dimensional torus
${\bf R}^4/L^{N-j} {\bf Z}^4$ of side $L^{N-j}$. A {\it
polymer\/}
$X$ is a possibly empty union of blocks of $\La$ where a {\it block} , $\Delta $, is
an open {\it unit} hypercube in
$\La$ centered on a point of the lattice ${\bf Z}^4/L^{N-j} {\bf Z}^4$. {\em Polymer
activities \/} are complex valued functions $K(X,\f)$ defined on polymers $X$ and
fields $\f$ (in fact, $K(X)$ will depend only on the restriction $\f|_X$). We include
the empty set as a polymer and assume, unless cautioned otherwise, that
$K(\emptyset) = 0$.
As we have mentioned, we shall regard the densities $\cZ_j$ as a
polymer exponential of
a polymer activity
$A_j(X,\f)$
$$\cZ_j= \cE xp(A_j) = \cI + A_j +\frac1{2!} A_j \circ A_j +...$$
The polymer exponential is that associated to the following commutative product on the
space of polymer functions
$A(X), B(X),
\dots
$
\cite{Rue69,GMM73}
$$(A \circ B)(X) = \sum_{Y \subset X} A(Y)B(X\setminus Y)$$
Here $\cI(\emptyset )=1$ and otherwise $\cI(X)=0$.
A local density can be written as the polymer expansion
$$\cE xp (\Box e^{-V})(\La)$$
where
\begin{equation}\label{2.2}
\Box (X) =
\left\{ \begin{array}{ll}
1 \mbox{ if } X \mbox{ is a unit block } \\
0 \mbox{ otherwise,}
& \end{array} \right.
\end{equation}
and $V(X,\f)$ is the local potential (\ref{potential}). In our problem,
the densities at every scale are nearly local, which means that they
can all be written in the form
$$A = \Box e^{-V} +K$$
where the activities $K(X,\f)$ are small in a sense we now describe.
\subsection{Polymer norms}
Our polymer activities $K(X,\f)$ will have
certain decay properties depending on the ``size'' of $X$, certain growth and
decay behaviour depending on the value of $\f$ and its derivatives, and finally
analyticity in the variable $\f$. This will be summarized by finiteness of a
certain type of norm for $K$.
First we suppose that for any $X$, $K(X,\f)$ is defined for $\f\in \cC^3(\bar
X)$, the Banach space of thrice differentiable functions on $\bar X$ with norm
\[ \|f\| =
\max_{|\beta|\leq 3 }\sup_{x}|\partial ^{\beta }f(x)|.
\]
The closure $\bar X$ here just means we assume that the
partial derivatives all have continuous boundary values. All $\f$--derivatives of
$K$ are assumed to exist: these are symmetric multi--linear functionals
defined by
\begin{eqnarray*}
{\pa \over \pa s_1} \cdots {\pa \over \pa s_n}
K(X,\phi + \sum s_i f_i)|_{s = 0}
&=&
K_{n }(X,\phi; f_1, \cdots, f_n).
\end{eqnarray*}
We further impose that $K(X,\f)$ should be Frechet-analytic in $\f$ in a complex
strip around the real space $\cC^3(\La )$.
The size of the derivative $K_n(X,\f)$ is measured by
the norm
\begin{equation}\label{natnorm}
\|K_{n}(X,\phi)\| =
\sup \{|K_{n}(X,\f;f_{1},\dots ,f_{n})|: f_j \in \cC^3(\bar X ) ,\
\|f_{j}\|_{\cC^3(X)} \leq 1\}.
\end{equation}
for $n > 0$ and $\|K_{0}(X,\phi)\| = |K_{0}(X,\phi)|$.
Actually, we need a localized version of this norm, and therefore
we consider derivatives restricted to neighborhoods
\be \tilde{\Delta } =
\{x: {\rm dist}(x, \Delta ) < 1/4 \} \ee
of blocks $\Delta $. Let $\Delta ^{\times n} =
(\Delta_1,\ldots,\Delta_n)$ be an n-tuple of blocks and define
\bea \label{norms-a}
&& \|K_{n}(X,\phi)\|_{\tilde{\Delta} ^{\times n}} \nn \\
&=& \sup \{|K_{n}(f_{1},\dots ,f_{n})|: f_j \in \cC^3(\bar X ), \
\|f_{j}\|_{\cC^3(X)} \leq 1, \
\supp f_{j} \subset \tilde{\Delta} _{j} \cap \bar X \}
\eea
A connection between the natural norm (\ref{natnorm}) and the
localized version is given if we select a smooth partition of unity
$\chi_{\De}$ indexed by unit blocks $\De$ such that $\supp \chi_{\De}
\subset \tilde \De$. We assume that each $\chi_{\De}$ is a translate
of a fixed function $\chi$. We define $\|\chi \|$ as the best constant
such that
\begin{equation}\label{chi-norm}
\|\chi _{\Delta }f\| \leq \|\chi \| \ \|f\|
\end{equation}
Then we have
\be
\|K_{n}(X,\phi) \|
\leq
\|\chi\|^n \sum _{\Delta^{\times n}}
\| K_{n}(X, \phi) \|_{\tilde{\Delta}^{\times n} }
\ee
The growth of $\|K_n(X,\phi)\|_{\tilde{\Delta} ^{\times n}} $ in $\f$ will
be controlled by a {\it large field regulator} which is some variation of the
standard choice $G = G(\kappa) = G(\kappa, X, \f )$ where
\be\label{Gchoice}
G(\kappa, X, \f )
=
\exp( \k \| \phi \|^2_{X,2,\si} )
\ee Here
\be
\| \phi \|^2_{X,a,b}=
\sum_{a \leq |\beta | \leq b}
\|\pa ^\beta \f \|_X^2 .
\ee
and $\| \f \|_X$ is the $L^2(X)$ norm. We take $\si=6$ which is large enough so
that this norm can be used in Sobolev inequalities for any low order derivative
$\pa ^{\al} \f $.
For any such $G$, we define a norm on derivatives $K_n(X,\f)$ by
\begin{equation} \label{norms-b}
\|K_{n}(X)\|_G
=
\sum_{\Delta ^{\times n}} \sup_{\phi \in \cC^3}
\|K_{n}(X,\phi)\|_{\tilde{\Delta} ^{\times n}} G^{-1}(X,\phi)
\end{equation}
To control decay in the ``size'' of $X$ we introduce
{\em large set regulators\/} $\Gamma_p(X)$ which are defined in
dimension $d=4$ by
\begin{eqnarray} \label{lsr}
\Gamma_p(X) &=& 2^{p|X|}\Gamma(X) \nn \\
\Gamma(X)
&=&
L^{(d+2)|X|}\Theta(X)
\nn \\
\Theta (X)
&=&
\inf_\tau\prod_{b\in\tau}\theta (|b|)
\end{eqnarray}
The {\it volume} $|X|$ of $X$ is the number of blocks in $X$. The infimum is over
trees
$\tau
$ composed of bonds
$b$ connecting the centers of the blocks in $X$, and the length $|b|$ of a bond
$b=xy$ is defined to be the $\ell^\infty$-metric $\sup_{1\le j\le d}
|x_{j}-y_{j}|$.
$\theta
$ is a polynomially increasing function chosen so that $\theta (s) = 1$ for
$s=0,1$ and
\be\label{thb}
\theta (\{s/L\}) \ge L^{-d-1} \theta(s),\ \ \ \ s = 2, 3, \dots
\ee
where $\{x\}$
denotes the smallest integer greater than or equal to $x$.
For any polymer function $K(X)$ we define
\begin{equation}
\|K\|_{\Gamma} =
\sup_{\Delta} \sum_{X \supset \Delta} |K(X)| \Gamma(X)
\end{equation}
Now we have the ingredients with which to assemble our norms.
For analyticity in a strip of width $h>0$, our preferred choice is
\bea \|K(X) \|_{G,h} & = &
\sum_n {h^n \over n!} \|K_n(X) \|_{G} \nn \\
\|K\|_{G,h,\Gamma} &=&
\| \ \|K(\cdot)\|_{G,h} \|_\Gamma
\eea
However, sometimes we change the order and take
\bea
\|K_n \|_{G,\Gamma} &=&
\| \ \|K_n(\cdot)\|_G \|_{\G} \nn \\
\|K \|_{G,\Gamma,h} &=&
\sum_n {h^n \over n!} \|K_n \|_{G,\Gamma}
\eea
There is also a limiting case of the
norms $\|K\|_{G,\G,h}$ in which $G^{-1}$ is concentrated at $\f =0$. These
are called {\em kernel norms\/} and are defined by
\bea
\label{kernorm} |K_{n}(X,0)|
&=& \sum_{\Delta ^{\times n}}
\|K_{n}(X,0)\|_{\tilde{\Delta} ^{\times n}} \nn \\
|K|_{h,\G} &=&
\| \sum_{n}{h^n\over n !} |K_n( \cdot, 0)|\ \|_{\Gamma } \nn \\
|K|_{\G,h}&=&
\sum_{n}{h^n\over n !}
\|\ |K_n( \cdot, 0)|\ \|_{\Gamma }.
\eea
\subsection{Bounds on $e^{-V}$}
We now state a bound on
$\|e^{-V(X)}\|_{G,h}$ for the $\f^4$ potential in $4$--dimensions:
\be
V(X) = V(X,\f, v; \la, \z ,\mu)
\ee
given in equation (\ref{cc}) with the regulator $G=G(\k)G_0^{-1}(\k_0)$ where
\be\label{gee0}
G_{0}(\k_{0},X,\phi) = \exp( \k_0 \| \f \|_{X}^2)
\ee
and $G(\k)$ is defined in (\ref{Gchoice}).
The bound is proved under the following hypotheses:
\benum\item
$\real(\la) h^4$
is positive and bounded by a sufficiently small constant,
\item $\imag (\la)/ \real (\la)$ is
bounded by a constant.
\item
$
|\mu | h^{2} \leq \real (\la) h^{4}, \ \
|\zeta| h^{2} \leq \real (\la) h^{4}, \ \
\k_0 h^{2} \leq \real (\la) h^{4} \ \
$
\item $ h^{-2}v(0),\ h^{-2}\partial ^{2}v(0),\
h^{-2}\k^{-1},\ h^{-2}\k_0^{-1} $
are all bounded by constants.
\eenum
In the above, constants are independent of $L$ and $|\zeta | = \max
{(|\zeta _{1}|, |\zeta _{2}|)}$.
\bthm \label{evthm} Under the above hypotheses
for any polymer $X$:
\be \label{evthm1}
\|e^{-V(X)}\|_{G(\k)G_{0}^{-1}(\k_0),h}
\leq 2^{|X|} ;
\ \ \
|e^{-V(X)}|_{h} \leq 2^{|X|}.
\ee
If $X$ is a subset of a unit block $\De$,
then
\be \label{block}
\|e^{-V(X)}\|_{G(\k,\Delta )G_{0}^{-1}(\k_0,\Delta ),h} \leq 2;
\ \ \
|e^{-V(X)}|_{h} \leq 2;
\ee
\ethm
\re We will need a generalization of this in which the fields $:\f^4(x):,
:\f^2(x):$, etc. in $V$ are multiplied by functions of x. The
results still hold provided the functions are pointwise bounded above and below
by suitable constants.
We define $P(X,\phi)$ to be a polynomial of degree $r$ if
derivatives of higher order than $r$ vanish. The following result will also be needed
when we come to control the perturbative part of the analysis
\blem \label{evthmP} Consider the regulator $G(\k)G_0^{-1}(\k_0)$
above with $\k = \cO(\lambda ^{1/2}), h = \cO(\lambda
^{-1/4})$. For any polynomial $P$ of degree $r$ there is a constant
$\one$ (depending on $r$) such that
\be
\| P e^{-V} \|_{G(\k)G_0^{-1}(\k_0),h,\G} \leq \one | P |_{h, \G_1}
\ee
\elem
\subsection{Results on a single RG map} \label{sec-fluc}
The RG map is the composition of fluctuation, extraction and
scaling. We discuss each of these steps in turn. The $j$th fluctuation step is
the map induced on polymer activities by Gaussian convolution with
respect to the measure with covariance $C=C_j(x-y)$ given in equation
(\ref{fourth}). These covariances are smooth functions, invariant
under the symmetries of the torus, which decay rapidly in the
separation $|x-y|$. Control of the fluctuation map in general depends
on smoothness of the fluctuation covariance $C$ and finiteness of the
following norm:
\begin{eqnarray}\label{C-norm}
\|C\|_{\theta}
&=& 3^{d}\sup_{\De_1} \sum_{\De_2} C(\De_1,\De_2)\theta\left(d(\De_1,\De_2)\right)\\
C(\De_1,\De_2) &=&
\|\chi_{\Delta_1 }C \chi_{\Delta_2 }\|_{\cC^{6}}
\end{eqnarray}
Here $\th(s)$ is the function given by (\ref{thb}), and $\chi_\De(x)$ is
the ``bump'' function chosen earlier. For theorem \ref{thm-flow2}
we also require the condition on $C$:
\be \label{C-diag}
\sup_{0 \leq |\beta| \leq 12} | \pa^{\beta}C(0)| \leq \one
\ee
The main fluctuation theorem refers to norms which involve a
one--parameter family of regulators
\begin{equation}\label{int}
G_t(X,\phi)
=
\left( 2^{|X|} G(2\k,X,\phi) \right)^{t}
G(\k,X,\phi)^{1-t}
\end{equation}
which has been constructed to satisfy
\[
\mu _{(t-s)C}\ast G_s(X,\phi)
\leq G_t(X,\phi)
\]
when $\k$ is small enough depending on a norm of $C$.
\bthm \label{thm-flow1} For any polymer activity $A$ and any $t\in [0,1]$,
there is a unique polymer activity $A(t)$ so that
\be \label{evolve}
\mu_{tC} \ast (\cE xp A)
=
\cE xp \bigl(A(t)\bigr)
\ee
The map $ \bar{\cF_t }(A) \equiv A(t)$ is analytic.
If $h' < h$ and
\be
\|A\|_{G(0), \G, h}
\leq
D\equiv \frac{(h-h')^2}{16\|C\|_{\theta}} \label{Abound}
\ee
then
\be
\|\bar{\cF_t }(A) \|_{G(t), \G, h'}
\leq
\|A\|_{G(0), \G, h} \label{fluc1}
\ee
\ethm
\bigskip
It turns out that the family $\bar{\cF }_t(A)= A(t)$ solves the following flow
equation
\be
E\bigl(A(t)\bigr)
\equiv
{\pa \over \pa t}A(t) -\De_C A(t)
- {1 \over 2}\cB_C\bigl( A(t), A(t)\bigr )
=
0
\ee
where the functional Laplacian $\De_C$ is the operator
\[ \De_C A(\f)=\frac{1}{2}\int \ d\mu_C(\z)\ A_2(\f;\z,\z)
\]
and $\cB_C$ is a
bilinear operator on activities: \[\cB_C(A,B)(\f)=\int \ d\mu_C(\z)
\ A_1(\f;\z)\circ B_1(\f;\z) \]
We will improve the estimates in the above theorem by constructing approximate
solutions to this flow equation using a perturbative analysis. Given any family
$t \rightarrow B(t)$, we measure how well it matches an exact evolution by the
{\it error} $E\bigl(B(t)\bigr)$. The following theorem tracks the growth of
the {\it remainder} $R(t)$ defined by
\be A(t) = B(t) + R(t) \ee
\bthm \label{thm-flow2}
\benum \item
Let $B(t)$ be a continuously differentiable function of $t \in [0,1]$.
Suppose $h>h'$ and $\|R(0)\|_{G(0),\G,h}, \ \sup_{0 \leq t \leq 1}\|
B(t)\|_{G(t),
\G,h}\le \frac 14 D$ where $D$ is defined in (\ref{Abound}). Then
\be
\| R(t) \|_{G(t), \G, h'}
\leq
2 (\|R(0)\|_{G(0), \G, h} +
t\sup_{ s \leq t}\| E\bigl(B(s)\bigr)\|_{G(s), \G, h}) \ee
\item Suppose further that $\|R(0)\|_{G(0),\G,h}, \ \sup_{0\le t\le 1}\|
B(t)\|_{G(t),
\G,h}\le h' /(2\| C \|_{\theta })$ and $h'\ge 2$. Then for any
$M\ge 0$,
\bea
| R(t) |_{ \G_{-1}, 1/2}
&\leq &
\one (|R(0)|_{\G_{-1},1} + (h')^{-M}\|R(0)\|_{G(0),\G, h}) \nn \\
& +&
\one \sup_{s \leq t}( |E\bigl(B(s)\bigr)|_{\G_{-1},1}
+(h')^{-M} \| E\bigl(B(s)\bigr)\|_{G(s),\G, h})
\eea
where $\one $ depends on $M$.
\eenum
\ethm
Now consider the extraction step.
Suppose that the polymer activity has the form $A = \Box e^{-V}
+K$. The extraction step consists in removing from $K(X) $ $\f$--independent terms
$F_0(X)$ and $\f$--dependent terms $F_1(X,\f)$
which are both assumed to satisfy a certain {\em localization\/} property:
$F(X,\f)$ has the
decomposition
\begin{equation} \label{loc}
F(X,\f) = \sum _{\Delta \subset X} F(X,\Delta ,\phi)
\end{equation}
where $\De$ is summed over open blocks, and $ F(X,\Delta ,\phi)$ has
the $\f$ dependence localized in $\De$, i.e. $ F(X,\Delta ,\phi)$ is a
functional on $\cC^3(\bar\De)$. The extraction step replaces the
potential $V$ by a potential $V(F)$ defined on a unit block $\De$ by
\begin{equation}\label{ex2}
(V\bigl(F)\bigr)(\De)=V(\De)-\sum_{Y\supset\De}F(Y,\Delta )
\end{equation}
The following theorem gives the essential properties of the extraction step.
The bounds are obtained when we have the following {\em stability}
of $V$ relative to the perturbation $F_1$:
there are
positive numbers
$f (X)$ independent of $\phi$ and a regulator $G$ such that
for all $\Delta $
\begin{equation} \label{ex4}
\|\exp \left\{
-V(\Delta ) - \sum _{X \supset \Delta} z(X) F_1(X,\Delta)
\right\} \|_{G,h} \le 2
\end{equation}
for all complex $z(X)$ with $|z(X)|f (X) \le 2$. The variation of Theorem
\ref{evthm} allowing variable coefficients is used to verify this condition.
\begin{theorem}\label{exthm2}
If $K$ is a polymer activity and $F_0(X),F_1(X,\f)$ satisfy
the localization hypothesis (\ref{loc}), then there
exists a new polymer activity
$\cE(K,F_0,F_1)$ so that:
\be\label{ex5}
\cE xp(\Box e^{-V}+K)(\La) =
e^{\sum _{X}F_{0}(X)}
\cE xp\bigl( \Box e^{-V(F_1)}+{\cE}(K,F_0,F_1)\bigr)(\La ),
\ee
where the linearization $\cE_{1}$ of $\cE$ in $K,F_0,F_1$ is
\be\label{ex7}
\cE_1(K,F_0,F_1)=K-(F_0 +F_1)e^{-V}.
\ee If in addition $F_1$ satisfies stability hypothesis (\ref{ex4}),
$\|f \|_{ \G_4},$ and $ \|K\|_{G,\G_2, h}$ are
sufficiently small, and $\sum_{Y \supset \De}|F_0(X,\De)| \leq \log 2$
then $\cE$ is jointly analytic in
$K,F_0,F_1$ and there is $\cO (1)$ such that
\begin{eqnarray}\label{ex8}
\|\cE( K,F_0,F_1)\|_{G,h,\G} &\leq &
\cO (1) ( \|K\|_{G,h,\Gamma_2} +
\|f\|_{ \Gamma_4} ); \\
|\cE( K,F_0,F_1)|_{h,\G} &\leq &
\cO(1) (|K|_{h,\G_2} + \|f\|_{\G_4} ).
\end{eqnarray}
Finally $\cE_{\ge 2} = \cE - \cE_{1}$ satisfies
\begin{eqnarray*}
\|\cE_{\ge 2}( K,F_0,F_1)\|_{G,h,\G}& \leq &\cO (1) \|K\|_{G,h,\Gamma_2}
\|f\|_{\Gamma_4}\\
|\cE_{\ge 2}( K,F_0,F_1)|_{h,\G}
&\leq &\cO(1) |K|_{h,\G_2} \|f\|_{\Gamma_4}.
\end{eqnarray*}
\ethm
Note the distinguished role of $\Lambda $ in equation (\ref{ex5}): the equation
does not generally hold for other polymers $X$.
\bigskip
Now consider the scaling step. The scaled field is
\begin{equation}\label{scaling-1}
\phi_{L^{-1}}(x)
=
L^{-(1-\ep/2)}\phi({x/ L})
\end{equation}
and functionals scale by
\be K_{L^{-1}}(X, \f)
=
K(LX, \f_{L^{-1}})
\ee
The scaling map on polymer activities $\cS(K)=\cS(K,V)$ is defined by the
equation
\begin{equation}\label{SKdef}
\cE xp(\Box e^{-V}+K) (LX,\phi_{L^{-1}})
=
\cE xp \left( (\Box e^{-V})_{L^{-1}}+\cS (K)\right)(X,\phi)
\end{equation}
from which one derives the explicit formula
\bea \label{rescaling1}
{\cS} (K)(Z,\f)
&=&
\sum_{\{X_j\} \to LZ}
\exp\bigl(-V(LZ \setminus X,\f_{L^{-1}})\bigr)
\prod_j K(X_j,\f_{L^{-1}})
\nn \\
&=&
\sum_{\{X_j\} \to LZ}
\exp\bigl(-V_{L^{-1}}(Z \setminus L^{-1}X,\f)\bigr)
\prod_j K_{L^{-1}}(L^{-1}X_j,\f) \eea
Here the sum is over disjoint 1--polymers $\{X_j\}$ with union $X$
such that the L-block closures $ \bar X_j^L $ are ``overlap
connected'' (see \cite{BrDiHu96a}) and have union $LZ$.
From theorem \ref{evthm}, we will verify that in our
model, $V$ satisfies the following stability bound: for all $L^{-1}$--scale polymers $ X \subset
\mbox{some block}\ \Delta$
\begin{equation}
\|(e^{-V})_{L^{-1}}(X) \|_{G,h} \le 2.
\label{stability}
\end{equation}
for $G=G(\k)$ with $\k$ small enough. This bound is needed for the main result
on scaling.
Define $\dim (\phi ) = 1 - \epsilon /2$ and
\bea h_L &=& L^{-\dim \f}h \nn \\
a &=& 2^4 \|\chi \| \eea
where $\|\chi \|$ is
the norm (\ref{chi-norm}) of the partition of unity bump function defined earlier.
\bthm \label{sthm} Let $V$ satisfy the stability assumption (\ref{stability}) and
suppose $\|K\|_{G_L,ah_L,\Gamma_{-5}}$ is sufficiently small. Then
\bea
\|\cS(K)\|_{G,h,\Gamma}
&\le& \cO (1) L^4 \|K\|_{G_L,ah_L,\Gamma_{-5}} \nn \\
|\cS (K) |_{h, \G}
&\leq&
\cO (1) L^4 |K|_{ah_L,\G_{-5}}
\eea
\ethm
We also need a sharper estimate on the linearization $\cS_1$ of $\cS$
\bea \cS_1 (K)\left(Z, \f)\right)
&=& \sum_{X: {\bar X}^L = LZ}
(e^{-V})(LZ\setminus X,\f_{L^{-1}})
K(X,\f_{L^{-1}})
\nn \\ &=& \sum_{X: {\bar X}^L = LZ}
(e^{-V})_{L^{-1}}(Z\setminus L^{-1}X,\f)
K_{L^{-1}}(L^{-1}X,\f)
\eea
The new estimate needs the stronger bound: for $L^{-1}$ scale polymers
$X$ contained in any block $\Delta $:
\begin{equation}
\|(e^{-V})_{L^{-1}} (X) \|_{g,h}
\le 2.
\label{newstability}
\end{equation}
where
\bea
g(X,\f) &=&
G_0^{-1}(\k_0, X, \f)G(\k /2, X, \f) \nn \\
&=& \exp\left[-\k_0 \|\f\|^2_X + \k/2 \|\pa\f\|^2_{X,2,\si}\right]
\eea
The {\it scaling dimension \/} of a polymer activity $K$ is defined by
\begin{defn}\label{def-dim}
\bea
\dim ( K_{n} )
&=&
r_n + n (1-\ep/2);
\nn \\
\dim (K)
&=&
\inf_{n} \dim ( K_{n})
\eea
where the infimum
is taken over $n$ such that $K_n(X,0) \neq 0$. Here $r_n$ is defined to be the
largest integer satisfying $r_n \leq 3$ and
$K_{n}(X,\phi=0;p^{\times n})=0$ whenever
$p^{\times n}$ is an $n$--tuple of polynomials of total degree less than $r_n$.
\end{defn}
Roughly $r_n$ gives the number of derivatives in $K_n$.
\bthm \label{s1theorem} Let $V$ satisfy (\ref{newstability}).
\benum \item
If
$K(X)$ is supported on large sets, then
\bea
\|\cS _1 (K) \|_{G,h,\G}
&\leq &
\cO(1) L^{-1} \|K\|_{G_L,ah_{L},\G_{-5}}
\nn \\
|\cS _1 (K) |_{h,\G}
&\leq &
\cO(1) L^{-1} |K|_{ah_{L},\G_{-5}}
\eea
\item If $K(X)$ is supported on small sets, and in addition
$\k_0h^2 \geq \one$ and $\k h^2 \geq \one$, then
\bea
\|\cS _1 (K) \|_{G,h,\G}
&\leq &
\one L^{4-\dim (K)} \|K\|_{G_L,h/2,\G_{-5}}
\nn \\
|\cS _1 (K) |_{h,\G}
&\leq &
\one L^{4-\dim (K)} |K|_{h/2,\G_{-5}}
\eea
\eenum
\ethm
\subsection{Infinite volume}
We have explained the RG transformation on finite tori with side
$L^N$. We now observe that every polymer formula makes sense when
evaluated on finite polymers in the infinite volume $\bR^4$. Indeed
the only explicit volume dependence is in the fluctuation step where
we would replace $C_j$ by $C$. (There is one exception to this claim
which is the extraction formula (\ref{ex5}). But this formula only motivates the
definition of the extraction map, and is not needed in the definition
in the sense that the formula for $\cE$ in \cite{BrDiHu96a} is well
defined for infinite volume.) Furthermore, every bound we have stated
is uniform for large volumes. Correspondingly there is an infinite
volume version of each of these results.
In the next two sections we treat finite volume and infinite volume
in parallel. In the final section we treat only the infinite volume flow.
We leave it to the companion paper
\cite{BrDiHu97} to discuss the relationship between the infinite volume
flow and the finite volume flow.
\section{The specific RG map}
A single RG transformation has been defined, and the important
estimates recorded. However
we have not yet capitalized on the freedom in the extraction step.
Our strategy now is to specify the extractions
$F$ so that in the rescaling step we have
activities $K$ with $\dim K>4$. Choosing the correct
extractions requires second order perturbation theory and control over
the higher order remainders.
We make the ansatz that after any number of
iterations the polymer activity
$A$ can be expressed in the following form:
\be \label{oldpoly}
A
=
A(\vec\la,R, w)
= B(\vec\la,v,w) + R
\ee
where
\be \label{Bdef} B(\vec\la,v,w)= \left(
\Box + Q(\vec\la,v,w)
\right) e^{-V(\vec\la,v)}
\ee
Here $\vec \la $ are effective coupling constants
\be
\vec\la =(\la,\z, \mu) = (\la,\z_1,\z_2, \mu),
\ee
$w=w(x-y)$ is a kernel and the remainder $R= R(X,\f)$ is a polymer
activity. The potential $V=V(\vec\la,v)$ is defined in
(\ref{cc}). The term
$Q=Q(\vec\la,v,w)$ is a polynomial in the field $\f$, and is the regular part of
the contribution of second order perturbation theory. The exact definition of
$Q$ is given below.
Once $Q$ is defined our goal is to exhibit a mapping $(\vec\la,R, w)
\to (\vec\la',R', w')$ such that the transform of the
activity $A=A(\vec\la,R,w)$ is $A' = A(\vec\la',R',w')$, and to demonstrate
that the remainder stays small.
\subsection{Definition and evolution of Q}\label{Qsection}
The discussion of second order perturbation theory for the $\phi^4_3$
model given in sections 3 and 7 of \cite{BrDiHu95} can be largely
repeated for our present model. Apart from the change in dimension,
and the introduction of the $\ep$ parameter, the other extra
ingredient here is the inclusion of the wave function coupling
constants $\z_1,\z_2$ and corresponding terms in $Q$ involving $\pa \f$. The
present version also has the improvement that leading order extractions are
not restricted to small sets. This means that our treatment of the RG flow
agrees with the standard perturbative treatment to second order in $\la$.
Let $v_{t} = v - tC$. Recall that the fluctuation step is an evolution
of polymer activities $A(t)$ such that $\mu _{v_{t}} \ast \cE xp
\left(A(t)\right)$ is constant in $t$. Then $A(t)$ solves
\begin{equation}\label{fl1}
E(A) \equiv \frac{\partial A}{\partial t} - \Delta _{C}A -
\frac{1}{2} \cB_C(A, A)
=
0
\end{equation}
We consider approximations $B(t)$ to $A(t)$.
We will say that $B(t)$ is a solution at order $\cO(\vec \la)$ if every term in
$E(B)$ is order $\cO(\vec \la^2)$.
\begin{lemma}\label{lem-Q1} The family
$B(t) = \Box \exp \left(-V(\vec{\lambda },v_t) \right)$ solves
(\ref{fl1} ) at order $\cO (\vec \la)$.
\end{lemma}
\pr
The $\cO(\vec \la)$ terms in
$E(B)$ are $(\pa / \pa t - \De_C)V$. This vanishes because of the following
``martingale'' property of Wick ordering
\begin{equation}\label{wick-ordering}
\left(\frac{\partial }{\partial t} - \Delta _{C}\right)
\colon \phi^{n} \colon_{v_{t}}
=
0
\end{equation}
\QED
\bigskip
We define $Q$ to improve this and get an $\cO(\vec \la^2)$ solution.
Actually we find we do not have to take all $\vec \la^2$ terms.
We will be interested in coupling constants of roughly the following
sizes: $ \la = \cO(\ep), \z_1 = \cO(\ep), \z_2 = \cO(\ep^2),
\mu = \cO(\ep^2)$. It is almost sufficient to restrict ourselves to
the $\cO(\ep^2) $ terms , i.e. $\la^2, \la\z_1, \z_1^2 $. This is not
quite true since $\z_2$ has to be a bit larger than $\cO(\ep^2)$ and so we include
also terms $\la\z_2, \z_1\z_2 $. The excluded terms are then
$\la\mu, \z_1 \mu, \z_2 \mu, \mu^2, \z_2^2$.
The functional $Q$ is made up of monomials of the form
\be
\cQ^{(m,n)}(v, w; X,\f)
:=
\int_{\tilde X }
\colon\f (x) ^m\colon_v w(x,y) \colon\f (y) ^n\colon_v
dxdy
\ee
where
\be \tilde X = \left\{ \barr{rcccl}
\De \times \De &&&& X = \De \\
\De \times \De ' \cup \De '\times
\De &&&& X = \De \cup \De ' \\
\emptyset &&&& {\rm otherwise}
\earr \right. \ee
Also a primed index indicates the field $\f$ is replaced by $\pa \f $
(and still Wick-ordered if appropriate) and a doubly primed index indicates that
$\f$ is replaced by $(-\De \f)$.
By convention $\colon
\phi^0 \colon = 1$.
Now define
\begin{eqnarray}
\cQ(\vec{\la}, v ,w)
&:=&\la^2 \left [
8\cQ^{(3,3)}(v, w)
+36 \cQ^{(2,2)}(v, w^{2} )
+48 \cQ^{(1,1)}(w^{3})
+ 12\cQ^{(0,0)}(w^{4})\right ]\nn \\
&& + \la\zeta_1 \left [
8 \cQ^{(3,1')}(v, \pa w)
+ 12\cQ^{(2,0)}(v, (\pa w)^{2}) \right ] \nn \\
&& + \zeta_1^2 \left [
2 \cQ^{(1',1')}(\pa \pa w)
+ \cQ^{(0,0)}\left( (\pa \pa w)^{2}\right) \right ] \nn \\
&& + \la \z_2 \left[ 4 \cQ^{(3,1)}(v,-\De w)
+ 4 \cQ^{(3,1'')}(v,w) + 12 \cQ^{(2,0)}\left(v,w(-\De w)\right) \right] \nn \\
&& + 2 \z_1 \z_2 \left[ \cQ^{(1',1)}(-\De \pa w)
+ \cQ^{(1',1'')}(\pa w) + \cQ^{(0,0)}\left(\pa w(-\De \pa w)\right) \right]
\label{Qnrdef}
\end{eqnarray}
\begin{lemma}\label{lem-Q2} Let $w_{t} = w + tC$. Then
\[
B(t)
=
\left(
\Box + \cQ(\vec{\la}, v_{t} ,w_{t})
\right) e^{-V(\vec\la,v_{t})}
\]
solves (\ref{fl1} ) at order $\cO(\vec \la^2)$, except for terms of order
$\la\mu, \z_1 \mu, \z_2 \mu, \mu^2, \z_2^2$.
\end{lemma}
\pr
The $\cO( \vec \la^2)$ terms in $E(B)$ are
\begin{equation}\label{cJ}
e^{-V(\vec{\lambda },v_{t})}\left( ({\pa \over \pa t} - \De_C
)\cQ(\vec{\la}, v_{t} ,w_{t}) - J(t)\right)
\end{equation}
where
\be \label{Jdef}
J(t;X,\f)
=
\left\{ \barr{rcccl} {1 \over 2}
\int_{\tilde X} C(x-y) {\pa V(t) \over \pa \f(x)}
{\pa V(t) \over \pa \f(y)} dx dy &&&& |X| \leq 2 \\
0 &&&& {\rm otherwise} \\
\earr \right.
\ee
But $\cQ$ is defined exactly so that
the terms in (\ref{cJ}) of order $\la^2, \la\z_1, \z_1^2, \la \z_2, \z_1\z_2 $
vanish. This is a calculation using identities like
\be
({\pa \over \pa t} - \Delta_C )\cQ^{(m,n)}(v_t,w_t)
=
\cQ^{(m,n)}(v_t, w_t' ) - mn \cQ^{(m-1,n-1)}(v_t,w_tC)
\ee
(See also equation (144) in \cite{BrDiHu95}).
\QED
\bigskip
This is not quite what we want because we will be repeatedly
extracting local terms from $\cQ(\vec{\la}, v_{t} ,w_{t})$.
This is handled by the following variation
\begin{lemma}\label{lem-Q3} Let $P_t(X,\phi)$ be any (integrated)
$v_{t}$-Wick ordered polynomial in $\phi$ and derivatives of $\phi$
whose coefficients are $\cO(\vec \la^2) $ and $t$-independent. Then
\[
B(t)
=
\left(
\Box + [\cQ(\vec{\la}, v_{t} ,w_{t})-P_t]
\right) e^{-V(\vec\la,v_{t})}
\]
solves (\ref{fl1} ) to order $\vec \la^2$ except for terms of order
$\la\mu, \z_1 \mu, \z_2 \mu, \mu^2, \z_2^2$.
\end{lemma}
\pr $(\pa/\pa t - \De_C)P_t = 0$ by (\ref{wick-ordering}).
\QED
\bigskip
Now we consider what extractions we want to make.
To avoid unbounded growth of $\cQ$ in the scaling step
we want to extract from it a local term $\cQ_{sing}$.
The basic criterion is that
\be \dim [\cQ-\cQ_{sing}] >4 \label{criterion} \ee
It is sufficient to arrange this for small sets, but we actually do
it for all sets.
We choose
$\cQ_{sing} =\cQ_{sing}\left(\la,v,w, v(0)\right)$
where
\bea \label{Q-sing}
&&\hspace{-.4in}
\cQ_{sing}\left(\vec{\la}, v ,w,v(0)\right) \nn \\
&=& \cQ^{(4,0)}\left (v, 36\lambda ^{2}w^{2} + 4\la \z_2(-\De w) \right ) \nn \\
&&+ \cQ^{(2',0)}
\left (v, 48 \lambda ^{2}w^{3}T +2 \zeta_1 ^{2} \pa\pa w +
2\z_1 \z_2 \ell (-\De \pa w) \right )
\nonumber \\
&&+ \cQ^{(2,0)}\left (v,
144\lambda ^{2} v(0)w^{2} + 48\lambda ^{2} w^{3}
+12\zeta_1 \lambda (\pa w)^{2} +12 \z_2 \la \left( v(0) + w \right)(- \De w)
\right )
\nonumber \\
&&+ \cQ^{(1+1',0)}\left(2\z_1\z_2(-\De \pa w) \right) \nn \\
&&+
\cQ^{(0,0)}(...) \eea
where the argument of $\cQ^{(0,0)}$ is
\bea
(...) &=& \la^2 \left( 72v(0)^2 w^{2} + 48(v(0) - (\pa\pa v)(0)T)w^{3}
+ 12 w^{4} \right) \nonumber\\
&&+ \z_1^2 \left(
2 (\pa\pa v)(0) \pa\pa w+ (\pa\pa w)^{2} \right) \nn \\
&&+2 \z_1\z_2 \left( (\pa \pa v)(0) \ell + 2\pa w )(-\De \pa w)
\right)
\eea
Here the superscript $1+1'$ indicates the field
$\f \pa \f =:\f \pa \f:_v $ and $\ell, T$ denote the kernels
\bea
\ell_a(x-y) &=& -(x-y)_a \nn \\
\label{Tdef}
T_{ab}(x,y) &=& -{1 \over 2} (x-y)_a (x-y)_b
\eea
(Actually $\cQ_{sing}\left(\vec{\la}, v ,w,v(0)\right)$ also depends on $(\pa \pa
v)(0)$ but we have suppressed it from the notation).
The demonstration that this satisfies the extraction criterion (\ref{criterion})
comes a little later.
Now we define the $Q$ that appears in our ansatz (\ref{Bdef}).
It is defined by
\be Q(\vec \la, v,w) = \cQ_{reg}(\vec \la, v,w) \equiv
\cQ(\vec \la, v,w) -
\cQ_{sing}\left(\vec \la, v,w, v(0)\right)
\ee
The objective is to show that $ \cQ_{reg}$ is RG covariant,
\begin{eqnarray}\label{Qsummary}
Q(\vec \la, v,w) - \cQ_{sing}\left(\vec \la, v,w, v(0)\right)
&\stackrel{\cF}{\longrightarrow}&
Q(\vec \la, v^{\#},w^{\#}) - \cQ_{sing}\left(\vec \la, v^{\#}, w, v(0)\right)\nn\\
&\stackrel{\cE}{\longrightarrow}&
Q(\vec \la, v^{\#},w^{\#}) -
\cQ_{sing}\left(\vec \la, v^{\#}, w^{\#}, v^{\#}(0)\right)\nn\\
&\stackrel{\cS}{\longrightarrow}&
Q(\vec \la, v, w) -
\cQ_{sing}\left(\vec \la, v, w, v(0)\right)
\end{eqnarray}
where $v^\# = v_1$ and $w^\# = w_1$. Introduce
\begin{equation}\label{Q-reg}
Q(t)
=
\cQ (\vec{\lambda },v_{t},w_{t}) -
\cQ_{sing}\left(\vec{\la}, v_{t} ,w,v(0)\right)
\end{equation}
This satisfies $Q(0) = Q$ and solves (\ref{fl1}) at order $\cO(\vec
\la^2)$ (with exceptions) by Lemma~\ref{lem-Q3}. At $t=1$ it equals
$Q(\vec \la, v^{\#},w^{\#}) - \cQ_{sing}\left(\vec \la, v^{\#}, w, v(0)\right)$
so we have the first step in (\ref{Qsummary}).
We also note that if we define
\be F_Q(t) = \cQ_{sing}\left(\vec{\la}, v_{t} ,w_{t},v_{t}(0)\right) -
\cQ_{sing}\left(\vec{\la}, v_{t} ,w, v(0)\right)
\ee
then we have the decomposition into regular and singular parts:
\be Q(t) = \cQ_{reg} (\vec \la , v_t, w_t) + F_Q(t) \ee
Extracting $F_Q(1)$ will give the second step in (\ref{Qsummary}).
The third step follows from the following scaling property:
\begin{lemma}\label{lem-Q4}
\[
\cS_{1}\left(
\cQ_{reg}(\vec{\lambda },v,w)
e^{-V(\vec\la,v)}
\right)
=
\cQ_{reg}(\vec{\lambda }_{L},v_{L},w_{L})
e^{-V(\vec{\la}_{L},v_{L})}
\]
where kernels scale by
\bea v_L(x-y) &=& L^{2-\ep}v\left(L(x-y)\right) \nn \\
w_L(x-y) &=& L^{2 -\ep}w\left(L(x-y)\right)
\eea
and the coupling constants scale by
\[ \vec\la_L= ( L^{2\ep}\lambda , L^{\ep}\z_1, L^{\ep}\z_2 ,L^{2+\ep}\mu) \]
\end{lemma}
\pr $\cS _{1}$ is linear so it suffices to prove that
\begin{eqnarray*}
&&
\cS_{1}\left(
\cQ(\vec{\lambda },v,w)
e^{-V(\vec\la,v)}
\right)
=
\cQ(\vec{\lambda }_{L},v_{L},w_{L})
e^{-V(\vec{\la}_{L},v_{L})}\nn \\
&&
\cS_{1}\left(
\cQ_{sing}\left(\vec{\la}, v, w, v(0)\right)
e^{-V(\vec\la,v)}
\right)
=
\cQ_{sing}\left(\vec{\la}_{L}, v_{L}, w_{L}, v_{L}(0)\right)
e^{-V(\vec{\la}_{L},v_{L})}
\end{eqnarray*}
These are all straightforward computations. For example we have
\be
\sum_{X: {\bar X}^L = LZ}
\cQ^{(m,n)}(v, w; X,\f_{L^{-1}})
= L^{d_{m,n} }
\cQ^{(m,n)}(v_{L}, w_{L}; Z,\f) .
\ee
where
\be
d_{m,n}
=
8-(1-\ep/2)(m + n+2) - \{ {\rm number\ of\ primed\ indices}\}
\ee
\QED
We shall now verify that the extraction criterion (\ref{criterion})
is satisfied with our choice of
$\cQ_{sing}$. For this we need to exhibit cancellations between $\cQ$ and
$\cQ_{sing}$. We rewrite (\ref{Q-sing}) to parallel equation
(\ref{Qnrdef}) by defining at $(X,\f)$
\bea
\cQ^{(2,2)}_{sing}(v,w^2)
&=&
\int_{\tilde X} :\f (x)^2:_v^2 w(x-y)^2 dxdy
\nn \\
&=&
\cQ^{(4,0)}(v,w^{2})+4\cQ^{(2,0)}(v,v(0)w^{2})
+ 2\cQ^{(0,0)} (v(0)^2 w^{2}) \label{q22} \\
&&\nn \\
\cQ^{(3,1)}_{sing}(v,-\De w) &=& \int_{\tilde X}
\colon \f(x)^3 \colon _v \f(x) (-\De w)(x-y) dx dy \nn \\
&=& \cQ^{(4,0)}(v, -\De w) + 3v(0)\cQ^{(2,0)} (v, - \De w)
\\ && \nn \\
\cQ^{(1,1)}_{sing}(w^3) &=&
{1 \over 2} \int_{\tilde X} [\f(x)^2+\f(y)^2
-[(x-y)\cdot\pa\f(x)]^2] w(x-y)^3 dx dy
\nn \\
&=&
\cQ^{(2,0)} (v, w^{3})
- \cQ^{(2',0)}(v, w^{3}T) + \cQ^{(0,0)}\left(
(v(0) - (\pa\pa v)(0)T)w^{3}\right) \nn \\
&&\label{q11} \\
\cQ^{(1',1')}_{sing}(\pa \pa w)
&=& \int_{\tilde X}
\left( \pa_a \f (x)\right)
\left( \pa_b\f (x)\right) (\pa_a\pa_b w)(x-y) dxdy
\nn\\
&=&
\cQ^{(2',0)} (v, \pa\pa w) + \cQ^{(0,0)} \left( (\pa\pa v)(0)\pa\pa w\right)
\label{qs1'1'}\nn\\
&& \\
\cQ_{sing}^{(1',1)}(-\De \pa w) &=& \int_{\tilde X} \pa \f(x)
[\f(x) + (y-x) \pa \f(x) ](-\De \pa w)(x-y) dx dy \nn\\
&=& \cQ^{(1+1',0)}(-\De \pa w )
+\cQ^{(2',0)} \left( \ell (-\De \pa w) \right)
+\cQ^{(0,0)}\left( (\pa \pa v)(0) \ell (-\De \pa w) \right)
\nn\\ &&\label{qs2'0}\eea
We also define
\be \cQ^{(2,0)}_{sing} = \cQ^{(2,0)} \ \ \ \ \ \ \ \ \ \
\cQ^{(0,0)}_{sing}
=\cQ^{(0,0)} \ee
with the remaining singular parts equal to zero.
Now defining $\cQ^{(m,n)}_{reg}=\cQ^{(m,n)}-\cQ^{(m,n)}_{sing}$ we find
at $(X,\f)$
\bea
\cQ^{(2,2)}_{reg}(v,w^2)
&=&
-{1 \over 2} \int_{\tilde X} (:\f (x)^2:_v
- :\f (y)^2:_v)^2 w(x-y)^2\ dxdy \nn\\
&& \nn \\
\cQ^{(3,1)}_{reg}(v,-\De w) &=& \int_{\tilde X}
\colon \f(x)^3 \colon _v (-\De w)(x-y) \left( \f(y)- \f (x)\right) dx dy \nn \\
&& \nn \\
\cQ^{(1,1)}_{reg}(w^3) &=& -{1 \over 2} \int_{\tilde X}[ \left(\f (x)
- \f (y)\right)^2 -[(x-y)\cdot\pa\f(x)]^2] w(x-y)^3 dx dy \nn\\
&& \nn \\
\cQ^{(1',1')}_{reg}(\pa \pa w) &=& -{1 \over 2} \int_{\tilde X}
\left(\pa_a \f (x)-\pa_a \f (y)\right)
(\pa_a \pa_b w)(x-y)\left(\pa_b \f (x)-\pa_b \f (y)\right) dx dy \nn \\
&& \nn \\
\cQ_{reg}^{(1',1)}(-\De \pa w) &=& \int_{\tilde X} \pa \f(x)
[\f(y) - \f(x) - (y-x) \pa \f(x) ](-\De \pa w)(x-y) dx dy \nn\\
&&\label{q1'1'}
\eea
We also have
\be \cQ^{(3,3)}_{reg}
= \cQ^{(3,3)} \ \
\cQ^{(3,1')}_{reg}
= \cQ^{(3,1')} \ \
\cQ_{reg}^{(3,1'')} = \cQ^{(3,1'')} \ \
\cQ_{reg}^{(1',1'')} = \cQ^{(1',1'')} \ee
with the remaining regular parts equal to zero.
A direct check shows that every non--zero $\cQ^{(m,n)}_{reg}$ has
$\dim > 4$.
Collecting terms we have
\bea \label{Qregular}
\cQ_{reg}(\vec{\la}, v ,w) & = & \la^2 \left[
8 \cQ^{(3,3)}(v,w) +36 \cQ^{(2,2)}_{reg}(v,w^2)
+48\cQ^{(1,1)}_{reg}(w^3) \right] \nn \\
&& +8\la \z_1 \cQ^{(3,1')}(v,\pa w) + 2\z_1^2 \cQ^{(1',1')}_{reg}(\pa \pa w) \nn \\
&& +\la \z_2 \left[ 4 \cQ^{(3,1)}_{reg}(v,-\De w)4 + \cQ^{(3,1'')}(v, w)
\right] \nn \\
&& +\z_1 \z_2 \left[ 2\cQ_{reg}^{(1',1)}(-\De \pa w)
+ 2\cQ^{(1',1'')}(\pa w)
\right]
\eea
\subsection{Fluctuation}
We summarize what happens when the fluctuation step is applied to the ansatz
(\ref{oldpoly}):
\bea
A &=& B+R
\nn \\
B &=& (\Box + Q) e^{-V}.
\eea
The fluctuation step constructs a new activity $A^{\#}$ such that $\mu
_{v^{\#}}\ast \cE xp (A^{\#}) = \mu _{v} \ast \cE xp(A)$ where
\[
v^{\#} = v - C.
\]
We define $A^\#=A(1)$ where $A(t)$ is determined by the requirement that
for $v_t = v-tC$ the quantity
$\mu_{v_{t}}\ast
\cE xp \left(A(t)\right)$ is constant in $t$ and therefore equal to $\mu _{v} \ast \cE
xp(A)$. Let $B(t)$ be the approximate solution
from the previous section and define $R(t)$ to be the remainder.
Thus we have
\bea
A(t) &=& B(t)+R(t)
\nn \\
B(t) &=& \left(\Box + Q(t)\right) e^{-V(t)}. \label{Aoft}
\eea
where $Q(t)$ is given by (\ref{Q-reg}) and $V(t) = V(\vec \la, v_t)$.
Then defining $B^\#=B(1)$ and $R^\# = R(1)$ we have
\bea A^\# &=& B^\#+R^\#
\nn \\
B^\# &=& (\Box + Q^\#) e^{-V^\#} \label{Rsharp}
\eea
where
\bea
V^{\#} &=& V(1) = V(\vec\la,v^\#)
\nn \\
Q^{\#} &=& Q(1)= \cQ_{reg}(\vec \la ,v^\#,w^\#)+F_Q
\nn \\
F_Q &=& F_Q(1) = \cQ_{sing}\left(\vec{\la}, v^\# ,w^\#,v^\#(0)\right) -
\cQ_{sing}\left(\vec{\la}, v^\# ,w, v(0)\right)
\nn \\
w^{\#} &=& w_1 = w +C. \label{fq1}
\eea
We also define $K^{\#}$ by
\[
A^\# = \Box e^{-V^\#} + K^\#
\]
so that
\[
K^\# = Q^\# e^{-V^\#} +R^\#.
\]
\subsection{Extraction}\label{Esection}
By theorem~\ref{exthm2}, the result of extracting a quantity
$F=F_0+F_1$ from
$K^\#$ is a new activity of the form
\be A^* = \Box e^{-V^*} + K^* \ee
where
\bea \label{Vstardef}
V^*(\De)
&=&
V^\#(\De) - \sum_{X \supset \De} F_1(X, \De)
\nn \\
K^*
&=&
\cE(K^\#,F) \nn \\
&=&
K^\# -F e^{-V^*} +\cE _{\geq 2 }(K^\#,F)
\eea
Here $F_1$ is the non-constant part of $F$, and
$F_{1}(X,\Delta )$ is related to $F_{1}(X)$ by the local decomposition:
\[
F_{1}(X) = \sum_{\De \subset X}F_{1}(X, \De)
\]
We choose $F$ of the form
\be
F=F_Q + F_ R
\ee
where
$F_Q $ is the singular part of $Q^\#$ defined above, and $F_R$ is an
extraction from $R^\#$ defined below. Then we have
\begin{eqnarray}
K^*
&=&
Q^{\#} e^{-V^{\#}} - F_{Q} e^{-V^{*}} + R^{\#} -
F_{R} e^{-V^{*}} + \cE _{2}(K^{\#},F)\nn \\
&=&
\cQ_{reg}(\vec \la ,v^\#,w^\#) e^{-V^*} + R^{**} \label{Kstar}
\end{eqnarray}
where
\be\label{rstarstar}
R^{**}
=
(R^\#-F_Re^{-V^*})+ Q^\#\left(e^{-V^\#}-e^{-V^*}\right)
+\cE_{\ge 2}(K^\#,F)
\ee
The local decomposition of $F_{Q,1}$ is obtained as follows.
The functional $F_{Q,1}$ is a sum
of monomials of the form
\be \label{F-form}
F(X) = \int_{\tilde X} :\f^n(x): f(x-y) dx dy
\ee
For each such monomial the local decomposition holds with $F(X,\Delta
)$ defined for $X = \De_{1} \cup \De_{2}$ by :
\be
F( X, \De)
=
\left\{\begin{array}{ll}
\int_{\De_1 \times \De_{2}} :\f^n(x): f(x-y) dx dy
& \mbox{ if } \Delta = \Delta_{1}\\
\int_{\De_2 \times \De_{1}} :\f^n(x): f(x-y) dx dy
& \mbox{ if } \Delta = \Delta_{2}\\
0&
\mbox{ otherwise}
\end{array}\right.
\ee
The next lemma is important because it shows that $F_{Q}$ is
responsible for the flow of the coupling constants $\vec{\lambda }$
and is the origin of the explicit formulas for this flow.
\begin{lemma}\label{lem-ex1}
\[
\sum _{X \supset \Delta }
F_{Q,1}(X,\Delta)
=
V(\Delta, \delta \vec{\la}_{Q}, v^{\#})
\]
where
\noindent \bea
\delta \la_Q
&=&
a(w) \la^2
=
36 \lambda^{2} \int_{\Lambda}
\left [
w^{\#2} - w^{2}
\right ] \ \nn \\
\delta \zeta_{1,Q}
&=&
b_1(w) \la^2
=
48 \lambda^{2} \int_{\Lambda}
\left [
w^{\#3} T- w^{3} T
\right ] \nn \\
\delta \zeta_{2,Q}
&=&
b_2(w) \la^2 =0 \nn \\
\delta \mu_Q
&=&
c(w) \la^2 - d(w) \la (\z_1 +\z_2) \nn \\
&=&
144 \lambda^{2} \int_{\Lambda}
\left [
v^{\#}(0)w^{\#2} - v(0)w^{2}
\right ] \nn \\
& &
+\ 48\lambda^{2} \int_{\Lambda}
\left [
w^{\#3}- w^{3}
\right ] \nn \\
&&
+\ 12 (\z_1 +\z_2)\lambda \int_{\Lambda}
\left [
(\pa w^{\#})^{2} - (\pa w)^{2}
\right ]
\eea
where $T$ was defined in (\ref{Tdef}).
\end{lemma}
\pr By (\ref{F-form})
\be
\sum_{X \supset \De} F(X, \De)
=
\left(\int_{\De} :\f^n(x):dx\right)\left( \int_{\La}f(x-y) dy \right)
\ee
which has the right form to be one of the monomials in $V(\Delta,
\delta \vec{\la}_{Q}, v^{\#})$. The formulas for $\delta
\vec{\la}_{Q}$ were obtained from ( \ref{Q-sing}), (\ref{fq1}).
Note that many contributions to $\de \vec \la_Q$ are zero since we integrate
a total derivative. For example the $\cO(\zeta _{1}^{2},\zeta
_{1}\zeta _{2})$ changes in the field strength vanish since
\be 2\z_1^2\int_{\La} (\pa_a \pa_b w^\#-\pa_a \pa_b w)=0 \ee
and since an integration by parts gives
\bea \z_1 \z_2 \int_{\La} (y-x)_a (-\De \pa_a w)(x-y) dy&& \nn \\
&&\hspace{-1in} =\ 4 \z_1 \z_2 \int_{\La} (-\De w)(x-y) dy \nn \\
&&\hspace{-1in}=\ 0
\eea
We have also used
\[ 12 \la \z_2 \int_{\La} \left[w^\#(-\De w^\#)-w(-\De w)\right]
= 12 \la \z_2 \int_{\La} \left[(\pa w^\#)^2 -(\pa w)^2\right] \]
to obtain the $\la \z_2$ contribution to $\de \mu_Q$.
\QED
\bigskip
\noindent {\it Construction of $F_{R}$:\/} $F_R(X,\phi)$ is zero if $X$ is not
a small set; otherwise it is a local polynomial of the form
\bea \label{fr}
F_R (X,\phi)
&=&
\al_{0}(X)|X| +
\al_{2,0}(X) \int_X \f^2 +
\sum _{a} \al_{2,1}(X,a) \int_X \f (\pa_a \f)
\nn \\
&+&
\sum _{a,b} \al_{2,2}(X,a,b) \int_X (\pa_a \f )( \pa_b \f ) +
\sum _{a,b} \al_{2,2}'(X,a,b) \int_X \f (\pa_a \pa_b \f) \nn \\
&+&
\al_{4}(X) \int_X \f^4
\eea
with coefficients $\al$ determined by the next lemma.
Let us introduce the notation
\be F_n=(F_R)_n(X,0) \ \; \ \ R_n=(R^\#)_n(X,0) \ \; \ \ V_n =V^\#_n(X,0)
\ee
\begin{lemma}\label{lem-F-R} There is a unique choice of coefficients
$\alpha $ in $F_R (X,\phi) $ such that
\begin{eqnarray}
&&
\dim(R^{\#}-F_{R}e^{-V^{\#}}) > 4 \nn \\
&&
\sum _{X \supset \Delta }
F_{R,1}(X,\Delta)
=
V(\Delta, \delta \vec{\la}_{R}, v^{\#}) \label{F-R}
\end{eqnarray}
Furthermore $\al_{m,n}(X)$ and $\delta \vec{\la}_{R}$ are polynomials
in $e^{V_0}R_n(p^{\times n})$ and $V_n(p^{\times n})$ where $n = 0,
2, 4$ and $p = 1, x_{a}, x_{b}, x_{a}x_{b}$. The $x$-origin can be
chosen arbitrarily for each $X$. The polynomials have $\one $
coefficients and are linear in $R_n$.
\end{lemma}
\pr Recall that $\dim $ was defined in definition (\ref{def-dim}).
The first four non-vanishing derivatives of $R^{\#}-F_{R}\exp(-V^{\#})$
evaluated at $\phi = 0$ are
\bea
&&
J_{0} =
R_0 - F_0e^{-V_0} \nn \\
&&
J_{2} =
R_2 - \left(
F_2 - F_0V_2
\right) e^{-V_0} \nn \\
&&
J_{4} =
R_4 - \left(
F_4 -6{\rm Sym}(F_2V_2) + 3F_0{\rm Sym}(V_2^2)
- F_0V_4
\right) e^{-V_0} \label{dimR}
\eea
Referring to the definition of dimension we find that the $\dim >4$
condition in (\ref{F-R}) is equivalent to the vanishing of these
derivatives when evaluated on the constant $1$ and monomials $x_{a}$,
$x_{a}x_{b}$ as follows
\bea \label{dim4}
&&
J_{0} = 0 \nn \\
&&
J_{2}(1^{\times 2}) = 0,
\ \ \
J_{2}(1,x_{a}) = 0,
\ \ \
J_{2}(x_{a},x_{b}) = 0,
\ \ \
J_{2}(1, x_{a}x_{b}) = 0 \nn \\
&&
J_{4}(1^{\times 4}) = 0
\eea
By using these equations in the order given we determine recursively
the coefficients $\alpha $ in (\ref{fr}) in the order
\[
\al_{0}(X), \
\al_{2,0}(X), \
\al_{2,1}(X,a), \
\al_{2,2}(X,a,b), \
\al_{2,2}'(X,a,b), \
\al_{4}(X)
\]
and easily see that $\alpha $ is a polynomial (linear) in
$e^{V_0}R_{n}$ and $V_{n}$ evaluated on $1, x_{a}, x_{b},
x_{a}x_{b}$. For example we have
\begin{eqnarray}
\al_{0}(X) &=& \frac{1}{|X|}e^{V_{0}}(X)R_{0}(X) \nn \\
\al_{2,0}(X) &=& \frac{1}{2|X|} e^{V_{0}}(X)
\left(
R_{2}(X;1^{\times 2}) +
R_{0}(X) V_{2}(X;1^{\times 2})
\right)\nn \\
\al_{2,1}(X,a) &=& e^{V_{0}}(X)
\Bigl[
R_{2}(X;1,x_{a}) + R_{0}(X)V_{2}(X;1,x_{a}) \nn \\
&& -\ \frac 1{|X|} \left(R_{2}(X;1^{\times 2})
+ R_{0}(X)V_{2}(X;1^{\times 2})\right)\int _{X}x_{a}\Bigr]
\label{alpha21}
\end{eqnarray}
Now $F_R$ is defined. Define $F_{R,1}(X, \De)$ by dropping
the $\al_{0}(X)$ term and restricting the
integrations in (\ref{fr}) to $\Delta $. It remains to establish
(\ref{F-R}).
Consider
\be
\sum _{\Delta }\left(
\sum_{X \supset \De} \al_{2,1}(X,a)
\right)
\int_{\De} \f (\pa_a \f)
\ee
Each iteration of the renormalization group map preserves the symmetry
of the initial data under reflections that map the lattice of unit
blocks to itself.
As a consequence of (\ref{alpha21}) with $x$-origin chosen at the
center of $\Delta $ one can show that
\be
\al_{2,1}(r_aX,a) =- \al_{2,1}(X,a)
\ee
where $r_a$ is the
reflection in the
$a$-direction through
$\De$. Hence the sum above is zero and there is no term of this form in the
potential. Similarly we conclude that
\bea
\sum _{X \supset \Delta } \al_{2,2}(X,a,b)
&=&
\al_{2,2} \de_{a,b}
\nn \\
\sum _{X \supset \Delta } \al_{2,2}'(X,a,b)
&=&
\al_{2,2}' \de_{a,b}
\eea
and these contribute to $\z_{1,R},\ \z_{2,R}$ respectively.
From these considerations we find that the changes in the coupling
constants are given by (taking into account the Wick ordering)
\bea
\de \la_R
&=&
\sum_{X \supset \De} \al_{4}(X)
\nn \\
\de \z_R
&=&
(\al_{2,2}, - \al_{2,2}')
\nn \\
\de \mu_R
&=&
\sum_{X \supset \De} \left(\al_{2,0}(X) + 6v^\#(0)\al_{4}(X)\right)
\eea
\QED
\bigskip
We introduce the notation
\bea
\de \la_R
&=&
- r(\vec{\la},w, R)
\nn \\
\de \z_{i,R}
&=&
- s_i(\vec{\la},w, R)
\nn \\
\de \mu_R
&=&
- t(\vec{\la},w, R)
\eea
Note that $r,s,t$ are linear functions of $R^\#$, which in turn is a
function of $(\vec{\la}, w, R)$.
The new potential is now $V^* = V(\vec{\la^*},v^\#)$
where $\vec{\la}^* = \vec{\la} - \de \vec{\la}_Q - \de \vec{\la}_R$.
Thus we have $\vec{\la^*}$ given by
\bea
\la^* &=& \la - a(w) \la^2 + r(\vec\la,w,R) \nn \\
\z_i^* &=& \z_i -b_i(w)\la^2 + s_i(\vec\la,w,R) \nn \\
\mu^* &=& \mu -c(w)\la^2 - d(w) \la (\z_{1}+\z_{2})
+ t(\vec\la,w,R)\label{lastar}
\eea
Referring to (\ref{Kstar}) we see that the coupling constants $\vec
\la^*$ in the potential do
not match the coupling constants $\vec \la$ in $Q^{**}= \cQ_{reg}(\vec
\la, v^\#,w^\#)$. We make a further adjustment by defining
\be
Q^* = \cQ_{reg}(\vec \la^*, v^\#,w^\#)
\ee
and have
\be
K^* = Q^* e^{-V^*} + R^* \label{kstar}
\ee
where
\be
R^* = (Q^{**} - Q^{*})e^{-V^*} + R^{**} \label{rstar}
\ee
\subsection{Scaling} \label{sec-scaling2}
After scaling we have a new activity
\be A' = \Box e^{-V'} + K' \ee
where $K'= \cS (K^*)$. By definition of $\cS $
\be
V'
=
V(\vec \la^*, v^\#)_{L^{-1}}
=
V(\vec \la^*_L, v^\#_L)= V(\vec \la',v')
\ee
where $v' = v^\#_L(x - y)$ and
\be \label{laprime}
\vec \la'
=
\vec \la^*_L
=
(L^{2\ep}\la^*, L^{\ep}\z^*, L^{2+ \ep}\mu^*)
\ee
Also
\be
K' = Q'e^{-V'} + R'
\ee
where
\bea \label{rprime}
Q'e^{-V'}
&=&
\cS_1 (Q^*e^{-V^*}) \nn \\
R'
&=&
\cS_1(R^*) + \cS_{\geq 2} (K^*)
\eea
$\cS_1$ is the linearization of $\cS$ and $\cS_{\geq 2} = \cS -\cS_1$.
From lemma~\ref{lem-Q4}
\be
Q' = \cQ_{reg}(\vec{\la'},v',w')=Q(\vec{\la'},v',w')
\ee
with $w' = w^{\#}_{L}$ which means that the form of our ansatz
(\ref{oldpoly}) has been preserved as we wanted.
\subsection{Summary}
Starting with the polymer activity (\ref{oldpoly}) we have found that
one RG step transforms the ansatz to
\be \label{newpoly}
A'
=
A(\vec\la',R', w')
=
\left(
\Box + Q(\vec\la',v',w')
\right) e^{-V(\vec\la',v')} + R'
\ee
The new coupling constants are obtained from
(\ref{lastar}),(\ref{laprime}) and are given by
\bea
\la'
&=&
L^{2\ep}\left(
\la - a(w) \la^2 + r(\vec \la, w,R) \right)
\nn \\
\z_i'
&=&
L^\ep \left( \z_i -b_i(w)\la^2 + s_i(\vec \la, w,R) \right)
\nn \\
\mu'
&=&
L^{2+ \ep} \left(
\mu -c(w)\la^2 - d(w) \la ( \z_1 +\z_2) + t(\vec \la, w,R) \right)
\eea
The new remainder $R'$ is given as a function
\be
R' = U(\vec \la ,R, w)
\ee
by equations (\ref{Rsharp}),(\ref{rstarstar}),(\ref{rstar}),(\ref{rprime}).
Finally the new kernel is
\be
w'(x-y)
=
L^{2-\ep} \left( w\left(L(x-y)\right) + C\left(L(x-y)\right) \right)
\ee
In infinite volume $v' = v$ and everything is reduced
to this single mapping in the variables $\vec \la , R, w$.
\section{Estimates}
We observe the convention that $\one $ stands for a constant
independent of $\ep$ and $L$ and that $C$ stands for a constant which
may depend on $L$ but is independent of $\ep$. The value of $C$ may
vary from line to line. This is consistent with the usage in
section~\ref{sec-review}.
\subsection{Estimates on $w, a(w), b(w),...$}
We begin with some estimates on $w$.
Recall that $w' = w_L + C_L$ where $ w_L(x) = L^{2-\ep} w(Lx) $ .
Starting with $w_0 = 0$ we get a sequence $w_j$
defined by $w_{j+1} = (w_j)_L + C_L$ or
\be w_j(x) = \sum_{l=1}^j L^{(2- \ep)l}C(L^l x) \ee
This can also be written
\be w_j(x) = \int_{L^{-2j}}^1 d\al \ \al^{\ep/2-2} e^{-x^2/4\al} \ee
For $x \neq 0$ this also makes sense for $j=\infty$. We sometimes
write $w_{\infty} = w_*$.
The basic properties of $C$ and $w_j$ are contained in:
\blem
For any multi-index $\beta$ there is a constant $\one $ such that
for all $x$ and $1 \leq j \leq \infty $
\bea
|(\pa^{\beta} C)(x)| &\leq& \one e^{-2|x|/L} \nn \\
|(\pa^{\beta} w_j)(x)| & \leq & \one e^{- \frac{3}{2}|x|}|x|^{-2-|\beta|+\ep}
\eea
\elem
\pr We have
\be |C(x)| \leq \one\int _1^{L^2}d \al \al^{\ep/2-2} e^{-2|x|/\sqrt{\al}}
\leq \one e^{-2|x|/L} \ee
and also
\beaa w_j(x)&\le& \one\int_{L^{-2j}}^1 \ d\alpha \ \alpha^{\ep/2-2}\
e^{- \frac{3}{2}|x|- \frac{1}{2}|x|/\sqrt{\alpha}}\\
&\le& \one e^{-\frac{3}{2}|x|} \ \int_0^\infty \ d\alpha \ \alpha^{\ep/2-2}\
e^{- \frac{1}{2}|x|/\sqrt{\alpha}}\\
&=&\one e^{- \frac{3}{2}|x|}\ |x|^{-2+\ep}
\eeaa
The bounds for the derivatives follow similarly.
\QED
\bigskip
We will find it useful to cast this in a Banach space setting.
Let $\cW$ be the Banach space of complex-valued functions on
$ \bf R^4 $ for which the
following norm is finite:
\be \|w\|=\sup_{|\beta| \leq 3}\sup_x (|x|^{9/4 +|\beta|}|\pa^{\beta}w(x)|e^{|x|}) \ee
By the lemma, the sequence $w_j$ starting at $w_0=0$ is in $\cW$
and there is a constant $k$ independent of $L$ such that for
$0 \leq j \leq \infty$
\be \|w_j\| \leq k/2 \ee
The point of taking the power $9/4$ here is to make a norm in which the
mapping
$w \to w'$ is a strong contraction.
It is easy to show that for any $w,\tilde w \in \cW$
\be \|w'-\tilde w' \| \leq L^{-1/4} \|w -\tilde w\| \ee
Thus, starting from any $w_0=w$ with $\| w \| \leq k/2$
we generate a sequence $w_j$ converging to the unique fixed point $w_*$ and
satisfying
\bea \|w_j - w_*\| &\leq & L^{-j/4}\|w-w_*\| \leq L^{-j/4}k \nn \\
\|w_j\| &\leq & k \label{kdef}\eea
Hereafter we consider general $w \in \cW$ with $\|w\|
\leq k$.
\blem \label{lem-abcd} There is a constant $C$ such that for $w \in \cW$
and $\| w \| \leq k $
\be |a(w)|, |b_{1}(w)|, |c(w)|, |d(w)| \leq C \ee
Furthermore there is a constant $k'$ independent of $L$ such that
\be |a(w_1) - a(w_2)| \leq k' \|w_1 - w_2 \| \ee
and similarly for $b,c,d$. \elem
\pr We have, from lemma~\ref{lem-ex1},
\bea a(w) &=& 36 \int ( 2wC + C^2) \nn \\
b_{1}(w) &=& 48 \int (3w^2C +3wC^2 + C^3)T \nn \\
c(w) &=& 48\int
\left(6v^\#(0)wC+3v^\#(0)C^2+3\left(C-C(0)\right)w^2+3wC^2+C^3\right) \nn \\ d(w)
&=& 12 \int \left(2\pa w \pa C + (\pa C)^2\right) \eea
The issue is the singularity at $x=0$.
First consider $a(w)$.
For $w \in \cW$ we have
\be |w(x)| \leq |x|^{-9/4} e^{-|x|}\|w\| \label{w1} \ee
Therefore $w$ is integrable and since $C \leq \one $ we have
that $wC$ is integrable
and $\int wC \leq \one$. Since $|C(x)| \leq \one e^{-2|x|/L}$
we also have that $C^2$ is integrable (but the bound depends on $L$).
For the Lipschitz bound only the first term contributes and so we can
use the bound (\ref{w1}) for $w_1-w_2$ to
get a bound independent of $L$.
For the bound on $b_{1}(w)$ we use that $|T(x)| \leq
\cO(|x|^{2})$ and
\be
|x|^{2}|w(x)|^2 \leq |x|^{-5/2}e^{-2|x|}\|w\|^2
\ee
to bound the first term and get the result. The Lipschitz
bound follows similarly.
For the bound on $c(w)$
we use $ |C(x) -C(0)| \leq \cO(|x|)$ to obtain the factor of $x$ needed
to integrate $w^2$. We also use $|v^\# (0)| \leq \one $.
Finally $d(w)$ is bounded since
\be |\pa w(x)| \leq |x|^{-13/4}e^{-|x|}\|w\| \ee
is integrable.
\QED
\bigskip
In fact we need a sharper bound on $a(w)$.
\blem Let $L$ be sufficiently large and $\ep < \ep (L)$.
Then there exist positive constants $c_1,c_2$ independent of $L$
such that for real $w \in \cW$
and $\| w \| \leq k $
\be c_1 \log L \leq a(w) \leq c_2 \log L \ee \elem
\pr As noted the first term in $a(w)$ is bounded independently of $L$.
Thus it suffices to show that
\be \one \log L \leq \int C(x)^2 dx \leq \one \log L \ee
This expression is written as
\[ \int C^2(x) dx = \int ^{L^2}_1 \ d\alpha\ \int ^{L^2}_1\ d\beta\ (\alpha\beta)^{\ep/2-2}
\int \ e^{-x^2/4\alpha}e^{-x^2/4\beta} dx
\]
To obtain an upper bound, we write this as twice the integral subject to $\beta\le\alpha$,
and bound $e^{-x^2/4\alpha}$ by $\one$. Then provided $\ep\log L\le 1$
\beaa \int C^2(x) dx &\le& 2\int ^{L^2}_1 \ d\alpha\ \int ^{\alpha}_1\ d\beta\
(\alpha\beta)^{\ep/2-2}
\int \ e^{-x^2/4\beta}d^4x\\
&\le& \one \int ^{L^2}_1 \ d\alpha\ \int ^{\alpha}_1\ d\beta\
(\alpha\beta)^{\ep/2-2} \beta^2
\\
&\le& \one \int ^{L^2}_1 \ d\alpha\
\alpha^{\ep-1}
\\
&\le&\one \log L
\eeaa
The integral is also bounded from below by the integral over the subregion
$\alpha\in (2,L),
\beta\in(\alpha/2,\alpha)$. On this region we can use the lower bound
$e^{-x^2/4\al}\ge e^{-x^2/4\beta}$
Then provided $\ep\log L\le 1$ we find
\beaa
\int C^2(x) dx &\ge& 2\int ^{L^2}_2 \ d\alpha\ \int ^{\alpha}_{\alpha/2}\ d\beta\
(\alpha\beta)^{\ep/2-2}
\int \ \left(e^{-x^2/4\beta}\right)^2 dx\\
&\ge& \one \int ^{L^2}_2 \ d\alpha\ \int ^{\alpha}_{\alpha/2}\ d\beta\
(\alpha\beta)^{\ep/2-2}\beta^2 \\
&\ge& \one \int ^{L^2}_2 \ d\alpha\
\alpha^{\ep-1} \\
&\ge&\one\log L
\eeaa
\QED
\bigskip
\subsection{The domain $\cD$} \label{sec-domain}
In the rest of this section we give estimates on the various quantities which
enter our flow equations.
The control of this flow is relegated to the next section.
To get useful estimates we must carefully specify
a domain $ \cD$ for the coupling constants
$\vec \la = (\la , \z , \mu )$, the remainder $R$, and the kernel $w$.
For the coupling constant we consider a region defined by the inequalities
\be |\la - \bar \la | < \ep^{\al} \ \;\ \ |\z_1| < \ep^{1-\de}
\ \;\ \ |\z_2| < \ep^{2-\De -\de}
\ \;\ \ |\mu| < \ep^{2-2\de}
\label{domainla} \ee
Here $\de , \De$ are small positive numbers with $\De > 4 \de$.
This domain is supposed to contain the fixed point we are searching for.
We will
choose $\bar \la > 0 $ to be a low order approximation to the fixed point
satisying $ \bar \la = \cO(\ep)$ and we take $1 < \al < 2$ so that $\la = \cO(\ep)$.
We also allow $\al = 1$ but then impose the condition
\be |\la - \bar \la | < d\ep \ee
with $d$ sufficiently small so that $\real(\la) >0$.
The choice of the integral powers of $\ep$ for $\z_1, \mu$ is also motivated
by a low order calculation of the fixed point.
It turns out to be convenient to reduce these powers slightly, which
gives a slightly larger domain.
For the remainder $R$ we consider the norms
\bea \| \cdot \| &=& \| \cdot \|_{G, h, \G} \nn \\
|\cdot | &=& | \cdot |_{1,\G} \eea
as defined in section 2.
We take $G=G(\k,X,\f)$ defined in (\ref{Gchoice}) with $\k =
\ep^{1/2}$.
We also choose $h = \ep^{-1/4}$.
Now define
\be
\trip\cdot\trip=\max\{\ep^2\| \cdot \| ,|\cdot |\}
\ee
We consider remainders $R$ which satisfy the bound
\be \label{domainR} \trip R \trip \leq \ep ^{3-\De} \ee
Finally $w$ should satisfy
\be \|w\| \leq k
\label{domainw} \ee
The region of complex valued $(\vec \la , R,w)$ satisfying (\ref{domainla}),
(\ref{domainR}), (\ref{domainw}) will be called
$\cD$.
Then we can prove the following result:
\bthm \label{mainestimate} Let $L$ be sufficiently large and $\ep$
sufficiently small (depending on $L$). Then $r,s,t,U$ are analytic
functions of $(\vec \la, R,w)$ on $ \cD $ and are
bounded there by
\bea
|r|,|s|,|t| &\leq & \one \ep^{3-\De} \nn \\
\trip U \trip & \leq & \cO(L^{-1+2\ep}) \ep^{3-\De}
\eea
\ethm
\subsection{Estimates on $Q$}
Recall our convention on $\one $ and $C$. We always assume that the
hypotheses of theorem \ref{mainestimate} are satisfied.
\blem \label{lem-Qreg} $\cQ_{reg}( \vec \la , v, w)$ satisfies
\bea \label{qreg} \|\cQ_{reg}e^{-V} \| &\leq & C\ep^{1/2 - 2 \de} \\
|\cQ_{reg}e^{-V}| & \leq & C\ep^{2 - 2\de} \eea \elem
\pr By lemma~\ref{evthmP} we have
\bea \|\cQ_{reg}e^{-V} \| &\leq & \one |\cQ_{reg}|_{h,\G_1} \nn \\
|\cQ_{reg}e^{-V} | &\leq & \one |\cQ_{reg}|_{1,\G_1} \eea
Now we bound the right side of these inequalities.
The main point is that the kernel of $\cQ_{reg}$
has integrable short distance singularities.
We examine $\cQ_{reg}$ term by term.
We first claim that
\be \label{33 bound}
\lambda ^{2}|\cQ^{(3,3)}|_{h,\G_1} \le C \ep^2 h^6\|w\|
\ee
Then the two choices $h=\ep^{-1/4}$ and $h=1$ lead to the bound we want.
To see this suppose we compute the sixth functional derivative of
$:\f(x)^3::\f(y)^3:$ at $\f =0$ by substituting
$\phi = \sum _{j} \alpha _{j}f_{j}$,
performing derivatives with respect to all $\alpha $, and setting
them all to zero. The result is
\[ 6! \ {\rm Sym} \left( f_1(x)f_2(x)f_3(x)f_4(y)f_5(y)f_6(y) \right) \]
This is bounded by a constant if $\|f_{\al}\|_{\cC^3} \leq 1 $.
This leads to the estimate
\bea
\|(\cQ^{(3,3)})_6(\De \cup \De',0)\|
& \leq &
\one \int_{\De\times\De'}| w(x-y)|dxdy \nn \\
&\le&
\one \|w \| \int_{\De\times\De'}|x-y|^{-9/4}e^{-|x-y|}dxdy\nn \\
&\le&
\one \|w\|e^{- d(\De,\De')}
\eea
The norm restricted to $\tilde{\Delta }^{\times 6}$ has the same bound.
The norm $|(\cQ^{(3,3)})_6(\De \cup \De',0)|$ defined in (\ref{kernorm})
also has the same bound since the sum over $\tilde{\Delta }^{\times 6}$
has $\one $ non-vanishing terms. (At most $\one$ block
neighborhoods $\tilde{\Delta }$ intersect $\De \cup \De'$.)
The same bound holds for lower derivatives and higher derivatives
vanish. Taking the $h$-norm gives a factor $h^6$ which combined with
the factor $\ep^2$ from the $\la^2$ gives the factor $\ep^2h^6$.
Finally the $\G$-norm is estimated by
\be
\sum_{\De'}\G(\De \cup \De')e^{- d(\De,\De')}\le C
\ee
and (\ref{33 bound}) is established.
Next we have
\be
\lambda ^{2}|\cQ^{(2,2)}_{reg}|_{h,\G_1}
\le
C \ep^2 h^4 \|w \|^2
\ee
For this we
evaluate the fourth functional derivative at $\phi=0$ of
\[
\left(
\colon \phi^{2}(x)\colon -
\colon \phi^{2}(y)\colon
\right) ^{2}
=
\left(
\phi^{2}(x) -
\phi^{2}(y)
\right) ^{2}
\]
with the result.
\[
4! \ {\rm Sym}
\left(
f_{1}(x)f_{2}(x) - f_{1}(y)f_{2}(y)
\right)
\left(
f_{3}(x)f_{4}(x) - f_{3}(y)f_{4}(y)
\right)
\]
This is bounded in absolute value by
$\one |x - y|^{2}$ for test functions in $\cC^3$.
Therefore
$|\lambda ^{2}\cQ^{(2,2)}_{reg}|_{h,\g\G}$ is bounded by $\one
\epsilon^{2}h^{4}$ times the $\G$ norm of
\bea
\int_{\De\times\De'} |x-y|^{2} | w(x-y)|^2 dxdy
&\le&
\|w\|^2 \int_{\De\times\De'}
|x-y|^{-5/2}e^{-2|x-y|}dxdy \nn \\
&\le&
\one \|w\|^2 e^{- d(\De,\De')}
\eea
which gives the bound we claimed.
To show
\be
\lambda ^{2}|\cQ^{(1,1)}_{reg}|_{h,\G_1}
\le
C \ep^2 h^2 \|w \|^3
\ee
we refer to (\ref{q11}) and use
\begin{eqnarray}
\left[
\f (x) - \f (y)
\right]^2 -
\left[
(x - y) \cdot (\pa \f)(x)
\right]^2 &=& \nn \\
&& \hspace{-2.7in}
\left[
\f (x) - \f (y) -
(x - y) \cdot (\pa \f)(x)
\right]
\left[
\f (x) - \f (y) +
(x - y) \cdot (\pa \f)(x)
\right]
\end{eqnarray}
to show that the second functional derivative is
\[
2 \ {\rm Sym } \left[
f_{1} (x) - f_{1} (y) -
(x - y) \cdot (\pa f_{1})(x)
\right]
\left[
f_{2} (x) - f_{2} (y) +
(x - y) \cdot(\pa f_{2})(x)
\right]
\]
This leads to a factor $\one (x-y)^{3}$ so that $\lambda
^{2}|\cQ^{(1,1)}_{reg}|_{h,\g\G} $ is bounded by $\one \epsilon^{2}
h^{2}$ times the $\G$ norm of
\bea
\int_{\De\times\De'}|x-y|^3| w(x-y)|^3 dxdy
&\le&
\|w\|^3 \int_{\De\times\De'}|x-y|^{-15/4}
e^{-3|x-y|}dxdy \nn \\
&\le&
\one \|w\|^3 e^{- d(\De,\De')}
\eea
To show
\be
\zeta_1 ^{2}|\cQ^{(1',1')}_{reg}|_{h,\G_1}
\le
C \ep^{2-2\de} h^2 \|w \|^2
\ee
we refer to (\ref{qs1'1'}). Here the derivative contains
\be
{\rm Sym} \ \left[
\pa_a f_{1} (x)-\pa_a f_{1} (y)
\right]
\left[
\pa_b f_{2} (x)-\pa_b f_{2} (y)
\right]
\ee
which leads to a factor $|x-y|^2$. Then we use
\bea
\int_{\De\times\De'}|x-y|^2|(\pa \pa w)(x-y)| dxdy
&\le&
\|w\| \int_{\De\times\De'}|x-y|^{-9/4}e^{-|x-y|}dxdy \nn \\
&\le&
\one \|w\| e^{- d(\De,\De')}
\eea
The other terms are bounded similarly.
\QED
\blem \label{qbound}
\bea \|F_Q e^{-V^\#} \| &\leq & C\ep^{1/2 - 2\de} \nn \\
|F_Q e^{-V^\#}| & \leq & C\ep^{2-2\de } \label{qsing} \eea \elem
\bigskip
\pr We have
\bea F_Q
&=&\cQ_{sing}\left(\vec \la,v^\#,w^\#,v^\#(0)\right) -\cQ_{sing}\left(\vec
\la,v^\#,w,v(0)\right) \nn \\
&=& \cQ^{(4,0)}\left (v^\#, 36\lambda ^{2}(2wC
+C^{2}) \right )\nn\\ &&+ \cQ^{(2',0)}\left (v^\#, 48 \lambda ^{2}(3w^2C + 3wC^2
+C^3)T +2 \zeta_1 ^{2} \pa\pa C \right )
\nonumber \\
&&+ \cQ^{(2,0)} \Bigl(v^\#,
48\lambda ^{2} (3v^\#(0)(2wC+C^2) + 3\left(C-C(0)\right)w^2 +3wC^2+C^3) \nn \\
&& +12\zeta_1 \lambda (2\pa C \pa w + 2 (\pa C)^2 ) \Bigr)
\nonumber \\
&&+ \cQ^{(0,0)} (\dots ) \nn \\
&&+ \cO(\la\z_2) + \cO(\z_1 \z_2)
\eea
where the argument of $\cQ^{(0,0)}$ is
\beaa \dots&=&\la^2\Bigl[
48w^3(C-C_2) + 72w^2\left(2v_2^\#C - 2v(0)^\#C(0) +C^2 - C(0)^2\right) \nn \\
&&+48w\left(C^3 +3v_2^\#C^2 +3 v^\#(0)^2C\right) +12 \left(C^4 +4v_2^\# C^3 +
6v^\#(0)^2C^2\right) \Bigr]
\nonumber\\
&&+ \zeta_1 ^{2}\Bigl[ 2 (\pa\pa v^\#)(0) (\pa\pa C) + 2 \left( (\pa\pa C)
-(\pa\pa C)(0)\right) \pa \pa w + (\pa\pa C)^{2} \Bigr]
\eeaa
Here we have introduced the notation
\be v_2(x) = v(0) - (\pa \pa v)(0)T(x) = v(0) + {1 \over 2}(\pa_a \pa_b v)(0)x_ax_b
\ee
Since reflection symmetry implies $ (\pa v) (0) =0 $ this is the expansion of
$v$ around $x=0$ to second order.
Now every term above has a kernel which is integrable. Note particularly
that $w^3$ which has a singularity $\cO(|x|^{-27/4})$ is multiplied by
$C-C_2$ which is $\cO(|x|^3)$ for an overall bound $\cO(|x|^{-15/4})$.
Also $ |\pa \pa w| = \cO(|x|^{-17/4}) $ is multiplied by $(\pa\pa C) -(\pa\pa C)(0)$
which is $\cO(|x|)$ for an overall bound $\cO(|x|^{-13/4})$.
Finally all the $w^2 =\cO(|x|^{-9/2})$ are multiplied by functions which are $\cO(|x|)$
for a bound $\cO(|x|^{-7/2})$. The terms $\cO(\la\z_2)$ and $\cO(\z_1 \z_2)$,
which we have not written out, are treated similarly.
Because the kernels are locally integrable and exponentially decreasing, bounds
of the type we have been discussing lead to the claimed estimate.
We omit the details.
Finally, the bounds on $\alpha_Q$ are straightforward.
\QED
\subsection{Estimates on $r,s,t,U$}
We introduce the norms
\bea \| \cdot \|_{\#} &=& \| \cdot \|_{G^\#, h^\#, \G^\#} \nn \\
|\cdot |_{\#} &=& | \cdot |_{1/2,\G^\#} \eea
where $G^{\#} = G^\#(\k, X,\f)$ equals $G(2\k, X,\f)2^{-|X|}$
where $G_{t}$ was defined in (\ref{int}), $\G^\# = \G_{-1}$,
and $h^\# = h/2$.
Note that our bound $\trip R \trip < \ep^{3-\De}$ implies that
\bea \label{Rbound}
\| R \| &\leq & \ep^{1-\De} \nn \\
|R| & \leq & \ep^{3-\De} \eea
\blem $R^\#=R^\#(\vec{\la},R,w)$ is analytic on $\cD$ and satisfies
there
\bea
\| R^\# \|_{\#} &\leq & \one \ep^{1-\De} \nn \\
|R^\#|_{\#} & \leq & \one \ep^{3-\De}
\eea
\elem
\re
Together with a weaker bound on $Q^\# e^{-V^\#}$ from
lemma~\ref{lem-Qreg} we obtain
\bea \| K^\# \| _ {\#} &\leq & C \ep^{1/2- 2 \de} \nn \\
| K^\# | _ {\#} &\leq & C \ep^{2- 2 \de} \label{Ksharp-bound}\eea
\pr Refer to (\ref{Aoft},\ref{Rsharp}) and theorem~\ref{thm-flow2}.
With $B(t) = \left(\Box + Q(t)\right)e^{-V(t)} $ and $E(t) = E\left(B(t)\right)$
we obtain
\bea \|R^\# \|_{\#} &\leq& 2 (\|R\| + \sup_t \|E(t) \|_{G_t,h,\G}) \nn \\
|R^\# |_{\#} &\leq& \one \left(|R| + \sup_t |E(t) | + \cO(\ep^{3})\right) \eea
taking into account
that $G(1)(X)=2^{|X|}G^\#(X)$, transfering the factor $2^{|X|}$ to
the $\G$ to get $\G_{-1}= \G^\#$ and choosing $M=12$.
Taking into account (\ref{Rbound})
we see that it suffices to show that
\bea \| E(t) \|_{G_t,h,\G} &\leq & C\ep^{1-3\de} \nn \\
|E(t)|_{1/2,\G} & \leq & C\ep^{3-3\de}
\eea
Working from the definitions (\ref{fl1}) for $E(t)$ and (\ref{Jdef}) for $J$,
one finds
\bea
E(t) &=& \left ({\pa \over
\pa t} - \Delta_C +J \right)\Box\ e^{-V(t)}\label{E1} \\
&+&Q(t) \cdot\left ({\pa \over
\pa t} - \Delta_C \right)e^{-V(t)} \label{E2} \\
&+&
C\left({\pa Q(t) \over \pa \phi}, {\pa V(t)\over \pa
\phi}\right) e^{-V(t)} \label{E3} \\
&+& \left [ \left({\pa \over
\pa t} - \Delta_C\right)Q(t) - J \right]e^{-V(t)} \label{E4} \\
&+&
\left[-{1 \over 2} \cB_C\left(
\Box e^{-V(t)}, \Box e^{-V(t)}\right)+J(1-\Box)e^{-V(t)}\right] \label{E5} \\
&-& \cB_C\left(\Box e^{-V(t)}, Q(t)e^{-V(t)} \right)
- {1 \over 2} \cB_C\left(
Q(t)e^{-V(t)}, Q(t)e^{-V(t)}\right) \label{E6}
\eea
We estimate each of these terms using lemma~\ref{evthmP} and
theorem~\ref{evthm}. Thus it suffices to bound the monomial parts
of these terms in the norms $| \cdot |_{h,\Gamma }$ with
either $h = \ep^{-1/4}$ or $h = 1/2$.
We use that for a monomial of degree $p$:
\begin{equation}\label{eps-count}
| \cdot |_{h,\Gamma }
\leq
C h^p \times |\mbox{coupling constants }|
\end{equation}
This is based on the fact that all kernels are integrable and contribute a
constant $C$ and that each field $\phi $ in the polynomial contributes a
factor $h $. The largest monomial in
$Q$ has degree $6$ and is $\cO (\lambda ^{2}) =
\cO(\epsilon ^{2})$. The largest monomial in $V$ has degree $4$ and is
$\cO(\lambda) = \cO(\epsilon )$. The largest coupling constant is $\zeta
_{1} = \cO(\epsilon ^{1-\delta })$.
Terms (\ref{E1}),(\ref{E5}) are both zero by the definition of
$J$. The definition of $Q$ is designed so that terms contained in
(\ref{E4}) have coefficients $\la\mu, \z_1 \mu, \z_2 \mu, \mu^2,
\z_2^2$ and so this term is bounded by $C \ep^{1-3\de}, C\ep^{3 -
3\de}$ for the choices $h = \ep^{-1/4}$ and $h = 1/2$ respectively.
(Terms $\la \z_2, \z_1\z_2$ would not quite have been small
enough).
Term (\ref{E2}) is equal to \be Q(t) C \left( {\pa V(t) \over \pa
\f},{\pa V(t) \over \pa \f} \right) e^{-V(t)} \ee We find that this
term is $C\ep , C\ep^{4 - 4\de}$. Term (\ref{E3}) has the estimate
$C\ep, C\ep^{3 - 3\de}$. Term (\ref{E6}) is $C\ep, C\ep^{3 -
3\de}$.
\QED \bigskip
Next we introduce norms
\bea \| \cdot \|_* &=& \| \cdot \|_{G^*, h^*, \G^*} \nn \\
|\cdot |_{*} &=& | \cdot |_{1/2,\G^*} \eea
where $G^* = G^\#, \G^* = (\G^\#)_{-4}=\G_{-5}, h^* = h^\#$.
Recall how dimension was defined in definition~\ref{def-dim}.
\blem $R^*$ is analytic on $\cD$ and can be written in the form
$R^* = R_1^* + R_2^*$ where $\dim \left(R_1^*(X)\right) >4$ for $X$ small and
\bea \| R_1^* \|_{*} &\leq & \cO(1)\ep^{1 - \De} \nn \\
|R_1^*|_{*} & \leq & \cO(1)\ep^{3- \De} \nn \\
\| R_2^* \|_{*} &\leq & C\ep^{1-4\de} \nn \\
|R_2^*|_{*} & \leq & C\ep^{3- 4\de} \eea
\elem
\re Combining this with a bound on $Q^* e^{-V^*}$
gives:
\bea \| K^* \| _ {*} &\leq & C \ep^{1/2-2\de} \nn \\
| K^* | _ {*} &\leq & C \ep^{2- 2\de} \eea
\pr From section~\ref{Esection}
\bea R^*_1 &=& R^\# - F_R e^{- V^*} \nn \\
R^*_2 &=& \cE _{\geq 2} (K^\#,F) \nn \\
&+& Q^\# (e^{-V^\#}- e^{-V^*})
+ (Q^{**} -Q^*)e^{-V^*} \eea
Then $R^*_1$ has dimension greater than 4 by construction.
Our bound $ |R^{\#}|_{\#} \leq \cO(1)\ep^{3-\De}$ implies
that $ | F_R|_{\#} \leq \cO(1)\ep^{3-\De}$, hence that
\be |(\al_R)_{n,m}(X)| \leq \cO(1)\ep^{3-\De} \G_{\#}(X)^{-1} \ee
for $(n,m) = (0,0), (2,0),...$ and hence that
\bea \| F_R e^{-V^*} \|_{*} &\leq & \cO(1)\ep^{2-\De} \nn \\
| F_R e^{-V^*}|_{*} &\leq & \cO(1)\ep^{3-\De} \eea
(Each field reduces the $\| \cdot \|_*$ norm by $\ep^{-1/4}$ and the maximum
number of fields in $F_R$ is 4.) These are sufficient to give the bounds
on $R_1^*$.
By theorem~\ref{exthm2}
\bea \| \cE_{\geq 2} (K^\#, F) \|_{*} &\leq &
\one \|f\|_{\G^\#} \|K^\#\|_\# \nn \\
| \cE_{\geq 2} (K^\#, F) |_{*} &\leq &
\one \|f\|_{\G^\#} |K^\#|_\#
\eea
where we take for some constant $C_f$
\be
f(X) = \left\{
\begin{array}{rcl} C_f\ep^{1-\de} & & X \ {\rm small} \\
\theta(X)^{-2} C_f \ep^{1-\de} & & X \ {\rm large}, |X|=2 \\
0 & & X \ {\rm large}, |X|\geq 3
\end{array}
\right.
\ee
To see that this $f$ satisfies the stability hypothesis (\ref{ex4}) of
the theorem:
The $\colon\phi^{4}\colon $ contribution to $F_Q$ has the form
$\int_{\De} \al (X,\De,x) \colon\phi^{4}(x) \colon dx$
where $|X| \leq 2$.
If $X = \De \cup \De'$ then from lemma \ref{qbound} $\sup_x | \al (X,\De,x)|
\leq C\ep^{2-\de} e^{-d(\De,\De')} $. Then we have that
\bea \sum_{|X| \leq 2, X \supset \De}\sup_x | \al (X,\De,x)| 2 / f(X)
& \leq& C\ep / C_f \sum_{\De'} \theta(\De,\De')^2 e^{-d(\De,\De')}
\nn \\ & \leq& C\ep / C_f \eea
By taking $C_f$ sufficiently large this is bounded by a
small multiple of $\ep$ so that stability holds by
theorem~\ref{evthm}: the $\lambda \colon\phi^{4}\colon$
part of $V$ remains dominant. The same arguments show that the other
terms in $F_{Q}$ are also compatible with stability.
The contributions to $F_R$ have the form $\al(X) \int_X p\left(\f(x)\right) dx$
where $\al$ is supported on small sets and $| \al(X)| \leq \cO(\ep^{3-\De}) $.
Then we have summing over $X$ small only
\be \sum_{ X \supset \De} | \al (X)| 2 / f(X) \leq C\ep^{2-\De+\de} / C_f \ee
Again the sum of these is less than a small multiple of
$\ep$.
With this choice of $f$ we have
\be \|f\|_{\G^\#} \leq C\ep^{1-\de} \ee
and combined with the bound (\ref{Ksharp-bound})
on $K^\#$ this gives
\bea \| \cE_{\geq 2} (K^\#, F) \|_{*} &\leq & C \ep ^{1 - 4 \de} \nn \\
| \cE_{\geq 2}
(K^\#, F) |_{*} &\leq & C \ep^{3- 4 \de} \eea
Using
\begin{equation}
e^{-V^\#}- e^{-V^*} = - \int _{0}^{1} e^{-tV^{\#} -
(1-t)V^{*}}(V^{\#}- V^{*}) dt
\end{equation}
one also shows
\bea \|Q^* (e^{-V^\#}- e^{-V^*})\|_*
+ \|(Q^{**} - Q^*)e^{-V^*}\|_*
&\leq &C\ep^{1-4\de} \nn \\
|Q^* (e^{-V^\#}- e^{-V^*})|_*
+ |(Q^{**} - Q^*)e^{-V^*}|_*
&\leq &C\ep^{3- 4\de} \eea
to complete the proof.
\QED
\blem $R'$ is analytic on $\cD$ and satisfies there
\bea
\| R' \| &\leq & \cO(L^{-1+2\ep})\ep^{1 - \De} \nn \\
|R'| & \leq & \cO(L^{-1+2\ep })\ep^{3- \De} \eea
\elem
\pr From section~\ref{sec-scaling2}:
\be
R' = \cS_1( R_1^*) +
\cS_1 ( R_2^*) + \cS_{\geq 2}(K^*) \ee
By theorem~\ref{sthm} a crude bound is
\bea
\|\cS(K^*) \|
& \leq &
\one L^4 \|K^*\|_{G_L, ah_L, \G_{-5} } \nn \\
& \leq &
\one L^4 \|K^*\|_* \nn \\
& \leq &
C\ep^{1/2 -2\de}
\nn \\
|\cS(K^*) |
&\leq &
C \ep^{2-2\de} \label{crude}
\eea
Here we have used $G_L ^{-1} \leq (G^*)^{-1}$ and
$ah_L \leq h^*$.
For $\cS_1( R_1^*)$ we use theorem~\ref{s1theorem}, break the estimate
into large and small sets and find:
\bea
\|\cS_1 ( R^*_1)\|
& \leq &
\cO( L^{-1+2\ep})\|R^*_1 \|_{*} \nn \\
& \leq &
\cO(L^{-1+2\ep}) \ep^{1- \De}
\nn \\
|\cS_1 ( R^*_1)|
& \leq &
\cO(L^{-1+2\ep}) \ep^{3- \De}
\eea
Here we have used $\dim(R^*) \geq 5-2\ep$.
Indeed from equation (\ref{dimR}) and the definition of dimension
we find that $\dim (R^{*}_{2}) = 2\dim (\phi)+3 = 5 - \epsilon$,
$\dim (R^{*}_{4}) = 4\dim (\phi)+1 = 5 - 2\epsilon$ and $\dim
(R^{*}_{6}) = 6\dim (\phi) = 6 - 3\epsilon$ .
We also have
\bea \|\cS_1 (R^*_2)\| &\leq & C\|R^*_2 \|_{*} \leq C\ep^{1-4\de} \nn \\
|\cS_1 (R^*_2)| &\leq & C|R^*_2 |_{*} \leq C\ep^{3- 4\de} \eea
which is sufficient because $\Delta > 4\delta $ and $\epsilon $ is
chosen after $L$.
Finally by Cauchy bounds and (\ref{crude}) we find
\bea \|\cS_{\geq2}(K^*)\| &\leq & C\ep^{1 - 4\de}
\nn \\
|\cS_{\geq2}(K^*)| &\leq & C \ep^{4-4\de}
\eea
to complete the proof.
\QED
\bigskip
\noindent{\bf Proof of Theorem \ref{mainestimate}}
That $r(\vec{\la},R,w),s(\vec{\la},R,w),t(\vec{\la},R,w)$ are analytic in
$\cD$ and satisfy there
\be |r|, |s|, |t| \leq \one \ep^{3- \De} \ee
follows from the same properties for the $\al_{n,m}(X)$ in
(\ref{fr}). See lemma~\ref{lem-F-R} and note that the bounds follow
from (\ref{domainR}).
The bound on $R' =U(\vec \la, R,w)$ is contained in the previous
lemma.
\QED
\section{Critical Models}
\subsection{The problem}
We now restrict to infinite volume so that the RG flow
is the iteration of a fixed mapping.
In this section we prove the existence of critical theories by
showing that one can pick $\mu_0 = \mu_0 (\la_0), \z_0 = \z_0(\la_0)$
so that the RG flow tends to a fixed point. This result is part of
the stable manifold theorem, but we have not been able to find a
version in the literature which is exactly applicable to our problem,
so we have to give an independent proof, at least in part.
So far we have proved that the renormalization group transformation
can be iterated as long as the parameters $ (\vec \la_j, R_j,w_j)$
that define the theory remain in the domain $\cD$ and the flow after
$j$ iterations is given by the equations
\bea \la_{j+1}&=&L^{2\ep}[\la_j - a(w_j)\la_j^2 +r(\vec \la_j, R_j,w_j)]\nn \\
\z_{j+1}&=&L^{\ep}[\z_j -b(w_j)\la_j^2 +s(\vec \la_j, R_j,w_j)]\nn \\
\mu_{j+1}&=&L^{2+\ep}[\mu_j -c(w_j)\la_j^2-d(w_j)\la_j(\z_{1,j} +\z_{2,j})
+t(\vec \la_j, R_j,w_j)]\nn \\
R_{j+1}&=&U(\vec \la_j, R_j,w_j) \nn \\
w_{j+1}&=& (w_j)_L + C_L \eea
The starting point is $(\vec \la_0, 0,0)$ with $\vec \la_0 = (\la_0
, \z_0, \mu_0)$, but we consider the more general case $ (\vec \la_0,
R_0, w_0)$.
If we ignore the higher order term $r$ and replace $w_j$ by its
limiting value $w_{*}$ we find the approximate fixed point for $\la$
\be \bar\la ={L^{2\ep}-1 \over L^{2\ep}a(w_{*})} \ee
In Section~\ref{sec-domain} we required that $\bar \la = \cO(\ep)$.
This is valid since for $\ep$ small we have
$L^{2\ep} \sim 1+ 2\ep \log L + \dots$ and we have seen that
$ a(w_*) \sim \cO(\log L)$.
(We continue to assume that $L$ is chosen large followed by $\ep$ small,
that $\one, \cO (\epsilon)$, etc. are $L$
independent constants, and that $C$ is a possibly $L$ dependent constant
which may change from line to line.)
We now rewrite the equations replacing $\la_j$ by the deviation
$\tilde \la_j =\la_j -\bar \la$.
With
\be \xi_j = (\tilde \la_j, \z_j, \mu_j, R_j, w_j) \ee
we have
\bea \label{newflow2} \tilde \la_{j+1}&=&(2-L^{2\ep})\tilde \la_j + \tilde r(\xi_j) \nn \\
\z_{j+1}&=&L^{\ep}\z_{j} + \tilde s (\xi_j) \nn \\
\mu_{j+1}&=&L^{2+\ep}\mu_j + \tilde t(\xi_j) \nn \\
R_{j+1}&=& U(\xi_j) \nn \\
w_{j+1}&=& (w_j)_L + C_L
\eea
where we have defined
\bea
\tilde r(\xi) &=& L^{2\ep}[-a(w_{*}) \tilde \la ^2
+ \left(a(w_{*}) - a(w)\right) \la^2 +r(\xi)] \nn \\
\tilde s_i(\xi) &=& L^{\ep}[ -b_i(w)\la^2 +s_i(\xi)] \nn \\
\tilde t(\xi) &=& L^{2+\ep}[ -c(w)\la^2-d(w)\la (\z_1 + \z_2) +t(\xi)]
\eea
In this equation $r(\xi) = r(\bar \la + \tilde \la , \z , \mu, R,w)$, etc.
Now $ \xi_0=(\tilde \la_0, \z_0, \mu_0, R_0, w_0)$ is the initial point. Note
that the shift has changed an expanding variable $\la$ into a contracting
variable $\tilde \la$. The equations
(\ref{newflow2}) are the iteration of a single mapping which we call $f$ so that
\be \xi_{j+1} = f(\xi_j) \ee
As a norm on $\xi$ we take
\be \label{norm} \|\xi\| =
\sup(\ep^{-\al}|\tilde \la|,\ep^{-1+\de}|\z_1|, \ep^{-2+\De +\de}|\z_2|,
\ep^{-2+2\de}|\mu|,\ep^{-3+\De}\trip R \trip, k^{-1} \|w\|)
\ee
where $\De > 4\de>0$ and $1 < \al$ and $k$ appeared in (\ref{kdef}).
This norm defines the Banach space $E$.
The unit ball $E(1)$ is the image of the domain $\cD$ under the shift
$ \tilde \la = \la - \bar \la$.
In section 5.3 to come, we will need to
replace the factor $\ep^{-\al}|\tilde \la|$ by $(d\ep)^{-1}|\tilde \la|$ with $d$
sufficiently small. The parameter $d$ will be chosen independently of
$L,\epsilon$. This will define a Banach space $F$.
We collect the bounds that we will need.
Assuming $\xi \in E(1)$ we have from the previous section:
\bea
|r(\xi)|,|s_i(\xi)| ,|t(\xi)| &\leq& \one \ep^{3-\De} \nn \\
\trip U(\xi) \trip & \leq & \cO(L^{-1+2\ep})\ep^{3-\De}
\eea
It follows that (recall $b_2 = 0$ )
\bea |\tilde r(\xi)| &\leq & \cO(\log L)\ep^{2\al} + \one \|w-w_*\| \ep^2
+\one \ep^{3-\De} \nn\\
|\tilde s_1(\xi)| &\leq & C \ep^{2}
\ \ ;\ \ |\tilde s_2(\xi)| \leq \one \ep^{3-\De} \nn \\
|\tilde t(\xi)| &\leq & C \ep^{2-\de}
\label{tildebds} \eea
Recall that $r$ is analytic in $E(1)$. Therefore we can use the
Cauchy representation for $r\left(\xi' + z(\xi - \xi ')\right)$ as a function of
the complex variable $z$ with $|z| < \cO (\|\xi - \xi'\|^{-1})$ to
obtain Lipschitz bounds: for $\xi,\xi' \in E(1/2)$:
\bea &&|r(\xi)- r(\xi')|, |s_i(\xi)- s_i(\xi')|, |t(\xi)- t(\xi')| \leq
\one \ep^{3-\De}\|\xi - \xi'\| \nn \\
&&\trip U(\xi) - U(\xi') \trip \leq \cO(L^{-1+2\ep})\ep^{3-\De}\|\xi
- \xi'\| \label{lip0}\eea
By Lemma~\ref{lem-abcd} this implies the bounds:
\[ | \tilde r(\xi)- \tilde r(\xi')| \leq
\left (\cO(\log L)\ep^{\al}+ \one \|w-w_*\|\ep \right) |\tilde \la - \tilde
\la'|\]
\be + \one \|w-w'\| \ep^2
+\one \ep^{3-\De}\|\xi - \xi' \| \label{lip1} \ee
and the same bound with $\ep^{\al}$ replaced by $d\ep$.
We also have
\bea
|\tilde s_1(\xi)-\tilde s_1(\xi')| &\leq &
C\left( | \tilde \la - \tilde \la'|\ep + \| w-w'\|\ep^2 + \ep^{3-\De} \| \xi -
\xi' \| \right)
\nn \\
|\tilde s_2(\xi)-\tilde s_2(\xi')| &\leq &
\one \ep^{3-\De} \| \xi - \xi' \|
\nn \\
|\tilde t(\xi)-\tilde t(\xi')| &\leq &
C\Biggl( | \tilde \la - \tilde \la'|\ep^{1-\delta } + |\zeta - \zeta'|\epsilon
\nn \\
&& \ \ \ + \| w-w'\|\ep^{2-\de} + \ep^{3-\De} \| \xi - \xi' \| \Biggr
)
\label{lipschitz} \label{lip2} \eea
\subsection{Existence of global solutions} \label{sec-existence}
We first want to prove the existence of bounded global solutions, that
is sequences $\xi_j \in E(1/2) $ defined for
all $j = 0, 1, \dots $
and satisfying $\xi _{j+1} = f(\xi _{j})$ for $j = 0,1,\dots $.
We start by looking for finite sequences $\xi_j$ with $0 \leq j \leq k$
with specified initial conditions $\la_0, R_0 $ and specified final conditions
$\z_k =\z_f, \mu_k =\mu_f$. It is straightforward to check that a
sequence $\xi_j$ is a solution if and only if it satisfies the
``integral'' equations:
\bea
\tilde \la_{j}
&=&(2-L^{2\ep})^j(\tilde \la_0 ) + \sum_{l=0}^{j-1}(2-L^{2\ep})^{j-l-1}\tilde
r(\xi_l); \ \ \ j = 1,\dots ,k \nn \nn\\
\z_{j}&=&L^{-\ep(k-j)}\z_f
-\sum^{k-1}_{l=j}L^{-\ep(l+1-j)}\tilde s(\xi_l); \ \ \ j = 0,\dots ,k-1\nn\\
\mu_{j}&=&L^{-(2 + \ep)(k-j)}\mu_f
-\sum^{k-1}_{l=j}L^{-(2+\ep)(l+1-j)}\tilde t(\xi_l); \ \ \ j = 0,\dots k-1\nn\\
R_{j} &=& U (\xi_{j-1}); \ \ \ j = 1, \dots , k
\eea
The imposition of a mixture of initial and final conditions has given
equivalent ``integral'' equations in which only exponentially decaying
factors appear. This is a standard device in the theory of hyperbolic
dynamical systems.
To obtain a global solution it is natural to consider the formal limit
as $k \to \infty$:
\bea
\tilde \la_{j}
&=&(2-L^{2\ep})^j(\tilde \la_0 ) + \sum_{l=0}^{j-1}(2-L^{2\ep})^{j-l-1}\tilde
r(\xi_l); \ \ \ j \geq 1 \nn\\
\z_{j}&=&
-\sum^{\infty}_{l=j}L^{-\ep(l+1-j)}\tilde s(\xi_l); \ \ \ j \geq 0 \nn\\
\mu_{j}&=&
-\sum^{\infty}_{l=j}L^{-(2+\ep)(l+1-j)}\tilde t(\xi_l); \ \ \ j \geq 0 \nn\\
R_{j} &=& U (\xi_{j-1}); \ \ \ j \geq 1
\label{fp} \eea
When $\xi_j \in E(1/2) $ these sums converge (see below) and one can
check that any solution of these equations is a bounded solution of
$\xi_{j+1} = f(\xi_j)$. Thus we focus on solving this equation.
Notice that the equations are now independent of $\z_f, \mu_f$.
Let
\[ \un \xi = (\xi_0, \xi _{1},\xi_2,... ) \]
Then the above equation can be written in the form
\[ \un \xi = F(\un \xi) \]
where
$F(\un \xi) = (F_0(\un \xi),F_1(\un \xi),\dots )$ and
\be F_j(\un \xi) =
\left(F_j^{\la }(\un \xi),F_j^{\z_1}(\un \xi), F_j^{\z_2}(\un \xi),
F_j^{\mu }(\un \xi),F_j^R(\un \xi)\right) \ee
is the right side of the flow equation augmented by the initial data
$F_0^{\lambda}(\un \xi) = \lambda _{0}, \ F_0^R(\un \xi) = R_{0}$.
We regard this as a fixed point equation
in the Banach space $\un E$ of all sequences $\un \xi$ with norm
$ \| \un \xi \| = \sup_j \|\xi_j \|$ . The ball $\un E(1)$ is all sequences
with entries in $ E(1)$. Since the $w_j$ component of $\xi _{j}$ must be
the solution of $w_{j+1} =(w_j)_L+ C_L$ starting with $w_0$, we
insert this solution $w_{j}$ into $\un \xi $ and only consider the first four
components $\xi_j = (\tilde \la_j, \z_j, \mu_j, R_j, w_j)$ to be
unknowns in the fixed point equation.
\bthm Let $ 1 < \al < 2-\De$ and let $\la_0, R_0,w_0$ be given such
that $(\tilde{\la}_0, 0, 0, R_0,w_0) \in E(1/4)$. Let the $w_{j}$
components of $\un \xi$ be fixed as described above. Then
\benum
\item If $\un \xi \in \un E (1/2)$ then
$F(\xi) \in \un E (1/2)$.
\item For $\un \xi,\un \xi'$ in $\un E (1/2)$ we have
$\| F (\un \xi)- F (\un \xi')\| \leq \| \un \xi - \un \xi' \|/2 $
\item For any $\un \xi^{(0)}$ in $\un E(1 /2)$ the iterates
$\un\xi^{(1)}=F(\un \xi^{(0)}),\un \xi^{(2)}=F(\un \xi^{(1)})$, etc. converge
to a limit $\un \xi = \lim_{n \to \infty}\un \xi^{(n)}$ in $\un E (1/2)$
which satisfies $\un \xi = F(\un \xi)$ and hence $\xi_{j+1}=f(\xi_j)$.
\eenum
\ethm
\pr
\noindent (1.) Note that because we have fixed $w_j$, by
(\ref{kdef}) the bound (\ref{tildebds}) can be written
\be |\tilde r(\xi_j)| \leq \cO(\log L)\ep^{2\al} + \cO(L^{-j/4}) \ep^2
+\one \ep^{3-\De} \ee
Then we have for $j \geq 1 $:
\bea
|F_j^{\la}|&\le& (2-L^{2\ep})^j|\tilde \la_0| +
\sum_{l=0}^{j-1}(2-L^{2\ep})^{j-l-1}|\tilde r(\xi_l) | \nn\\
&\le& (2-L^{2\ep})^j( \ep^{\al}/4) +
\sum_{l=0}^{j-1}(2-L^{2\ep})^{j-l-1} \one (\ep^{2\al} + L^{-l/4}\ep^2 + \ep^{3-\De})\nn\\
&\le & \ep^{\al}/4 +\one ( \ep^{2 \al -1} + \ep^2 + \ep^{2-\De}) \nn\\
&\le & \ep^{\al}/2. \eea
Next we have
\bea
|F_j^{\z_1}|&\le&
\sum_{l=j}^{\infty} L^{-\ep (l+1-j)}|\tilde s_1(\xi_l) | \nn\\
&\le&
\sum_{l=j}^{\infty} L^{-\ep (l+1-j)}C \ep^{2} \nn\\
&\le & C\ep\nn\\
&\le & \ep^{1-\de}/2 \eea
and similarly
\be
|F_j^{\z_2}| \leq \ep^{2-\De-\de}/2
\ee
Next we have
\bea
|F_j^{\mu}|&\le &
\sum^{\infty}_{l=j}L^{-(2+\ep)(l+1-j)}| \tilde t(\xi_l)|\nn\\
&\le &
\sum^{\infty}_{l=j}L^{-(2+\ep)(l+1-j)} C \ep^{2-\de}\nn\\
& \leq & C\ep^{2 - \de} \nn\\
& \leq & \ep^{2-2\de}/2 \eea
Finally we have
\be \trip F^R_j \trip = \trip U(\xi_{j-1}) \trip \leq \cO(L^{-1+2\ep})\ep^{3-\De}
\leq {1 \over 2}\ep^{3-\De} \ee
\noindent (2.) We denote $\de\un \xi =\un \xi - \un \xi'$ and
$\de F = F(\un\xi ) - F(\un\xi')$, etc.
Since $w_j$ is fixed there is no variation in $w$ and the estimates
(\ref{lip1}), (\ref{lip2}) become
\bea |\de \tilde r(\xi_j)| &\leq &
(\cO(\log L)\ep^{\al}+\cO(L^{-j/4})\ep ) |\de \tilde \la_j|
+\one \ep^{3-\De}\|\de \xi_j \| \nn \\
|\de \tilde s_1(\xi_j)| &\leq &
C(\ep |\de \tilde \la_j|+ \ep^{3-\De}\|\de \xi_j \|)\nn \\
& \leq & C \ep^{1+\al}\|\de \xi_j \| \nn \\
|\de \tilde t(\xi_j)| &\leq &
C( | \de \tilde \la |\ep^{1-\delta } + |\de \zeta |\epsilon +
\ep^{3-\De} \| \de \xi_j \| ) \nn \\
& \leq &
C \ep^{2-\delta } \| \de \xi_j \|
\eea
Then we have
\bea
\ep^{-\al}|\de F_j^{\la}| &\le &
\sum_{l=0}^{j-1}(2-L^{2\ep})^{j-l-1} \ep^{-\al}|\de \tilde r(\xi_l)| \nn\\
&\le &
\sum_{l=0}^{j-1}(2-L^{2\ep})^{j-l-1}
(\cO(\log L)\ep^{\al}+\cO(L^{-j/4})\ep +\one \ep^{3-\De - \al}) \|\de \xi_l\| \nn\\
& \leq & (\cO(\log L)\ep^{\al-1}+ \one \ep
+ \one \ep^{2-\De - \al}) \|\de \un \xi \|\nn\\
& \leq & \|\de \un \xi \|/2
\eea
Next we have
\bea
\ep^{-1+\de}|\de F_j^{\z_1}|&\le&
\sum_{l=j}^{\infty} L^{-\ep (l+1-j)}\ep^{-1+\de}|\de \tilde s_1(\xi_l) | \nn\\
&\le& \sum_{l=j}^{\infty} L^{-\ep (l+1-j)}C \ep^{\al+\de}\|\de \xi_l \| \nn\\
&\le & C\ep^{\al - 1 +\de}\|\de \un \xi \| \nn\\
&\le & \|\de \un \xi \|/2.
\eea
and similarly
\be
\ep^{-2+\De +\de}|\de F_j^{\z_2}| \leq \|\de \un \xi \|/2 \ee
Next we have
\bea \ep^{-2+2\de}|\de F_j^{\mu}|
&\le & \sum^{\infty}_{l=j}L^{-(2+\ep)(l+1-j)}\ep^{-2+2\de}|\de \tilde t(\xi_l)|\nn\\
&\le & \sum^{\infty}_{l=j}L^{-(2+\ep)(l+1-j)} C\ep^{\de}\| \de \un \xi \| \nn\\
&\leq &\|\de \un \xi \|/2 \eea
Finally by (\ref{lip0}) we have
\be \ep^{-3+\De} \trip \de F_j^R \trip = \ep^{-3+\De} \trip \de U(\xi_{j-1}) \trip
\leq \cO(L^{-1+2\ep}) \| \de \un \xi \| \leq {1 \over 2}\|\de \un \xi \| \ee
Combining all these gives
\[ \| \de F \| \leq 1/2 \|\de \xi \| \]
\noindent (3.) This is standard.
\QED
\bigskip
\re Thus we have shown that for
each $(\la_0, R_0, w_0) $ in $E(1/4)$ there exists $(\z_0, \mu_0)$ (given by
(\ref{fp}))
such that the trajectory $\xi_j$ starting at
that $\xi_0=(\la_0, \z_0, \mu_0, R_0,w_0)$ stays in $E(1/2)$ for all $j$.
In the next section we will see that $(\mu_0,\z_0)$ is unique.
\subsection{The stable manifold and the fixed point}
The existence results of the previous section refer to the Banach space $E$ for
which the norm $\| \xi \|$ is given by (\ref{norm}) with weight factor
$\ep^{-\al} | \tilde \la |$ for fixed $1< \al < 2-\De $. For uniqueness we
consider the Banach space $F$ with norm $\| \xi \|$ still given by (\ref{norm})
but now with weight factor $ (d\ep)^{-1} | \tilde \la |$. Since $\ep^{\al} < d\ep$
the norm in $F$ is dominated by the norm in $E$ and so $E( {1 \over 2}) \subset
F( {1 \over 2})$, etc.
We now return to regarding $w$ as a dynamical variable so that the problem is
again to study the iterations of the mapping $f$.
The Banach space $F$ can be regarded as a product
$F_1 \times F_2 $
where $F_1$ is all triples $(\tilde \la, R, w )$ and $F_2$ is all pairs
$(\z, \mu)$. Then $F_1$ corresponds to contracting data and $F_2$ corresponds to
expanding data. Let $p_i$ be the projection
onto $F_i$, let $\xi_i = p_i \xi$, and let $f_i = p_i \circ f$.
We follow the analysis of Shub, Irwin (\cite{Shu87}, p51).
\blem Suppose $\|\xi_2 -\xi_2' \| > \|\xi_1 -\xi_1'\|$ so that
$\|\xi -\xi'\|=\|\xi_2 -\xi_2'\|$. Then
\bea
\|f_1(\xi)- f_1(\xi')\| &\leq &(1-\ep)\| \xi - \xi' \| \nn \\
\|f_2(\xi)- f_2(\xi')\| &\geq &(1+\ep)\| \xi - \xi' \|
\eea \elem
\re Note that with $\ep ^\al$ replaced by $d\ep$
the bounds (\ref{lip1}), (\ref{lip2}) can be written
\bea (d\ep)^{-1} | \tilde r(\xi)- \tilde r(\xi')| &\leq &
\left(\cO(\log L)d+ \one d^{-1}\right)\ep \|\xi - \xi' \| \nn \\
\ep^{-1+ \de}|\tilde s_1(\xi)-\tilde s_1(\xi')| &\leq &
C \ep^{1+ \de} \| \xi - \xi' \| \nn \\
\ep^{-2+\De + \de}|\tilde s_2(\xi)-\tilde s_2(\xi')| &\leq &
C \ep^{1+ \de} \| \xi - \xi' \| \nn \\
\ep^{-2+ 2\de}|\tilde t(\xi)-\tilde t(\xi')| &\leq &
C \ep^{\de} \| \xi - \xi' \|
\eea
\pr
We estimate
\be f_1 = (f_{\la}, f_{R}, f_{w})=
\left( (2-L^{\ep}) \tilde \la + \tilde r,\ U,\ w_L + C_L\right ) \ee
We have
\bea
(d\ep )^{-1}|f_{\la}(\xi) -f_{\la}(\xi')|&\leq & \left( (2-L^{2\ep}) +(\cO(\log
L)d+ \one d^{-1} )\ep )\right) \|\xi -
\xi'
\|
\nn \\ &\leq&
\left(1 + \ep (-2\log L +\cO(1)d \log L + \one d^{-1} )\right) \|\xi -
\xi'\| \nn
\\ &\leq &( 1 - \ep) \|\xi - \xi'\| \eea
where we first choose $d$ small enough so
that $-2 + \one d < 0$ and then
choose $L$ large enough for that $(-2 + \one d^{-1} )\log L + \one < -1$.
We also have
\be
\ep^{-3+\De}\trip f_{R}(\xi) -f_{R}(\xi') \trip
\leq \cO(L^{-1+2\ep})\|\xi - \xi'\|
\leq (1- \ep)\|\xi - \xi'\| \ee
and
\bea k^{-1}\|f_{w}(\xi) -f_{w}(\xi')\|
& = & k^{-1}\|w_L -w'_L \| \nn \\
& \leq & L^{-1/4} \|\xi - \xi'\| \nn \\
& \leq & (1- \ep )\|\xi - \xi'\| \eea
This completes the proof of the first bound.
For the second bound we estimate
\be f_2 = (f_{\z},f_{\mu}) = (L^\ep \z + \tilde s, \ L^{2+ \ep }\mu + \tilde t )
\ee
Note that one of $ \ep^{-1+\de}|\z_1 - \z_1'|$ and $ \ep^{-2+\De +\de}|\z_2 - \z_2'|$
and
$\ep^{-2+2\de}|\mu - \mu'|$ must be equal to $\|\xi_2-\xi_2' \|= \|\xi-\xi' \|$.
In the first case we have
\bea
\|f_2(\xi) -f_2(\xi') \| & \geq &
\ep^{-1+\de}|f_{\z_1}(\xi) -f_{\z_1}(\xi')| \nn \\
&\geq & L^{\ep} \ep^{-1+\de}|\z_1 - \z_1'|
- \ep^{-1+\de}\|\tilde s_1 (\xi)-\tilde s_1 (\xi')\| \nn \\
& \geq & (L^{\ep}- C \ep^{1+\de})\|\xi-\xi' \| \nn \\
&\geq & (1 + \ep)\|\xi-\xi' \|
\eea
The second case is similar, and in the last case we have
\bea
\|f_2(\xi) -f_2(\xi') \| & \geq &
\ep^{-2+2\de}|f_{\mu}(\xi) -f_{\mu}(\xi')| \nn \\
&\geq & L^{2+\ep} \ep^{-2+2\de}|\mu - \mu'|
- \ep^{-2+2\de}\|\tilde t (\xi)-\tilde t (\xi')\| \nn \\
& \geq & (L^{2+\ep}- C \ep^{\de})\|\xi-\xi' \| \nn \\
&\geq & (1 + \ep)\|\xi-\xi' \|
\eea
\QED
\bigskip
Now we can state our main result.
Consider the RG transformation $f$ on $F$ and
we define the stable manifold to be
\be M = \{ \xi \in E(1/2) : \xi_j= f^j(\xi) \in F(1/2) \ \mbox{for all\ }
j=1,2,...\} \ee where $f^{j}$ is the $j$-fold composition of $f$ with itself.
\bthm The stable manifold $M$ is the graph of a function $\xi_2 = h (\xi_1)$
which is Lipschitz continuous with Lipschitz
constant 1. The RG mapping $f$ is a contraction on $M$ and thus has a unique
fixed point $ \xi^*=(\xi_1^*,\xi_2^*)$ where $\xi_2^*=h(\xi_1^*)$. \ethm
\bigskip
\pr
Suppose $\xi,\xi'$ are two points on M. We claim that
\be \| \xi_2 - \xi_2'\| \leq \| \xi_1 - \xi_1'\| \ee
If not then $ \| \xi_2 - \xi_2'\| > \| \xi_1 - \xi_1'\|$
and the previous lemma is applicable.
It follows that
\bea
&&\|f_2(\xi) - f_2(\xi') \| \geq (1+\ep) \|\xi - \xi'\| \nn \\
&&> (1-\ep) \|\xi - \xi'\| \geq \|f_1(\xi) - f_1(\xi') \|
\eea
whence $\|f(\xi) - f(\xi') \| \geq (1+\ep) \|\xi - \xi'\|$.
Now apply the lemma again to $f(\xi), f(\xi')$ and get
\bea
&&\|f_2\left(f(\xi)\right) - f_2\left(f(\xi')\right) \| \geq (1+\ep) \|f(\xi) - f(\xi')\| \nn \\
&&> (1-\ep) \|f(\xi) - f(\xi')\| \geq \|f_1\left(f(\xi)\right) -
f_1\left(f(\xi')\right) \|
\eea
whence $\|f^2(\xi) - f^2(\xi') \| \geq (1+\ep)^2 \|\xi - \xi'\|$.
Continuing in this fashion we get
\[\|f^n(\xi) - f^n(\xi') \| \geq (1+\ep)^n \|\xi - \xi'\| \]
which gives a contradiction as $n \to \infty$.
Because of the inequality we have that $\xi_1 = \xi_1'$ implies that $\xi_2 = \xi_2'$.
Thus there is at most one $\xi_2$ for each $\xi_1$ on $M$. Recall from
the remark at the end of Section~\ref{sec-existence} that we already
know that
there is at least one $\xi_2$ and hence there is exactly one $\xi_2$. Thus $M$
is the graph of a function $\xi_2=h(\xi_1)$ and the inequality says that
$\|(h(\xi_1) -h(\xi_1')\| \leq \|\xi_1 -\xi_1'\|$.
On $M$ we have $\|\xi -\xi'\|= \|\xi_1 -\xi_1'\|$ and hence
$\|f(\xi) - f(\xi')\| = \|f_1(\xi)- f_1(\xi')\|$.
We have already seen that $\|f_1(\xi) - f_1(\xi')\| \leq (1-\ep)
\|\xi -\xi'\|$ and so $f$ is a contraction on $M$.
\QED
\bigskip
\res To express $M$ in terms of the original variables, write $h$ as a pair of
functions \bea
\z_0 &=& h_{\z}(\tilde \la_0,R_0,w_0) \nn \\
\mu_0 &=& h_{\mu}(\tilde \la_0,R_0,w_0) \eea
The case of interest is $ R_0=0, w_0=0$ which gives our functions
$\mu_0 = \mu_0 (\la_0), \z_0 = \z_0(\la_0)$.
When $ R_0=0, w_0=0$ the point $\left(\z_{0,1}(\la_0),\z_{0,2}(\la_0)\right)$
describes the same density as the point
$(\z_{0,1}(\la_0) + \z_{0,2}(\la_0), 0)$ so we can get on the stable
manifold with a potential of the form (\ref{originalcc}).
Note that the fixed
point has the form
$\xi^*=(\tilde \la ^*,\z^*, \mu^*, R^*,w^*)$
where $\z^* = h_{\z}(\la^*,R^*,w^*),
\mu^* = h_{\mu}(\la^*,R^*, w^*)$.
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