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Gross-Neveu model, Renormalization Group
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%   Construction of the renormalized GN$_{2-\epsilon}$ trajectory   %
%               M. Salmhofer and C. Wieczerkowski                   %
%                        June, 1999                                 %
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\begin{document}

\begin{titlepage}

\hsize=16.0truecm\vsize=22.5truecm\vglue6.3truecm
%$\phantom{\text{Preprint number}}$\hfill MS-TP1-99-03

\begin{center}
{\Large\bf 
   Construction of the renormalized GN$_{2-\epsilon}$ trajectory 
}\\[10mm]
{\Large 
   M.~Salmhofer$^{1}$ and Chr.~Wieczerkowski$^{2}$
}\\[10mm]
${}^{1}$ Mathematik, ETH Z\"urich\\
CH-8092 Z\"urich, Switzerland\\
manfred@math.ethz.ch\\[2mm]
${}^{2}$ Institut f\"ur Theoretische Physik I,
Universit\"at M\"unster\\
Wilhelm-Klemm-Stra\ss e 9, D-48149 M\"unster\\
wieczer@uni-muenster.de\\[2mm]
\end{center}

\begin{abstract}
\noindent
We construct the renormalized Gross-Neveu trajectory in 
$2-\epsilon$ dimensions. 
Our construction uses a contraction mapping for an extended 
renormalization group. 
The extension is a running coupling with linear 
step $\beta$-function.
The contraction mapping relies on norm estimates for a 
fermionic momentum space renormalization group. 
\end{abstract}

\end{titlepage}

\section{Introduction}

In this paper, we construct the renormalized trajectory 
of the (chiral) Gross--Neveu model in $2-\epsilon$ dimensions.
The two dimensional model was introduced by Gross and Neveu
\cite{Gross/Neveu:1974} and by Mitter and Weisz 
\cite{Mitter/Weisz:1973}, both as a model for asymptotic 
freedom and for dynamical mass generation.
In this paper, we consider a super-renormalizable
deformation of its renormalization flow. 
The deformation mimicks a dimensional continuation, without 
being a regularization. 
We use it to illustrate how rigorous control of a fermionic 
ultraviolet limit can be gained by general norm bounds 
on fermionic renormalization groups. 
Our construction relies on a cumulant bound, which was 
proved first by Gawedzki and Kupiainen in 
\cite{Gawedzki/Kupiainen:1985}. 
A simplified proof by Lesniewski appeared later in 
\cite{Lesniewski:1987}.
In an accompanying paper \cite{Salmhofer/Wieczerkowski:1999},
we will give a fresh proof of the cumulant bound and its 
implications on norm estimates for fermionic 
renormalization groups. 

Our construction is furthermore based on a non-perturbative 
implementation of the beta function method of 
\cite{Wieczerkowski:1997a,Wieczerkowski:1997b}.
In this approach, one computes renormalized field theories 
directly as invariant curves emerging from a renormalization 
fixed point. The fundamental dynamical equations are 
the condition of renormalization invariance and a tangent 
(or first order) condition, which selects a particular curve.   

The two dimensional Gross-Neveu model has attracted a lot of attention 
by rigorous renormalization theorists, both because of its 
simplicity and its interesting non-perturbative features. 
We mention the work of Gawedzki and Kupiainen 
\cite{Gawedzki/Kupiainen:1985}; Feldman, Magnen, Rivasseau, and 
Seneor \cite{Feldman/Magnen/Rivasseau/Seneor:1985};
Iagolnitzer and Magnen \cite{Iagolnitzer/Magnen:1987,
Iagolnitzer/Magnen:1988}; Kopper, Magnen and Rivasseau 
\cite{Kopper/Magnen/Rivasseau:1995}. 
Recent work of Disertori and Rivasseau 
\cite{Disertori/Rivasseau:1998} simplifies earlier 
constructions by avoiding the use of phase space expansion 
technology.  
Our work has the same intention although it proceeds along a 
different route.
Another simple, and conceptually rather different, approach, 
which organizes perturbation theory in a {\em ring expansion}, 
was developed in \cite{Feldman/Knoerrer/Trubowitz:1998}.
Although it has not been applied to a construction of the 
Gross-Neveu model, it is an alternative to our bounds.

In the following, we briefly describe the model and our main 
result, leaving detailed definitions for later sections. 
The $\epsilon$ comes from a modification of the massless free 
propagator, which reads 
\begin{equation}
\label{propa}
\widehat{C}(p) = \frac{\zeta\sla{p}}{|p|^{2+\epsilon}}
\end{equation}
in momentum space. 
Here $\zeta\in\mathbb{C}$; 
$\sla{p}=p_1 \gamma_{1} + p_2 \gamma_{2}$, 
where $\gamma_{1}$ and $\gamma_{2}$
are two--dimensional hermitean Dirac matrices, with  
$\{\gamma_\mu,\gamma_\nu\}=2\delta_{\mu,\nu}$; and 
finally $0<\epsilon<\frac{2}{3}$.
As an interaction, we can take any chirally invariant 
four-fermion interaction. 
For instance, the two choices
\begin{equation}\label{GNdisc}
\mathcal{O}_{GN}(\psi) 
=\int\D x \Big(\bar\psi\psi (x)\Big)^{2}
\end{equation}
and
\begin{equation}\label{GNcont}
\mathcal{O}_{GN} (\psi) 
=\int\D x 
\left\{
   \Big(\bar\psi\psi (x)\Big)^{2} 
   +\Big(\bar\psi\I \gamma_5\psi (x)\Big)^{2}
\right\},
\end{equation}
which correspond to the Gross--Neveu model with discrete or
continuous chiral symmetry, are allowed. 
Here $\bar\psi\psi(x)=\sum_{a,\sigma}
\bar\psi_{\sigma,a}(x)\psi_{\sigma,a}(x)$, where 
$\sigma\in\{1,2\}$ is the spin and 
$a\in\{1,\ldots,N\}$ is the colour index. 
A function of $F(\psi,\bar{\psi})$ has a continuous chiral 
invariance if 
\begin{equation}\label{chinv}
F(\psi,\bar{\psi}) 
=F\big(\e^{\I\alpha\gamma_{5}}\psi, 
\bar{\psi}\e^{\I\alpha\gamma_{5}}\big)
\end{equation}
for all $\alpha\in\mathbb{R}$,
regarding $\psi$ as a column vector and $\bar{\psi}$ as a row 
vector with respect to the spin indices. 
$\gamma_{5}$ is a hermitian matrix that anticommutes with the 
matrices $\gamma_{1}$ and $\gamma_{2}$
and whose square is $(\gamma_{5})^2=1$. 
The interaction \Ref{GNdisc} is only invariant under the above
transformation if $\alpha =\pi$. But even this discrete symmetry
forbids a mass term $m \int\D x\; \bar{\psi}(x) \psi(x)$. 

A construction of the model can be obtained by iteration of a 
renormalization group transformation $R_{L}$, which combines 
an integration over fluctuations with a rescaling step.
We use the scaled momentum space renormalization group 
as in \cite{Gawedzki/Kupiainen:1985}, given by
\begin{equation}
\label{trafo}
R_{L}(V)(\Psi)=
\log \int \D\mu_{C_{L}}(\Phi)
\;\exp\Big(V(S_{L}\Psi+\Phi)\Big)-\text{const.},
\end{equation} 
Here $C_{L}$ is a two sided regularization of \Ref{propa},
with unit ultraviolet cutoff and infrared cutoff $L^{-1}$ in 
units of mass,  
$\D\mu_{C_{L}}(\Phi)$ is the corresponding fermionic 
Gaussian measure \Ref{Graga}, 
$S_{L}$ is a dilatation by a scale factor of $L$,
and $V(\Psi)$ is a fermionic potential, which perturbs 
the free model (details follow below). The constant is 
subtracted to make $R_L(V)(0) = 0$.
Eq.\ \Ref{trafo} satisfies the semi-group law $R_{L}R_{L^\prime}=
R_{L\,L^\prime}$. 
Consequently, an $n$-fold iteration of \Ref{trafo} is equivalent 
to a single step with scale $L^n$. 
For technical reasons, we prefer a discrete renormalization group 
with a rather large scale $L$.    
The coupling constant $g$ in front of the interaction will have
to be small, its maximal value $\gamma$ depending on $L$.
It may be complex, but since our bounds involve only $|g|$ 
and the beta function will
only amount to multiplication by a real scale factor, we may 
take $g > 0$ without loss of generality. 

\bigskip\noindent
{\bf Theorem. }{\em
There are $L > 1$ and $\gamma \leq 1$ such that, for all 
$0 \leq g \leq \gamma$, the following holds.
\begin{enumerate}

\item
Let $g_{N}=L^{-2\epsilon N} g$, and
$V_0^{(N)} = g_{N}\,\mathcal{O}_{GN}(\psi)$, with 
$\mathcal{O}_{GN}(\psi)$ given by \Ref{GNdisc} or \Ref{GNcont}. 
Then the limit 
\begin{equation}
V(\psi,g) = \lim_{N \to \infty} (R_{L})^N (V_0^{(N)})(\psi) 
\end{equation}
exists. 

\item
Let $\beta_{L}(g)=L^{-2\epsilon} g$. 
Then the composition of $R_{L}$ with the application of the 
step beta function $\beta_{L}$ has a fixed point. 
For all $g < L^{-2\epsilon} \gamma $,
\begin{equation}
R_{L}\big(V(\psi, g)\big)
=V(\psi,\beta_{L^{-1}}(g)).
\end{equation}

\item
\begin{equation}
V(\psi ,g)
=g\,\mathcal{O}_{GN}(\psi) 
+g^{{7 \over 4}}\mathcal{V}(\psi,g),
\end{equation}
where $\mathcal{V}$ is small in a norm that depends on $L$ 
(the details will be given in Section \ref{ball}).

\end{enumerate}
} % end of \em

\bigskip\noindent
We prove the Theorem by showing that, in an appropriate
Banach space of coupling constants $g$ and interactions, the  
extended renormalization group $T_L$, defined by
\begin{equation}
V(\psi,g)\mapsto R_{L}(V)\big(\psi,\beta_L(g)\big),
\end{equation}
is a contraction mapping on a cone emerging from the free 
field fixed point, which corresponds to a ball of second order 
perturbations $\mathcal{V}$. The interactions in this Banach space are 
analytic in the fields, chirally invariant, and have exponential 
spatial decay. Their decay length is determined by that of 
the fluctuation covariance $C_L$, and is of the order $O(L)$.
\footnote{
In the scaled renormalization group, one iterates the same 
transformation, and localization properties depend on this iterated 
transformation rather than on a flowing scale. 
Translated to a non-scaled renormalization group, where fluctuation 
propagators come on different scales, the localization scale becomes
proportional to the ultraviolet cutoff.
}

\subsection{Setup}

We consider continuum functional integrals with ultraviolet and 
infrared cutoff.
\footnote{
The continuum regularized functional integral again can be defined 
by discretizing the regularized field theory to a finite lattice. 
One then performs both its infinite-volume and zero lattice spacing 
limit in the presence of continuum cutoffs.
We remark that the bounds given in Section \ref{template} imply 
that the effective action converges as the lattice cutoff is 
removed. 
} 
Our cutoffs will be built into the propagator. 
The model is then defined by a regularized propagator together 
with an effective (inter-) action. 
The details of this standard setup are, for example, given in 
\cite{Salmhofer:1999}. 
It is also possible to regard the Grassmann variables merely as 
a convenient way of organizing infinite systems of equations for 
antisymmetric functions. 

Our fermionic fields $\Psi$ are indexed by 
$\mathbb{X}=\mathbb{R}^{2}\times\mathbb{\Lambda}$, where 
$\mathbb{\Lambda}$ is a discrete set, 
in our case 
$\mathbb{\Lambda}=\{1,-1\}\times\{1,2,\ldots,N\}\times\{1,-1\}$,
where the first index is the spin index, the second a colour index,
and the third distinguishes between $\psi$ and $\bar{\psi}$ according to
$\Psi (x,\sigma,a,1) = \bar{\psi}_{\sigma,a}(x)$ and 
$\Psi (x,\sigma,a,-1)=\psi_{\sigma,a}(x)$. 
The Grassmann Gaussian integral corresponding to a free theory with
a propagator $C$ is determined by
\begin{equation}
\label{Graga}
\int \D\mu_C (\Psi)\; \e^{(\eta,\Psi)} 
= \e^{{1\over 2} (\eta, C \eta)}.
\end{equation}
Here the $\eta (X)$ are Grassmann source fields labelled by 
$X \in \mathbb{X}$, and $(\eta,\Psi)$ is an abbreviation for
$\int_{\mathbb{X}} \D X\,\eta(X)\, \Psi(X)$, 
the integral over $\mathbb{X}$ meaning 
$\int\D X F(X)=\int\D^{2}x\sum_{\lambda} F(x,\lambda)$.

The fluctuation integral in \Ref{trafo} is well--defined if
the covariance $C(X,X')$ is a bounded function of $X$ and $X'$.
The inverse Fourier transform of \Ref{propa} is not bounded;
the construction proceeds by first replacing it by an 
ultraviolet cutoff covariance which is a finite sum of bounded covariances. 
The ultraviolet cutoff is removed by taking the number of terms 
in the sum to infinity, and at the same time letting the coupling 
constant flow in the way described in the Theorem.
%
%Eq.\ \Ref{Graga} is well-defined if the covariance $C(X,X')$
%is a bounded function of $X$ and $X'$, which is the case in 
%presence of an ultraviolet cutoff. 
%The construction of the limit in the Theorem works by 
%writing an unbounded $C$ as
%an infinite sum of bounded ones and taking a limit (with
%an appropriate adjustment of the coupling constant, as described 
%in the Theorem). 
The terms in the sum are given by the single--scale covariance of our model, 
{\small\begin{equation}
C_{L}((x,\sigma,a,-1),(x',\sigma',a',1)) 
= \mathbf{C}_{1,L}(x,\sigma,a;x',\sigma',a')
=-C_{L}((x',\sigma',a',1),(x,\sigma,a,-1)),
\end{equation}}
and zero when the charge indices coincide: $C_L ((\cdot,j),(\cdot,j)) = 0$.
It is given by the following Dirac propagator
\begin{equation}
\label{fourC}
\mathbf{C}_{L,L'}(x,\sigma,a;x',\sigma',a')=
\delta_{aa'} \int \dbar p \;
\e^{\I p (x-x')} 
\;\frac{\sla{p}_{\sigma,\sigma^\prime}}{|p|^{2+\epsilon}}
\;\Big(
\hat \chi (Lp) - \hat \chi(L'p)
\Big),
\end{equation}
which is two-sided regularized in momentum space with the help of
the cutoff function 
\begin{equation}
\hat \chi (p) = 
\frac{1}{\Gamma(1+\frac{\epsilon}{2})} \int_{p^2}^\infty
\D t \; \e^{-t} t^{\epsilon \over 2}.
\end{equation}
(This particular regulator has the advantage that the 
cutoff propagator \Ref{fourC} becomes analytic in momentum 
space.)

The covariance with unit infrared cutoff and ultraviolet cutoff $L^N$ 
can be written as a telescope sum 
\begin{equation}\label{addup}
\mathbf{C}_{L^{-N},1} 
=  \sum_{m=1}^N \mathbf{C}_{L^{-m},L^{-m+1}};
\end{equation}
in terms of self-similar $\mathbf{C}$s, which are supported on narrow 
momentum slices,  
\begin{equation}\label{skax}
\mathbf{C}_{L^{-m},L^{-m+1}} (x,\sigma,a;x',\sigma',a') =
L^{2m\sigma}
\mathbf{C}_{1,L} (L^m x,\sigma,a;L^m x',\sigma',a').
\end{equation}

\subsection{The RG transformation}
\label{RGT}

The exponent $\sigma$ denotes the scaling dimension of the 
massless free fermionic field. 
In our model, $\sigma=\frac{1}{2}(1-\epsilon)$.
The associated scale transformation of fields reads
\begin{equation}
S_{L}(\Psi)(x,\lambda)=L^{-\sigma}\,\Psi\left(L^{-1}\,x,
\lambda\right).
\end{equation}
With its help, the self-similarity property of the telescoped 
covariances \Ref{skax} becomes 
\begin{equation}
\mathbf{C}_{L^{-m},L^{-m+1}} =
S_{L^{-m}} \mathbf{C}_{1,L} (S_{L^{-m}})^T
\end{equation}
in operator notation, where $T$ denotes the transposition. 
The additive decomposition 
\Ref{addup} of the covariance implies that the exponential of 
the effective action at scale $1$ is
\begin{equation}
\label{theory}
\int \D \mu_{C_{L^{-N},1}} (\Psi) \e^{\mathbf{V}(\Psi+\Phi)}
= \int \prod_{m=1}^N \D \mu_{C_{1,L}} (\Psi_m) 
\e^{\mathbf{V}\big(S_{L^{-1}}\Psi_1+\ldots+ S_{L^{-m}}\Psi_m+\Phi\big)}
\end{equation}
and is thus equal to  
$\e^{(R_L  \circ \cdots \circ R_L) (V) (\Phi)}$,
%the exponential of an $m$-fold iteration of \Ref{trafo},
provided that $V$ is a scaled version of the bare potential,
namely $V(\Psi) = \mathbf{V}(S_{L^{-m}}\Psi)$.
Notice that because of this rescaling,
the infrared cutoff of the theory, defined by
the left hand side of (\ref{theory}), is one and not $L^{-N}$, 
and is not changed by the scaling of the bare potential. 
More generally, one obtains the scaled renormalization group 
by a multi-scale transformation, where each multi-scale 
component is rescaled to a unit scale. 

Conversely, one reconstructs the non-scaled renormalization flow 
by the introduction of a (physical) renormalization scale,
often together with a renormalization condition on a coupling 
parameter. 
We emphasize that the converse step thus requires an additional 
datum.
\footnote{
%There seems to persist time independent confusion 
%between scaled and non-scaled renormalizers about the value of 
%cutoffs in the scaled renormalization group. 
The scaled 
renormalization group is best thought of as a block spin 
transformation on lattice theories, which live on an infinite 
unit lattice, but encode exact continuum information.} 

We may decompose $R_{L}$ into two parts, 
$R_{L}=S_{L}\circ F_{L}$, with $F_{L}$ an 
integration over the fluctuations
\begin{equation}
F_L (V) (\Phi)  = \log \int \D \mu_{C_{1,L}} (\Psi) 
\e^{V(\Psi+\Phi)} - \text{const.}
\end{equation}
We will derive an estimate on the renormalization group in terms 
of estimates on these two parts.
A field independent constant, which is proportional to the 
volume, is subtracted in order to preserve the condition
$V(0)=0$ in the RG flow. 
In statement 1 of the Theorem, the rescaling of the initial
coupling constant as a function of the renormalized coupling constant
$g$ is given. In the next section, we specify a set of potentials
$V$ to which the RG transformation can be applied. 

An important property is that the cutoff covariances
are of the form $\sla{a}$, so that
\begin{equation}
\e^{\I \alpha \gamma_5} 
\;\mathbf{C}_{L^{-m},L^{-m+1}} 
\;\e^{\I \alpha \gamma_5}
= \mathbf{C}_{L^{-m},L^{-m+1}}.
\end{equation}
Thus, any discrete or continuous chiral invariance of $V$
is preserved under the RG transformation. In other words,
if $V$ obeys \Ref{chinv}, then the same holds for $F_L(V)$
and $R_L(V)$. 

\section{Estimate on the renormalization flow}\label{template}

We now give a norm estimate on the renormalization
group transformation $R_{L}=S_{L}\circ F_{L}$ built from 
two separate norm estimates, an estimate on the scale transformation 
$S_{L}$ and an estimate on the fluctuation integral $F_{L}$. 
It will serve as a template for the refined estimates presented 
thereafter. 

\subsection{Banach space $\mathbb{V}_{h,\kappa}$}

We consider potentials of the following general (power series) type. 
Let $V(\Psi)$ be given by an infinite sum 
$V(\Psi)=\sum_{f=1}^{\infty}V_{f}(\Psi)$ 
of $f$--point vertices
\begin{equation}
V_{f}(\Psi)=\int\D X_{1}\Psi(X_{1})\cdots\int\D X_{f}\Psi(X_{f})
\,V_{f}(X_{1},\ldots,X_{f}),
\end{equation} 
where the vertices are distributional kernels.
We will restrict our attention to vertex functions of the general 
form 
\begin{equation} 
V_{f}(X_{1},\ldots,X_{f})=
\sum_{l=0}^{k-1}\overline{V}_{f,l}(X_{1},\ldots,X_{f})
\,\sum_{\pi\in\mathfrak{S}_{f}}\prod_{i=1}^{l}
\,\delta (x_{\pi(1)}-x_{\pi(1+i)}),
\end{equation}
where $\overline{V}_{f,l}\in L^{1}_{loc}(\mathbb{X}\times\cdots
\times\mathbb{X},\mathbb{C})$, and has the usual properties
of a fermionic theory (anti-symmetry, Euclidean covariance).
We also assume that $V$ is even, that is,
$V_{f}(\Psi)=0$ for $f\in 2\mathbb{N}+1$.
Then $R_L(V)$ is also even. 
Temporarily, $\Psi$ denotes the fermionic field without derivatives. 
Later, we will encorporate derivative fields by enlarging
$\mathbb{\Lambda}$ to an appropriate multiplet. 
Let $\|V\|_{h,\kappa}=\sum_{f=1}^{\infty}h^{f}\,
\|V_{f}\|_{\kappa}$, where 
\begin{equation}
\|V_{f}\|_{\kappa}=\sup_{x_{0}\in\mathbb{R}^{2}}
\int\D X_{1}\cdots\D X_{f}\,\delta(x_{0}-x_{1})
\,\vert V_{f}(X_{1},\ldots,X_{f})\vert
\,\exp\Big(\kappa\,\mathcal{L}(x_{1},\ldots,x_{f})\Big).
\end{equation}
Here $\mathcal{L}(x_{1},\ldots,x_{f})$ denotes the tree distance 
of $(x_{1},\ldots,x_{f})$. The tree distance on an $f$-tuple is 
defined as 
\begin{equation}
\mathcal{L}(x_{1},\ldots,x_{n})=
\inf_{\tau\in\mathcal{T}_{n}}
\sum_{b\in\tau}\|x_{b_{1}}-x_{b_{2}}\|,
\end{equation}
where $\mathcal{T}_{n}$ is the set of trees on $\{1,2,\ldots,n\}$, 
and where $b = (b_1,b_2) \in \tau$ are the bonds of $\tau$. 
The potentials with $\Vert V\Vert_{h,\kappa} < \infty $ 
form a Banach space $\mathbb{V}_{h,\kappa}$. 
It depends on two parameters $h$ and $\kappa$, where 
$h$ can be thought of as an inverse radius of convergence in field 
space and $\kappa$ as an inverse exponential rate of decay.
We will show that there exists a choice such that 
the action of $R_{L}$ is well--defined on a suitable ball around zero 
in $\mathbb{V}_{h,\kappa}$.

\subsection{Estimate on $S_{L}$}

Let $S_{L}(V_{f})(\Psi)=V_{f}\Big(S_{L} (\Psi)\Big)$. 
%Since 
%\begin{equation}
%S_{L}(V_{f})(\Psi)=\int\D X_{1}\Psi(X_{1})\cdots
%\int\D X_{f}\Psi(X_{f})
%\,L^{f\,(2-\sigma)}\,V_{f}\Big((L\,x_{1},\lambda_{1})
%,\ldots,(L\,x_{f},\lambda_{f})\Big),
%\end{equation}
%it follows that 
Then
\footnote{The exponent $2-f\sigma$ is an old friend from perturbative 
renormalization theory, namely the scaling dimension of $V_{f}$.} 
$\|S_{L}(V_{f})\|_{\kappa}=L^{2-f\sigma}\,\|V_{f}\|_{L^{-1}\kappa}$, 
so $S_{L}$ performs the following simple scale 
transformation on our norm
\begin{equation}
\|S_{L}(V)\|_{h,\kappa}=L^{2}
\,\|V\|_{L^{-\sigma}h,L^{-1}\kappa}.
\label{scalingI}
\end{equation} 
Derivatives produce additional inverse powers of $L$. 
Because $V_{f}(\Psi)=0$ for $f\in 2\mathbb{N}+1$, 
\begin{equation}
\|S_{L}(V)\|_{h,\kappa}
\leq L^{1+3\epsilon}
\;\|V\|_{L^{-\frac{\epsilon}{4}}h,L^{-1}\kappa},
\label{scalingII}
\end{equation}  
at least under the wasteful condition that 
$0<\epsilon <\frac{2}{3}$. 
Here we saved a small amount of the scale factor to control 
an anticipated shift of the field, which will come about in 
the integral over fluctuations. 

\subsection{Estimate on $F_{L}$}
As shown in the Appendix, the one--scale propagator $C_L$ satisfies
\begin{equation}
\vert C_{L}(X,Y)\vert\leq O(1)\,L^{-2\,\sigma}
\,\exp\Big(-L^{-1}\,\|x-y\|\Big)
\end{equation}
(here and in the following, $O(1)$ denotes constants which are
independent of $L$), and it has a Gram representation 
\begin{equation}
\mathbf{C}_{1,L}(x,\sigma,a;x',\sigma',a')
=\left\langle \varphi_{L} (x,\sigma,a)\vert 
\tilde\varphi_{L}(x',\sigma',a')\right\rangle
\end{equation} 
where $\norm{\varphi_{L} (x,\sigma,a)}$ and 
$\norm{\tilde\varphi_{L}(x',\sigma',a')}$ are $O(1)$ uniformly in $X$. 
The fluctuation transformation is defined as follows. 
%If the cumulant expansion converges, then
In an expansion in the fields, 
$F_{L}(V)(\Psi)=\sum_{f=1}^{\infty}F_{L}(V)_{f}(\Psi)$ 
with 
\begin{gather}
F_{L}(V)_{f}(\Psi)=\int\D Z_{1}\Psi(Z_{1})\cdots\int\D Z_{f}\Psi(Z_{f})
\nonumber\\
\sum_{n=1}^{\infty}\frac{1}{n!}
\sum_{f_{1}=1}^{\infty}\sum_{e_{1}=0}^{f_{1}-1}\binom{f_{1}}{e_{1}}
\cdots
\sum_{f_{n}=1}^{\infty}\sum_{e_{n}=0}^{f_{n}-1}\binom{f_{n}}{e_{n}}
\,\delta_{f,e_{1}+\cdots e_{n}}\,\Theta_{f,1}
\,(-1)^{\alpha_{n} (f_{1},e_{1},\ldots,f_{n},e_{n})}
\nonumber\\
\int\D Y_{1,1}\cdots\D Y_{1,i_{1}}
\,V_{f_{1}}(X_{1,1},\ldots,X_{1,e_{1}},Y_{1,1},\ldots,Y_{1,i_{1}})
\nonumber\\
\cdots
\int\D Y_{n,1}\cdots\D Y_{n,i_{n}}
\,V_{f_{n}}(X_{n,1},\ldots,X_{n,e_{n}},Y_{n,1},\ldots,Y_{n,i_{n}})
\nonumber\\
\Big\langle
   \Phi(Y_{1,1})\cdots\Phi(Y_{1,i_{1}});\cdots;
   \Phi(Y_{n,1})\cdots\Phi(Y_{n,i_{n}})
\Big\rangle^{T}_{C_{L}},
\label{cumulant}
\end{gather}
where $(Z_{1},\ldots,Z_{f})=(X_{1,1},\ldots,X_{1,e_{1}},
\ldots,X_{n,1},\ldots,X_{n,e_{n}})$, $(-1)^{\alpha_{n}}$ is a 
sign factor, and where we use the notation $f_{l}=e_{l}+i_{l}$. 
($f_{l}$ is the power of fields of the $l$'th vertex, 
$e_{l}$ is the number of external fields chosen therefrom, 
and $i_{l}$ is the number of the remaining internal fields.)
The fluctuation integral produces effective vertices 
$F_{L}(V)_f(Z_{1},\ldots,Z_{f})$ as the anti-symmetrized 
kernels given by the integrand of the expression (\ref{cumulant}). 
They are infinite sums of convolutions of the original kernels 
with propagators. The norm estimates in the remainder of this section
imply that these infinite sums converge if $\Vert V\Vert_{h,\kappa}$ is small 
enough. 

\subsubsection{Estimate on partially truncated correlators}

The cumulant expansion (\ref{cumulant}) involves partially 
truncated correlators. They obey the following beautiful bound 
due to Gawedzki and Kupianen \cite{Gawedzki/Kupiainen:1985} and 
Lesniewski \cite{Lesniewski:1987}. There are positive constants
$\kappa_{1}$, $C_{1}$ and $C_{2}$, all independent of $L$, such that
\begin{gather}
\label{estimate}
\left\vert\Big\langle
   \Phi(Y_{1,1})\cdots\Phi(Y_{1,i_{1}});\cdots;
   \Phi(Y_{n,1})\cdots\Phi(Y_{n,i_{n}})
\Big\rangle^{T}_{C_{L}}\right\vert
\\ \leq
n! %(n-1)!
\;C_{1}^{i_{1}+\cdots+i_{n}}
\;\Big(L^{-2\sigma}\,C_{2}\Big)^{n-1}
\;\exp\Big(-L^{-1}\kappa_{1}
\mathcal{L}(y_{1,1},\ldots,y_{1,i_{i}}\trenner
   \cdots\trenner y_{n,1},\ldots,y_{n,i_{n}})\Big)
\nonumber
\end{gather}
In an accompanying paper, we derive these bounds, 
and the norm bounds that follow from them, 
in a simplified way \cite{Salmhofer/Wieczerkowski:1999}.
Here $\mathcal{L}(\underline{y_{1}}\trenner\ldots\trenner\underline{y_{n}})$
denotes the inter-tuple tree distance of the tuples  
$\underline{y_{l}}=(y_{l,1},\ldots,y_{l,i_{l}})$ defined as 
\begin{equation}
\mathcal{L}(\underline{y_{1}}\trenner\ldots\trenner\underline{y_{n}})
=\inf_{j_{l}\in\{1,\ldots,i_{l}\}}
\mathcal{L}(y_{1,j_{1}},\ldots,y_{n,j_{n}}).
\end{equation}
One selects a point in each tuple and computes the ordinary 
tree distance for this selection. The selection with a minimal 
tree distance defines the inter-tuple tree distance. 
It will be important that $\kappa_{1}$, $C_{1}$, and $C_{2}$
do not depend on $L$ because we shall use a large $L$ argument
later on. The constant $C_1$ is proportional to the Gram constant
(see the Appendix). 

\subsubsection{Estimates on tree distances}

Since the inter-tuple distance is a tree distance with 
respect to one particular tree, which may or may not be the 
minimal one, we have that
\begin{equation}
\sum_{l=1}^{n}\mathcal{L}(\underline{x_{l}},\underline{y_{l}})
+\mathcal{L}(\underline{y_{1}}\trenner\ldots\trenner\underline{y_{n}})
\geq \mathcal{L}(\underline{x_{1}},\underline{y_{1}},\ldots,
\underline{x_{n}},\underline{y_{n}}).
\label{treedistanceI}
\end{equation}
In addition to (\ref{treedistanceI}), we need a bound which
tells how tree distances behave under the removal of points.  
\footnote{For this reason, Gawedzki and Kupiainen use 
a different tree distance in \cite{Gawedzki/Kupiainen:1985}.}
There exists a constant $\alpha$, with $1<\alpha\leq 2$, 
such that 
\begin{equation}
\alpha\,\mathcal{L}(\underline{x_{1}},\underline{y_{1}},\ldots,
\underline{x_{n}},\underline{y_{n}})\geq
\mathcal{L}(\underline{x_{1}},\ldots,\underline{x_{n}}).
\label{treedistanceII}
\end{equation}
Simple examples show that \Ref{treedistanceII} cannot hold with
$\alpha =1$.  For  $\alpha=2$, 
(\ref{treedistanceII}) is readily proved by grouping the 
removed points into trees and reconnecting the 
connected components in a way that can be estimated by 
twice the length of the removed trees.

Let  
\begin{equation}
\label{Lkaann}
\kappa=L^{-1}\,\kappa_{1} \mbox{ and } L\geq 4 \ge 2\alpha.
\end{equation}
Then
\begin{gather}
\frac{\kappa}{L}
\,\mathcal{L}(\underline{x_{1}},\ldots,\underline{x_{n}})
-\frac{\kappa_{1}}{L}
\,\mathcal{L}(\underline{y_{1}}\trenner\ldots\trenner\underline{y_{n}})
%\nonumber\\
\leq\kappa\sum_{l=1}^{n}
\mathcal{L}(\underline{x_{l}},\underline{y_{l}})
-\frac{\kappa_{1}}{2\,L}\mathcal{L}(\underline{x_{1}},
\underline{y_{1}},\ldots,\underline{x_{n}},\underline{y_{n}})
\label{treedistanceIII}
\end{gather}   
The estimate (\ref{treedistanceIII}) is the only
property of tree distances which we need in our bound for 
the fluctuation integral. 

\subsubsection{Estimate on the $f$-vertex}

We proceed under the assumption \Ref{Lkaann}. %s of (\ref{treedistanceIII}).
Return to (\ref{cumulant}). From the estimates (\ref{estimate}) 
and (\ref{treedistanceIII}), it follows that 
\begin{gather}
\|F_{L}(V)_{f}\|_{L^{-1}\,\kappa}
\leq
\sum_{n=1}^{\infty}
\Big(L^{2-2\sigma}\,C_{2}\,C_{3}\Big)^{n-1}
\nonumber\\
\sum_{f_{1}=1}^{\infty}\sum_{e_{1}=0}^{f_{1}-1}\binom{f_{1}}{e_{1}}
\,C_{1}^{i_{1}}\,\|V_{f_{1}}\|_{\kappa}
\cdots
\sum_{f_{n}=1}^{\infty}\sum_{e_{n}=0}^{f_{n}-1}\binom{f_{n}}{e_{n}}
\,C_{1}^{i_{n}}\,\|V_{f_{n}}\|_{\kappa}
\;\delta_{f,e_{1}+\cdots e_{n}}.
\label{resultI}
\end{gather}
For each vertex, one chooses a point to anchor its tree. 
One then pulls out the vertex norms. The remaining integral 
over the anchors is estimated using the spared exponential decay, 
\begin{equation}
\sup_{x_{0}}\int\D^{2} x_{1}\cdots\int\D^{2} x_{n}
\,\delta(x_{0}-x_{1})
\,\exp\left(-\frac{\kappa_{1}}{2\,\alpha\, L}
\mathcal{L}(x_1,\ldots,x_n) \right)
\leq \Big(L^{2}\,C_{3}\Big)^{n-1}.
\end{equation}
To obtain a bound on the $(h,\kappa)$-norm from this, we have 
to sum over the powers of fields. 
This yields the geometric series
\begin{gather}
\label{resultII}
\|F_{L}(V)\|_{L^{-\sigma}h,L^{-1}\kappa}
\leq\sum_{n=1}^{\infty}
\Big(L^{2-2\sigma}C_{2}C_{3}\Big)^{n-1}
\Big(\|V\|_{L^{-\sigma}h+C_{1},\kappa}\Big)^{n}
\end{gather}
which converges if 
$q\|V\|_{L^{-\sigma}h+C_{1},\kappa} < 1$, where 
$q= L^{2-2\sigma}C_{2}C_{3}$;
then
\begin{equation}
\label{resultIIa}
\|F_{L}(V)\|_{L^{-\sigma}h,L^{-1}\kappa}
= \frac{\|V\|_{L^{-\sigma}h+C_{1},\kappa}}{1-q\|V\|_{L^{-\sigma}h+C_{1},\kappa}}.
\end{equation}
This shows that the RG transform is well--defined on a ball of 
potentials that are analytic in the fields. 

\subsection{Estimate on $R_{L}$}

Let $h$ satisfy
\begin{equation}\label{hchoice}
h=L^{-\frac{\epsilon}{4}}h+C_{1}.
\end{equation}
Both $h$ and $\kappa$ now depend on $L$. 
Let $V(\Psi)$ then be an element of the ball
$B_{r}=\Big\{V(\Psi)\in\mathbb{V}_{h,\kappa}
\Big\vert \|V\|_{h,\kappa}\leq r\Big\}$
with sufficiently small radius $r$. 
Then (\ref{scalingII}) and (\ref{resultIIa}) imply together that 
$R_{L}:B_{r}\rightarrow B_{f_{L}(r)}$ with a flow of 
radii given by 
\begin{equation}
f_{L}(r) = \frac{L^{1+3\epsilon}r}{1-qr}
%=\frac{L^{1+3\epsilon}}
%{L^{2-2\sigma}\,C_{2}\,C_{3}}
%\;\sum_{n=1}^{\infty}
%\Big(L^{2-2\sigma}\,C_{2}\,C_{3}\,r\Big)^{n}.
\label{radiusflow}
\end{equation} 
Unfortunately, this bound is not sufficient for an iteration of $R_L$
because small potentials tend to grow.
\footnote{
The largest eigenvalue of the linearized 
renormalization group is here $L^{1+\epsilon}$. The extra 
factor of $L^{2\epsilon}$ is due to the 
non-linear corrections.
}
This behavior indicates the necessity of renormalization.
The factor $L$ in (\ref{radiusflow}) will be 
removed by restricting to a subspace of potentials 
with vanishing mass vertex. 

\section{Two point vertex}
The scaling dimension of an $2f$--vertex is 
$2-f\sigma$. Because 
all vertices with an odd number of fields $f$
vanish, the lowest non-vanishing vertex is a two point vertex. 
Its scaling dimension is $1+\epsilon$, 
which is also the scaling dimension of a local mass vertex. 
The two point vertex is the most relevant vertex of our flow. 
In this section, we will split it into a local and a non-local part. 
The non-local part will have an improved scaling dimension.
The local part is zero for chirally invariant interactions.

\subsection{Localization operator $\mathbf{L}$} 
%
The localization operator amounts to a Taylor 
expansion with remainder in momentum space. 
For the purposes of this paper, a lowest order 
expansion of the two point vertex suffices. 
\footnote{
In the case when $\epsilon=0$, one has to expand the two point 
vertex to third order and the four point vertex to first order, 
as is done in \cite{Gawedzki/Kupiainen:1985} and 
\cite{Feldman/Magnen/Rivasseau/Seneor:1985}.
The formulas are immediate generalizations of those presented 
here.
}
In real space, we define 
$\mathbf{L}$ by the decomposition
\begin{equation}
V_{2}(\Psi)=\int\D X_{1}\int\D X_{2}\,\Psi(X_{1})
\,\Big(\mathbf{L}(V_{2})(X_{1},X_{2})
+(\mathbf{1}-\mathbf{L})(V_{2})(X_{1},X_{2})\Big)\,\Psi(X_{2})
\end{equation}
into a local part 
\begin{equation}
\mathbf{L}(V_{2})\Big((x_{1},\lambda_{1}),
(x_{2},\lambda_{2})\Big)=
\delta(x_{1}-x_{2})\,\int\D^{2}y_{2}\;
V_{2}\Big((x_{1},\lambda_{1}),(y_{2},\lambda_{2})\Big)
\end{equation}
and a non-local part
\begin{gather}
(\mathbf{1}-\mathbf{L})(V_{2})\Big((x_{1},\lambda_{1}),
(x_{2},\lambda_{2})\Big)
\nonumber\\
=\sum_{\mu=1}^{2}
\int_{0}^{1}\D t\,
t^{-2}\,V_{2}\left((x_{1},\lambda_{1}),
\left(x_{1}+\frac{x_{2}-x_{1}}{t}\right),\lambda_{2}\right)
\,\frac{x_{2}^{\mu}-x_{1}^{\mu}}{t}
\,\frac{\partial}{\partial x_{2}^{\mu}}.
\end{gather}
The $t$-integral converges at $t=0$ 
because the vertex decays exponentially fast at infinity.
This splitting follows from a Taylor expansion with remainder
term of the second field 
\begin{equation}
\Psi(x_{2},\lambda_{2})
=\Psi(x_{1},\lambda_{2})+\int_{0}^{1}\D t\,
(x_{2}-x_{1})\cdot(\partial\Psi)(x_{1}+t(x_{2}-x_{1}),
\lambda_{2})
\end{equation}
around the position of the first one. After a change of 
integration variables, one obtains an expression of the form

\begin{gather}
V_{2}(\Psi)=\int\D^{2}x\sum_{\lambda_{1},\lambda_{2}}
\,\Psi(x,\lambda_{1})\,m_{\lambda_{1},\lambda_{2}}
\,\Psi(x,\lambda_{2})
\nonumber\\
+\sum_{\mu}\int\D X_{1}\int\D X_{2}
\Psi(X_{1})\,V_{2,\mu}(X_{1},X_{2})
\,(\partial_{\mu}\Psi)(X_{2})
\label{taylor}
\end{gather}

\subsection{Redefinition of the norm $\|V_{2}\|_{\kappa}$}

Let us represent the two point vertex as in (\ref{taylor}).
Then we may redefine its norm into
$\dreinorm V_{2}\dreinorm_{\kappa}=\|\mathbf{L}V_{2}\|
+\|(\mathbf{1}-\mathbf{L})V_{2}\|_{\kappa}$ 
with
$\|\mathbf{L}V_{2}\|=\sum_{\lambda_{1},\lambda_{2}}
\vert m_{\lambda_{1},\lambda_{2}}\vert$
and
\begin{gather}
\|(\mathbf{1}-\mathbf{L})V_{2}\|_{\kappa}=
\kappa\;\sup_{x_{0}}
\int\D X_{1}\int\D X_{2}
\sum_{\mu}\,\vert V_{2,\mu}(X_{1},X_{2})\vert
\,\exp\Big(\kappa\,\| x_{1}-x_{2}\|\Big),
\label{newnorm}
\end{gather}
which is the old norm of the non-local part times $\kappa$.
For the higher vertices, we use the old norm. We can now
redo the above estimates with this redefined norm.

\subsubsection{Estimate on $S_{L}$}

The remainder term has an improved scaling dimension. 
$S_{L}$ and $\mathbf{L}$ commute. Therefore, 
the local term scales according to
$\|\mathbf{L}S_{L}V_{2}\|=L^{1+\epsilon}\,\|\mathbf{L}V_{2}\|$,
while the non-local remainder scales as
\begin{equation}
\|(\mathbf{1}-\mathbf{L})S_{L}V_{2}\|_{\kappa}
=L^{\epsilon}\,\|(\mathbf{1}-\mathbf{L})V_{2}\|_{L^{-1}\kappa}
\end{equation}
because of its derivative field. In our model, the local 
mass term is zero because of the chiral symmetry (\ref{chinv}). 
The net gain of the localization procedure is a factor 
of $L^{-1}$ for the redefined norm, since
\begin{equation}
\dreinorm S_{L}V\dreinorm_{h,\kappa}
=L^{\epsilon}\,h^{2}\,\dreinorm V_{2}\dreinorm_{L^{-1}\kappa}
+\sum_{n=2}^{\infty}
L^{2-n(1-\epsilon)}\,h^{2n}\,\dreinorm V_{2n}\dreinorm_{L^{-1}\kappa}.
\end{equation}
This gives the following refinement of (\ref{scalingII}).
If $0<\epsilon <\frac{2}{3}$, then
\begin{equation}
\dreinorm S_{L}V\dreinorm_{h,\kappa}
\leq L^{3\epsilon}
\;\dreinorm V\dreinorm_{L^{-\frac{\epsilon}{4}}h,L^{-1}\kappa}
\label{scalingIII}
\end{equation}

\subsubsection{Estimate on $F_{L}$}

The norm of the non-local term can be bounded by the 
norm of the non-differentiated vertex. By definition
\begin{gather}
\|(\mathbf{1}-\mathbf{L})V_{2}\|_{\kappa}=
\kappa\,\sup_{x_{1}}\int\D^{2}x_{2}
\sum_{\lambda_{1},\lambda_{2},\mu}
\,\e^{\kappa\|x_{1}-x_{2}\|}
\nonumber\\ 
\left\vert
\int_{0}^{1}\D t\,
t^{-2}\,V_{2}\left((x_{1},\lambda_{1}),
\left(x_{1}+\frac{x_{2}-x_{1}}{t},\lambda_{2}\right)\right)
\,\frac{x_{2}^{\mu}-x_{1}^{\mu}}{t}
\right\vert
\end{gather}
it follows that
\begin{gather}
\|(\mathbf{1}-\mathbf{L})V_{2}\|_{\kappa}\leq
\sup_{x_{1}}\int\D^{2}x_{2}
\sum_{\lambda_{1},\lambda_{2}}
\,\e^{\kappa\|x_{1}-x_{2}\|}
\,\left\vert 
V_{2}\Big((x_{1},\lambda_{1}),(x_{2},\lambda_{2})\Big)
\right\vert
\nonumber\\
\kappa\,\sum_{\mu}\vert x_{1}^{\mu}-x_{2}^{\mu}\vert
\,\int_{0}^{1}\D t\,\e^{t\,\kappa\,\|x_{1}-x_{2}\|}.
\end{gather}
For our convenience, we define 
$\|x\|=\sum_{\mu}\vert x^{\mu}\vert$.
Then we have the promised estimate
\begin{equation}
\|(\mathbf{1}-\mathbf{L})V_{2}\|_{\kappa}
\leq \|V_{2}\|_{\kappa}.
\end{equation}
Consequently, we find the following estimate for the 
effective non-local two point vertex 
\begin{equation}
\|(\mathbf{1}-\mathbf{L})S_{L}F_{L}(V)_{2}\|_{\kappa}
= L^{\epsilon}
\;\|(\mathbf{1}-\mathbf{L})F_{L}(V)_{2}\|_{L^{-1}\kappa}
\leq L^{\epsilon}\;\|F_{L}(V)_{2}\|_{L^{-1}\kappa}
\end{equation} 
computed as the image of one renormalization group 
transformation.  
The local mass term is zero by the chiral invariance.

%Let us proceed under the assumption that the 
%renormalization group preserves the property that the 
%local mass term be zero. Then we 
We now do the estimate of 
the fluctuation step exactly as in Section \ref{template}.
This is possible because (\ref{newnorm}) is 
of the same form as the old norm up to a 
factor $\kappa$. For each factor $V_{2}$ we pick up
a factor $\kappa^{-1}$. But we also get one derivative
field for each factor $V_{2}$. Fortunately, 
\begin{equation}
\left\vert
\frac{\partial}{\partial x^{\mu}}
C_{L}(X,Y)
\right\vert
\leq O(1)\,L^{-2\sigma-1}
\,\exp\Big(-L^{-1}\,\|x-y\|\Big)
\end{equation}
comes with an additional factor $L^{-1}$, which 
compensates the $L$-factor in $\kappa^{-1}$. 
The remaining constant is easily accommodated since it 
is of the order $O(1)$. We shift it into a modified 
cumulant bound. Thus $C_{1}$ and $C_{2}$ are now 
understood to be redefined such that the cumulant 
bound holds for the enlarged multiplet $\Psi$, which 
includes derivative fields. 

\subsection{Estimate on the massless renormalization group}
Summing over powers of the field, we get
\begin{equation}
\dreinorm R_{L}(V)\dreinorm_{h,\kappa}
\leq L^{\epsilon}\,h^{2}\,\|F_{L}(V)_{2}\|_{L^{-1}\kappa}
+\sum_{n=2}^{\infty}L^{2-n(1-\epsilon)}\,h^{2n}
\,\|F_{L}(V)_{2n}\|_{L^{-1}\kappa}.
\end{equation}
The largest scale factor is now $L^{2\epsilon}$. 
For $0<\epsilon<\frac{2}{3}$, it follows that 
\begin{equation}
\dreinorm R_{L}(V)\dreinorm_{h,\kappa}
\leq L^{3\epsilon}
\sum_{n=1}^{\infty} \Big(L^{-\frac{\epsilon}{4}}\,h\Big)^{2n}
\,\|F_{L}(V)_{2n}\|_{L^{-1}\kappa}.
\end{equation}
As before, $h=L^{-\frac{\epsilon}{4}}h+C_{1}$.
Then we can sum the series as above.  
The result is the estimate
\begin{equation}
\dreinorm R_{L}(V)\dreinorm_{h,\kappa}
\leq L^{3\epsilon}
\frac{\dreinorm V\dreinorm_{h,\kappa}}{1 - q \dreinorm V\dreinorm_{h,\kappa}}
%\dreinorm R_{L}(V)\dreinorm_{h,\kappa}
%\leq L^{3\epsilon}
%\sum_{n=1}^{\infty}\Big(L^{2-2\sigma}\,O(1)\Big)^{n-1}
%\,\Big(\dreinorm V\dreinorm_{h,\kappa}\Big)^{n}.
\end{equation}
with $q = L^{2-2\sigma}\,O(1)$. 
Thus, also in the massless renormalization group, small 
potentials tend to grow. But the pace is reduced.  

In the following, we shall only work with $\dreinorm \; \cdot \; \dreinorm$;
for simplicity, we denote it by the usual norm symbol $\norm{\; \cdot \;}$. 

\section{Invariant ball}
\label{ball}

We turn our attention from points in the space of chirally invariant
even potentials to parametrized continuous curves 
$V(\Psi\vert g)$, $g\in[0,\gamma]$, which are of the form 
\begin{equation}
\label{curveform}
V(\Psi\vert g)=g\,\mathcal{O}_{GN}(\Psi)+
g^{\frac{7}{4}}\,\mathcal{V}(\Psi\vert g).
\end{equation}
Here $\mathcal{O}_{GN}(\Psi)$ denotes the normal ordered Gross-Neveu vertex 
and $\mathcal{V}(\Psi\vert g)=O(g^{\frac{1}{4}})$
denotes a second order correction to it. 
For any fixed $g$, the potentials of the type (\ref{curveform}) form a linear space.
We shall estimate the remainder in 
%This space of curves is then equipped with the supremum norm 
\begin{equation}
\|\mathcal{V}\|_{\gamma,h,\kappa}=\sup_{g\in[0,\gamma]}
\|\mathcal{V}(\cdot\vert g)\|_{h,\kappa}.
\end{equation}
The additional parameter $\gamma$ denotes the maximal 
admissible value of the coupling constant $g$ in our estimates. 

\subsection{Step $\beta$-function}

The linearization of $R_{L}$ at the free field 
fixed point $V^{\star}(\Psi)=0$ is the first term 
of the cumulant expansion
\begin{equation}
DR_{L}(V)(\Psi)=\int\D\mu_{C_{L}}(\Phi)
\,V(S_{L}\Psi+\Phi)-\text{const.}
\end{equation}
The normal ordered Gross-Neveu vertex is an 
eigenvector of the linearized renormalization group
\begin{equation}
DR_{L}(\mathcal{O}_{GN})(\Psi)
=L^{2\epsilon}\,\mathcal{O}_{GN}(\Psi)
\label{eigen}
\end{equation}
with eigenvalue $L^{2\epsilon}$. For $\epsilon>0$, it 
is a relevant perturbation. We use the inverse of the 
eigenvalue in (\ref{eigen}) to define our step 
$\beta$-function as the linear function
\begin{equation}
\beta_{L}(g)=L^{-2\epsilon}\,g.
\end{equation}

\subsection{Extended renormalization group}

We then define an extended renormalization group 
transformation as the composition 
$T_{L}=\beta_{L}\circ R_{L}$ of a linear coupling 
transformation 
$\beta_{L}(V)(\Psi\vert g)=V(\Psi\vert \beta_{L}(g))$
and the renormalization group $R_{L}$. 
The additional step $\beta$-function turns the 
Gross-Neveu vertex into a fixed point
\begin{equation}
DT_{L}(g \mathcal{O}_{GN})(\Psi)=
g \mathcal{O}_{GN}(\Psi)
\end{equation}
of the linearized extended renormalization group.
Our desire is a non-linear extension thereof. For 
this purpose, we consider the transformation of the 
second order correction
\begin{equation}
\mathcal{T}_{L}(\mathcal{V})(\Psi\vert g)
=g^{-\frac{7}{4}}
\,\beta_{L}S_{L}
\mathcal{F}_{L}(\mathcal{V})(\Psi\vert g)
=g^{-\frac{7}{4}}
\,\Big(T_{L}(V)(\Psi\vert g)-g\,\mathcal{O}_{GN}(\Psi)\Big)
\label{subtract}
\end{equation}

\subsubsection{Estimate on $\beta_{L}$}

The flow of the coupling constant yields an extra small factor. 
It will turn out to be sufficient to renormalize the 
theory. We have that 
\begin{equation}
\|\mathcal{T}_{L}(\mathcal{V})\|_{\gamma,h,\kappa}
=\,L^{-\frac{7\epsilon}{2}}\,\sup_{g\in[0,L^{-2\epsilon}\gamma]}
g^{-\frac{7}{4}}
\,\|S_{L}\mathcal{F}_{L}(\mathcal{V})(\cdot\vert g)\|_{h,\kappa}
\label{estimate0}
\end{equation}
Since $L>1$ and $\epsilon >0$, we have that 
$[0,L^{-2\epsilon}\gamma]\subset [0,\gamma]$ and therefore
\begin{equation}
\|\mathcal{T}_{L}(\mathcal{V})\|_{\gamma,h,\kappa}
=\,L^{-\frac{7\epsilon}{2}}\,\sup_{g\in[0,\gamma]}
g^{-\frac{7}{4}}
\,\|S_{L}\mathcal{F}_{L}(\mathcal{V})(\cdot\vert g)\|_{h,\kappa}.
\label{estimate1}
\end{equation}
In the following, we do not need the small scale factors 
coming with terms of higher order than $g^{2}$.

\subsubsection{Estimate on $S_{L}$}

As a payoff of our general massless estimate (\ref{scalingIII})
%$\|S_{L}V\|_{h,\kappa}\leq
%L^{3\epsilon}\,\|V\|_{L^{-\frac{1}{4}}h,L^{-1}\kappa}$
it follows that the right hand side of 
(\ref{estimate1}) itself can be further estimated by 
\begin{equation}
\|\mathcal{T}_{L}(\mathcal{V})\|_{\gamma,h,\kappa}
\leq\,L^{-\frac{\epsilon}{2}}\,\sup_{g\in[0,\gamma]}
g^{-\frac{7}{4}}
\,\|\mathcal{F}_{L}(\mathcal{V})(\cdot\vert g)
\|_{L^{-\frac{\epsilon}{4}}h,L^{-1}\kappa}
\label{estimate2}
\end{equation}
The prefactor $L^{-\frac{\epsilon}{2}}<1$ will become 
responsible for the contraction property. 

\subsubsection{Estimate on $\mathcal{F}_{L}$}

In this renormalization group, we track the transformation
of the non-linear corrections to a pure Gross--Neveu vertex.
By its definition (\ref{subtract}), the subtracted 
fluctuation step reads, in a selfexplanatory notation, 
\begin{gather}
\mathcal{F}_L(\mathcal{V})(\Psi\vert g)
=g^{\frac{7}{4}}
\,\Big\langle
\mathcal{V}(S_{L}\Psi+\Phi\cdot\vert g)
\Big\rangle_{C_{L}}
\nonumber\\
+\sum_{n=2}^{\infty}\frac{1}{n!}
\Big\langle
\prod_{i=1}^{n}
\Big[
   g\,\mathcal{O}_{GN}(S_{L}\Psi+\Phi)
   +g^{\frac{7}{4}}\,\mathcal{V}(S_{L}\Psi+\Phi)
;\Big]
\Big\rangle^{T}_{C_{L}}.
\label{subtract2}
\end{gather}
The estimate of
(\ref{subtract2}) goes exactly as in Section \ref{template}.
With the choice (\ref{hchoice}), the result is
%Put $h=L^{-\frac{\epsilon}{4}}h+C_{1}$. The result is 
\begin{gather}
\|\mathcal{F}_{L}(\mathcal{V})(\cdot\vert g)
\|_{L^{-\frac{\epsilon}{4}}h,L^{-1}\kappa}
\leq g^{\frac{7}{4}}\|\mathcal{V}(\cdot\vert g)\|_{h,\kappa}
\nonumber\\
+\sum_{n=2}^{\infty}
\Big(L^{2-2\sigma}\,C_{2}\,C_{3}\Big)^{n-1}
\;\Big(g\,\|\mathcal{O}_{GN}\|_{h,\kappa}
+g^{\frac{7}{4}}\,\|\mathcal{V}(\cdot\vert g)\|_{h,\kappa}\Big)^{n}
\label{favorite}
\end{gather} 
When this estimate is plugged into (\ref{estimate2}), 
the factor $g^{-\frac{7}{4}}$ cancels, and 
\begin{gather}
\|\mathcal{T}_{L}(\mathcal{V})\|_{\gamma,h,\kappa}
\leq\,L^{-\frac{\epsilon}{2}}
\,\sup_{g\in[0,\gamma]}
\bigg\{
   \|\mathcal{V}(\cdot\vert g)\|_{h,\kappa}
\nonumber\\
   +\sum_{n=2}^{\infty}
   \Big(L^{2-2\sigma}\,C_{2}\,C_{3}\,g^{\frac{1}{4}}\Big)^{n-1}
   \;g^{\frac{3}{4}(n-2)}
   \;\Big(\|\mathcal{O}_{GN}\|_{h,\kappa}
   +g^{\frac{3}{4}}\,\|\mathcal{V}(\cdot\vert g)\|_{h,\kappa}\Big)^{n}
\bigg\}
\label{estimate3}
\end{gather}
We have chosen $h$ and $\kappa$ to depend on $L$. We now also
choose $\gamma$ to depend on $L$. We demand that $\gamma\leq 1$ 
be so small that 
\begin{equation}
L^{2-2\sigma}\,C_{2}\,C_{3}\,\gamma^{\frac{1}{4}}
\leq\frac{1}{2\,\|\mathcal{O}_{GN}\|_{h,\kappa}}.
\end{equation}
With this, we have the following estimate on 
the extended renormalization group
\begin{gather}
\|\mathcal{T}_{L}(\mathcal{V})\|_{\gamma,h,\kappa}
\leq\,L^{-\frac{\epsilon}{2}}
\left\{
   \|\mathcal{V}\|_{\gamma,h,\kappa}
   +2\,\|\mathcal{O}_{GN}\|_{h,\kappa}
   \;\sum_{n=2}^{\infty}
   \left(
      \frac{1}{2}
      +\frac{
%\Big(L^{-2\epsilon}\gamma\Big)^{\frac{3}{4}}
         \|\mathcal{V}\|_{\gamma,h,\kappa}}
         {2\,\|\mathcal{O}_{GN}\|_{h,\kappa}}
   \right)^{n}
\right\}.
\label{estimate4}
\end{gather}

\subsection{Invariant ball}
The only parameter which has not been fixed yet is $L$. 
This last parameter in our map can be 
used to find an invariant ball of second order perturbations. 
Let 
\begin{equation}
B=\left\{
   \mathcal{V}\in\mathbb{V}_{\gamma,h,\kappa}
   \bigg\vert 
   \|\mathcal{V}\|_{\gamma,h,\kappa}
   \leq\frac{\|\mathcal{O}_{GN}\|_{h,\kappa}}{2}
\right\}
\label{invariant}
\end{equation}
Let $L$ be so large such that
\begin{equation}
L^{-\frac{\epsilon}{2}}\leq\frac{1}{10}.
\label{largeLcondition}
\end{equation}
Then (\ref{estimate4}) implies that the ball of second 
order perturbations (\ref{invariant}) is invariant under
the extended renormalization group transformation 
$\mathcal{T}_{L}$.

\section{Contraction property}

The last property to be shown is that any pair of points
in the invariant ball move closer under an extended 
renormalization group transformation. 

\subsection{Estimates on $\beta_{L}$ and $S_{L}$}

The treatment of $\beta_{L}$ and $S_{L}$ remains the 
same as in the previous section. The result is 
\begin{gather}
\|\mathcal{T}_{L}(\mathcal{V}_{1})
-\mathcal{T}_{L}(\mathcal{V}_{2})\|_{\gamma,h,\kappa}
\nonumber\\
\leq L^{-\frac{\epsilon}{2}}\,
\sup_{g\in [0,\gamma]}
g^{-\frac{7}{4}}
\,\|\mathcal{F}_{L}(\mathcal{V}_{1})(\cdot\vert g)
-\mathcal{F}_{L}(\mathcal{V}_{2})(\cdot\vert g)
\|_{L^{-\frac{\epsilon}{4}}h,L^{-1}\kappa}
\label{starter}
\end{gather}

\subsection{Estimate on $\mathcal{F}_{L}$}

The difference on the right hand side of (\ref{starter})
leads to a cancellation of the $\mathcal{V}$-independent 
term, as is best seen from the formula 
\begin{gather}
\mathcal{F}_{L}(\mathcal{V}_{1})(\Psi\vert g)
-\mathcal{F}_{L}(\mathcal{V}_{2})(\Psi\vert g)
=g^{\frac{7}{4}}
\,\Big\langle
\mathcal{V}_{1}(\cdot\vert g)-\mathcal{V}_{2}(\cdot\vert g)
\Big\rangle_{C_{L}}(\Psi)
\nonumber\\
+\sum_{n=2}^{\infty}\frac{1}{(n-1)!}
\int_{0}^{1}\D s\;
\bigg\langle
\Big[
   g\,\mathcal{O}_{GN}
   +g^{\frac{7}{4}}\,\mathcal{V}_{2}
   +s\,g^{\frac{7}{4}}\,
   \Big(
      \mathcal{V}_{1}(\cdot\vert g)
      -\mathcal{V}_{2}(\cdot\vert g)
   \Big)
;\Big]^{n-1}
\nonumber\\
;g^{\frac{7}{4}}
\,\Big(\mathcal{V}_{1}(\cdot\vert g)-\mathcal{V}_{2}(\cdot\vert g)\Big)
\bigg\rangle^{T}_{C_{L}}(\Psi)
\label{formulaI}
\end{gather}
In complete analogy to (\ref{favorite}), we conclude that 
the following estimate holds
\begin{gather}
\|\mathcal{F}_{L}(\mathcal{V}_{1})(\Psi\vert g)
-\mathcal{F}_{L}(\mathcal{V}_{2})(\Psi\vert g)
\|_{L^{-\frac{\epsilon}{4}}h,L^{-1}\kappa}
\leq g^{\frac{7}{4}}
\;\|\mathcal{V}_{1}(\cdot\vert g)-\mathcal{V}_{2}(\cdot\vert g)
\|_{h,\kappa}
\nonumber\\
+g^{\frac{7}{4}}
\;\|\mathcal{V}_{1}(\cdot\vert g)-\mathcal{V}_{2}(\cdot\vert g)
\|_{h,\kappa}
\;\sum_{n=2}^{\infty}
\Big(L^{2-2\sigma}C_{2}C_{3}g^{\frac{1}{4}}\Big)^{n-1}
n\,g^{\frac{3}{4}(n-1)}
\nonumber\\
\int_{0}^{1}\D s\;
\Big(
   \|\mathcal{O}_{GN}\|_{h,\kappa}
   +g^{\frac{3}{4}}
   \,\|(1-s)\mathcal{V}_{1}(\cdot\vert g)
      +s\mathcal{V}_{2}(\cdot\vert g)\|_{h,\kappa}
\Big)^{n-1}
\label{proI}
\end{gather}
On top of (and consistent with) the above choices of $h$,
$\kappa$, $\gamma$, and $L$, we 
demand that $\gamma$ be so small that 
$n\gamma^{\frac{3}{4}(n-1)}\leq \left(\frac{4}{3}\right)^{2}$ 
for all $n\in\{2,3,4,\ldots\}$. 
Then we have that
\begin{gather}
\|\mathcal{T}_{L}(\mathcal{V}_{1})
-\mathcal{T}_{L}(\mathcal{V}_{2})\|_{\gamma,h,\kappa}
%\nonumber\\
\leq L^{-\frac{\epsilon}{2}}\,
\Bigg\{
1+\left(\frac{4}{3}\right)^{2}
\;\sum_{n=2}^{\infty}\left(\frac{3}{4}\right)^{n}
\Bigg\}
\;\|\mathcal{V}_{1}-\mathcal{V}_{2}\|_{\gamma,h,\kappa}
\label{proII}
\end{gather}
But $L^{-\frac{\epsilon}{2}}\leq\frac{1}{10}$, so 
(\ref{proII}) implies that
\begin{equation}
\|\mathcal{T}_{L}(\mathcal{V}_{1})
-\mathcal{T}_{L}(\mathcal{V}_{2})\|_{\gamma,h,\kappa}
\leq
\frac{1}{2}
\;\|\mathcal{V}_{1}-\mathcal{V}_{2}\|_{\gamma,h,\kappa}.
\label{proIII}
\end{equation}
Thus our extended RG %renormalization group
transformation $\mathcal{T}_{L}$ is indeed a contraction 
mapping on the ball $B$.

\section{Conclusions}

In this paper, we have constructed the renormalized 
Gross-Neveu trajectory as an invariant curve in the 
unstable manifold of the free field fixed point. 
We have chosen a parametrization, in which the 
renormalization group acts on the curve in a normal
form. The normal form of super-renormalizable models 
is a linear step $\beta$-function. It can be used
in models whose differential $\beta$-function
\begin{equation}
\dot{\beta}(g)=\partial_L\beta_L(g)\vert_{L=1}
=\beta_{1}g+\beta_{2}g^2+\beta_{3}g^{3}\cdots
\end{equation}
has a non-vanishing coefficient $\beta_{1}$
(and is regular enough for the first coefficent to be leading).
%aside of an appropriate regularity. 
In our model $\beta_1=-2\epsilon$. 

In the non-deformation limit $\epsilon=0$,
the model stays renormalizable due to the sign of the 
second order correction. 
Its construction is slightly different from the 
super-renormalizable case. 
The normal form of the differential $\beta$-function is 
now cubic, that is, $\dot{\beta}(g)=\beta_2 g^2+\beta_3 g^3$
with the well known universal constants $\beta_2$ 
and $\beta_3$. As $L$ is increased, the coupling 
flows logarithmically rather than powerlike. 
Therefore, we cannot extract inverse powers of $L$ 
from the flowing coupling. 
One deals with this situation in the usual way by
imposing renormalization conditions the non-irrelevant
vertices, namely the Gross-Neveu vertex and 
the wave function vertex. (The mass vertex is still 
forbidden.) 
But the infinite series of higher monomials in the fields $\psi$
can be treated exactly as in this paper.

An interesting extension of the present work 
would be to gain complete control of the 
renormalized trajectory all the way from the 
ultraviolet to the expected infrared fixed point. 
In our model, this would require control of the 
large coupling limit. 

\bigskip\noindent
{\large\bf Acknowledgements}

\bigskip\noindent
We thank the {\sl Forschungsinstitut f\"ur Mathematik} at 
the ETH Z\"urich for support. 

\appendix

\section{Propagator properties}

The decay properties follow immediately from the integral 
representation 
\begin{equation}
\mathbf{C}_{1,L} (x,a,\sigma;x',a',\sigma')
= \delta_{a,a'}\frac{\I}{4 \pi\,\Gamma(1+{\epsilon\over 2})}
\int_1^{L^2} \D \alpha \;\alpha^{{\epsilon\over 2}-1}
\sla{\partial}_{\sigma,\sigma'} \e^{-{(x-x')^2 \over 4\alpha}}.
\end{equation}
The Gram representation holds by the Fourier representation \Ref{fourC}
of $\mathbf{C}_{1,L}$: with the spectral decomposition  
\begin{equation}
\gamma_\mu = \sum_{\rho} 
\lambda_\rho |\mu,\rho \rangle \langle \mu,\rho |,
\end{equation}
we have
\begin{equation}
(\gamma_\mu)_{\sigma,\sigma'} = 
\sum_\rho \lambda_\rho \langle 
\sigma|\mu,\rho \rangle \langle \mu,\rho |, \sigma'\rangle.
\end{equation}
Thus
\begin{eqnarray}
\varphi_L(x,a,\sigma) (\rho,p,\mu,c) &=&
\delta_{a,c}
\;\e^{-\I px}
\;|\lambda_\rho p_\mu|^{1\over 2}
\;\langle\sigma\vert\mu\rho\rangle
f_L(p)
\\
\tilde\varphi_L(x',a',\sigma')(\rho,p,\mu,c) &=&
\delta_{c,a'}
\;\e^{-\I px'}
\;\frac{\lambda_\rho p_\mu}{|\lambda_\rho p_\mu|^{1\over 2}}
\;\langle\sigma^\prime\vert\mu\rho\rangle
f_L(p)
\end{eqnarray}
with 
\begin{equation}
f_L(p) = \left(
\frac{\hat\chi (p) - \hat \chi(Lp)}{|p|^{1+{\epsilon\over 2}}}
\right)^{1/2}.
\end{equation}
The norms of $\varphi$ and $\tilde\varphi$ are bounded by
\begin{equation}
(|\lambda_1|+|\lambda_2|) 
\int_{\mathbb{R}^2} \frac{\D^2 p}{|p|^{1+\epsilon}} 
(\hat\chi (p) - \hat \chi(L\,p))
\leq 2
\int_{\mathbb{R}^2} \frac{\D^2 p}{|p|^{1+\epsilon}} 
\hat\chi (p)
\end{equation}
Because $\hat \chi(p) \leq O(1) |p|^\epsilon \e^{-p^2}$ for 
$|p| \geq 1$, 
the integral converges at infinity. The integral over the unit disk
is finite for $\epsilon < 1$. 

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\end{thebibliography}

%\tableofcontents

\end{document}
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