Structural stability of the RG flow in the Gross-Neveu model

J<EMAIL_ADDRESS>and Cheng<EMAIL_ADDRESS>
Dept. of Mathematics, SUNY at Buffalo

Buffalo, NY 14260, USA

We study flow of renormalization group (RG) transformations for the  massless Gross-Neveu model in a non-perturbative formulation.   The model is defined  on a two dimensional Euclidean space with  a finite volume.
The quadratic approximation to the flow stays bounded after suitable renormalization.  We show that for weak coupling  this property
also is true for the complete flow.   As an application we prove an ultraviolet stability bound for the model.   Our treatment is an application of a method of Bauerschmidt, Brydges, and Slade.  The method was  developed for an
infrared problem, and is  now applied to an ultraviolet problem.

§ INTRODUCTION

In a quantum field theory model  a renormalization group transformation replaces the given action by an effective action on a larger length scale.
When iterated  one gets a flow of the effective actions which distill  information about  the large scale structure of model.    At each stage one identifies
an effective coupling constant   and a few other parameters like field strength  and mass.   The flow of the coupling constant is particularly important
as it  drives the other parameters.     In weakly coupled models in critical dimension
it has the form

g_k+1  = g_k  ±β_k g_k^2  + (g_k^3)  β_k >0
Structural stability is the statement that  this quadratic flow $g_{k+1}  = g_k  \pm \beta_k g_k^2$ , together with the quadratic flow of the other parameters,  is the
dominant  behavior  in complete  flow of the effective actions.  In particular with a minus   sign  the flow  is toward zero,  the effective actions become free,   and infrared (long distance) problems  become tractable.   With a plus sign the flow
is away from the origin and,  after renormalization to tiny initial values,  ultraviolet  (short distance)  problems  become tractable.

Bauerschmidt, Brydges, and Slade [5] have developed a method for proving structural stability in the infrared case.   They are
interested in an $n$-component  scalar field $\phi$ on a four dimensional  unit lattice with a $| \phi |^4$  interaction term [2] ,
as well as the self-avoiding random walk which is formally the case $n=0$ [3], [4].
In these  models  the minus sign occurs in (<ref>) they are able to use the structural stability result
to prove results on the bulk behavior of the model.

In this paper we show that this method  works  for an  ultraviolet problem  and in the continuum.   We consider  the massless Gross-Neveu model in dimension two.    It features
anti-commuting  fermi fields $\psi, \bar \psi$
with  $n$ internal  components.  The   action is
S(ψ) = ∫(  - + (1+z)   ψ̅ψ-  g  ( ψ̅ψ)^2    ) dx
with  coupling constant $g$,   field strength $(1+z)$, and energy density $\vep$.
We work in a finite volume and add a high  momentum cutoff  to provide a short distance  regularization of   the model.
We start with initial values   $g_0, z_0, \vep_0$,  depending on the cutoff, and make a sequence  of RG transformations.   After $k$-transformations we will
have a new effective action which will include terms $ \int ( - \vep_k + (1+z_k)  \ \bar \psi \slpa \psi     -  g_k \ ( \bar \psi \psi )^2   )  $  as well as
psuedo-scalar terms   $p_k \int     \ ( \bar \psi \ga_5 \psi )^2  $  and vector terms    $v_k   \int  \ ( \bar \psi \ga_{\mu}  \psi )^2  $, and also
an infinite dimensional piece $E_k$.     Each RG transfomation reduces the cutoff and we continue until we reach unit cutoff and an tame
expression.     The variables $g_k,z_k, \vep_k, p_k,v_k, E_k$  and    evolve as a nonlinear discrete dynamical system (see(<ref>).    We assume
$g_k$  and all other variables are small.      At first we ignore terms which are   $\cO(e_k^3)$  which includes the infinite dimensional piece $E_k$.
Then we have the quadratic equations

g_k+1   =      g_k + β_k g_k^2

z_k+1 =     z_k + θ_k g_k^2

with $\beta_k>0$ and bounded above and below and $\theta_k$ bounded.    It is straightforward to show that
one can choose initial values  $g_0,z_0$ such that  $g_k$ increases  and  takes any sufficiently small final value $g_f$
and $z_k = \cO( g_k)$.          Structural stability is the statement that this remains true for the full system.   That is for the
full flow  $g_k$ increases  and  takes any sufficiently small final value $g_f$  and the other variables     $z_k, \vep_k, p_k,v_k, E_k$
are bounded by some power of $g_k$.
In particular one has bounds on the effective actions uniform in the cutoff,  and this  gives a uniform
bound on the partition function
=   ∫e^-S(ψ)  D ψThe method could lead to a complete construction of the model which would involve   taking the limit as the cutoff is removed both in the partition function
and correlation functions,   and taking  the infinite volume limit.

Here is  some history of work on the model.  The model  was first considered by Mitter, Weisz  [23], and then taken up by
by Gross and Neveu  [16] who established dynamical mass generation for large $n$.      Rigorous treatments were given somewhat later  by Gawedzki and Kupiainen
[18]  and by Feldman, Magnen, Rivasseau, and Seneor [13].
In both  these cases the authors work directly in infinite volume and give a renormalization group analysis based on perturbation
theory to all orders.  They take  advantage of the fact the the perturbation theory for this model  is convergent.   Further results
along these lines  can be found in
[19],   [20], [21], [22]  and there  are also results  in d=3 [10].

Our non-perturbative   results are an alternative to   these perturbation theory methods.   We  mention some specific advantages.
(1.) At each stage of the iteration we have an exact expression for the effective action,   rather than a full power series expansion of the correlation functions.
The higher order  terms in the action  are treated as a polymer gas,   a general technique which works well   for other models.    The perturbation theory treatments are special to  Gross-Neveu, and do not generalize.
(3.)   We only have to consider Feynman diagrams of second order.    The perturbation theory  treatments consider Feynman diagrams of arbitrarily high order.

§ PRELIMINARIES

§.§ notation

We start in a finite volume  by working on the torus $
\bbT_M = \bbR^2/ L^M   \bbZ^2
with volume $L^{2M}$.  Here  $L$ is a fixed positive integer and $M$ is
the long distance cutoff. The fields are elements on a  Grassmann  algebra generated by  independent. elements
$\bpsi^i _a(x),\bpsi^i _a(x) $ where $x \in \bbT_M$,   the spinor index  is $a  = 1,2$,   and $i=1, \dots,  n$ is the internal index.
This  is an infinite dimensional Grassmann   algebra; we are more precise about its exact
definition in  section <ref>.

The partition function is interpreted as an  integral on the Grassmann algebra and has the form (up to an overall constant)
∫exp(  ∫[   - z ψ+ g ( ψ̅ψ)^2  )  ] )  dμ_G (ψ)
where bilinears have the form  $(\bpsi \psi )(x)  = \sum_{i=1}^{n}  \sum_{a = 1,2} \bpsi^i_{a} (x)\psi^i_a (x) $, etc.
With self-adjoint Dirac matrices satisfying  $\{ \ga_{\mu}, \ga_{\nu} \} = \de_{\mu \nu} $,
the Euclidean  Dirac operator  is   $\slpa = \ga_{\mu} \pa_{\nu}  = \ga_0 \pa_0 + \ga_1 \pa_1$.
Further  $\int [ \cdots ] d\mu_G$ is the Grassmann Gaussian integral
with covariance $G = \slpa ^{-1} $.
Formally the kernel $G(x-y) $ is defined by the Fourier series  [ See Appendix <ref> for our  Fourier series conventions]
G (x-y) =   L^-2M   ∑_ p  ∈_M^*  e^ip(x-y)  -i/p^2
where $\bbT_M^* = 2 \pi   L^{-M } \bbZ^2$.    More precisely if we include internal indices it is $G (x-y) \de_ {ij}$.

We regularize  this  by introducing  a smooth momentum cutoff  at approximately $|p| = L^N$
and we  also exclude $p=0$.    The modified covariance is

G (x-y) =   L^-2M   ∑_ p  ∈_M^*, p ≠0  e^ip(x-y)  -i/p^2    e^-   p^2/L^2N
is now well defined pointwise.   Our goal is to get uniform bounds  as $N \to \infty$.

It is convenient to scale up to the torus $\bbT_{N+M}$ with a unit momentum cutoff.
This is accomplished    replacing $\psi$ on $\bbT_{M} $ by    $\psi_{L^{-N}  }$  for $\psi$ on $\bbT_{M+N}$ where
ψ_ L^- N (x)   = L^ N/2   ψ( L^N x )   _ L^- N  (x)  =   L^ N/2   ( L^N x )
Then we find (up to an overall constant)
=  ∫e^S_0( ψ)   d μ_G_0   ( ψ)
The action becomes
S_0 =   ∫( _0  -     z_0 ψ+  g_0   ( ψ̅ψ)^2   )
with new parameters      $ \vep_0  = L^{-3N} \vep, z_0 = z,  g_0 = g$.
The covariance is now
G_0  (x-y) =   L^-2(M+N)    ∑_ p  ∈_M+N^*, p ≠0    e^ip(x-y) -i/p^2   e^- p^2
Then $N  \to \infty$ limit is now a long distance problem.   Instead of potential singularities as   $p \to \infty$,   we have potential singularities as $p \to 0$.

§.§ renormalization group

The renormalization group method  is a technique to find alternate  expressions for $\sZ$ on coarser momentum lattices,  thereby softening the
singularities.     The improvement comes in a sequence of steps.  After $k$ steps one has fields $\psi$  on $\bbT_{M+N -k} $   and the expression
= ∫e^S_k(ψ)  d μ_G_k ( ψ)
Here the new covariance is essentially the same as  $G_0$ and is given

G_k(x - y)  =    ∑'_p  ∈^*_N+M-k    e^ip(x-y)  -i/p^2    e^- p^2
The  primed sum is a weighted sum with $p\neq 0$   defined by

∑'_p  ∈^*_N+M-k[ ⋯] =    L^ -2(M+N-k)     ∑_p  ∈^*_N+M-k , p ≠0 [⋯]
The new    $S_k$  is yet to be specified,  but should have leading terms  similar to $S_0$.

Given such an expression on $\bbT_{M+N -k} $  we generate the expression on $\bbT_{M+N -k-1} $ as follows.
We split off the highest momentum piece of the covariance by
G_k  =   G_k+1,L   + C_k
where   $G_{k+1,L } (x, y)   = L^{-1} G_{k+1} (L^{-1}x, L^{-1} y )$.   Then we have

G_k+1,L  (x-y)  =      ∑' _ p ∈_M+N-k ^*  e^ip(x-y)  -i/p^2   e^- L^2p^2

C_k(x-y)  =      ∑' _ p  ∈_M+N-k ^*  e^ip(x-y)  -i/p^2  (  e^- p^2  -  e^- L^2p^2   )

The integral splits  as
=   ∫[  ∫e^   S_k ( ψ+ η) dμ_C_k (η) ]   d μ_G_k+1,L ( ψ)
The expression in brackets is the fluctuation integral,  and one searches for a new action $\tilde S_{k+1}$
such that
e^   S̃_k+1(ψ)      =   ∫e^  S_k ( ψ+ η) dμ_C_k (η)
Replace  $\psi(x)$ on   $\bbT_{M+N -k }$       by  $\psi_{L } (x) = L^{-\frac12} \psi(L^{-1} x)$ now with  $\psi$ on  $\bbT_{M+N - k -1}$.
Under this scaling $G_{ k+1,L } $ scales to $G_{k+1} $  and  $\tilde S_{k+1} $ scales to
$S_{k+1}(\psi)   = \tilde S_{k+1} ( \psi _L  ) $.
We have then
=    ∫e^  S_k+1 (ψ)   d μ_  G_k+1   ( ψ)
We are back to a unit momentum  cutoff,  but the volume has been reduced.  The issue is to keep track the effective actions $S_k$.

In the main text will  use a modification of the above in which the field strength is absorbed into
the measure before studying the fluctuation integral.

§.§ the Grassmann algebra

We give a more precise definition of the Grassmann algebras (see also  [15],  [8] ).
The algebras  are formally   generated by    elements  $\psi(\xi)$ satisfying    $\psi(\xi) \psi(\xi') =  -\psi(\xi') \psi(\xi)$.    Here the index   $\xi = (x, a, i, \om)$ consists of  a point
$x$  in  some   torus  $\bbT_j=  \bbR^2/  L^{j} \bbZ^2$, a Dirac index   $a$, an internal index   $1 \leq i \leq n$,   and
$\om =0,1$ which  distinguishes between $\psi, \bpsi$.   We have
ψ(x, a, i, 0 )   =  ψ_a, i (x)    ψ(x, a, i, 1 )  = _a, i (x)
A general element of the Grassman algebra is given by the formal expression

F =  ∑_ℓ=0^∞ 1/ℓ!  ∫F_ℓ ( ξ_1,  ⋯,  ξ_ℓ   ) ψ(ξ_1) , ⋯ψ( ξ_ℓ )  d ξand we assume that  the kernels $ F_{\ell} $ are anti-symmetric to match the anti-symmetry of $\psi(\xi_1) , \cdots  \psi( \xi_{\ell} ) $.  So we assume
$\textrm{Alt} F_{\ell} = F_{\ell}$
( Alt F)(ξ_1, …, ξ_n) = 1/n! ∑_π   π   F( ξ_π(1) , …, ξ_π( n))
and  the sum is over a permutations $\pi$  of $1,2, \dots, n$.

The kernels  $F_{\ell}$ are allowed to be distributions and we  assume they are in  a Banach space of distributions $\cC'$
to  be specified.    Then we define a norm  depending on a parameter $h>0$  by

F _h   = ∑_ℓ=0^∞ h^ℓ /ℓ!    F_ℓ _'
Let     $\cG_h$    be    Banach space  of  all formal expressions (<ref>) such that   $\|F\|_h $ is finite.

A product of two elements is formally

FG =   1/n!m!  ∫F_n(ξ_1, …, ξ_n   ) G_m(ξ_n+1 , …, ξ_m  )   ψ(ξ_1) , ⋯ψ( ξ_n+m )  d ξ

But $ \psi(\xi_1) , \cdots  \psi( \xi_{n+m} ) $ is anti-symmetric.  So if we  take account  that
<  H_1, AltH_2>    = < Alt H_1,H_2>
we can formally transfer an $\textrm{Alt}  $ operation from the generators to the kernel.    The precise  definition of the product is  then
FG  =∑_n,m   1/n!m!  ∫(   Alt(  F_n ⊗G_m) ) (ξ_1, …, ξ_n+m   )  ψ(ξ_1) , ⋯ψ( ξ_n+m  )  d ξ

The next result shows that   under certain conditions on the space $\cC'$  the space  $\cG_h $ is a Banach algebra.

Suppose that
Alt(  F_n ⊗G_m))_'   ≤F_n_'  G_m_'

Then if  $F \in \cG_h$  and $G \in \cG_h$ then $FG  \in \cG_h$ and $\| FG \|_h \leq \| F\|_h \|G\|_h $

Let $H  = FG$.
H  =∑_ℓ  1/ℓ!   ∫H_ ℓ  (ξ_1, …, ξ_ℓ  )  ψ(ξ_1) , ⋯ψ( ξ_ℓ  )  d ξ

H_ℓ   = ∑_n + m = ℓ  ℓ!/n!m!       Alt(  F_n ⊗G_m)

H_ℓ _' ≤     ∑_n + m = ℓ  ℓ!/n!m!       Alt(  F_n ⊗G_m) _'
≤∑_n + m = ℓ  ℓ!/n!m!         F_n _'   G_m _'

and  we have the announced

H _h   =
∑_ℓ h^ℓ/ ℓ!    H_ℓ _'    ≤∑_n,ml   h^n+m /n!m!         F_n _'   G_m _'    =  F_hG_h
Now we define the space  $\cC' = \cC'( \bbT_j ^{\ell } )  $.     It is the dual space to a Banach space $\cC = \cC( \bbT_j ^{\ell } )  $ defined as
all functions   $f$   on    $\bbT_j ^{\ell } =  \bbT_j  \times \cdots \times \bbT_j  $  such that   $f \in \cC^3(\bbT_j)$  in each of the $\ell $ variables
separately.  With    $\al = (\al_1, \dots \al_{\ell} )$  and  $\xi = ( \xi_1 \cdots, \xi_{\ell} ) $
the  norm is
f _   =    sup_ ξ, |(  ^   f)( ξ)   |
Here  the multi-index  $\al$  has entries $\al_i = (\al_{i,0}, \al_{i,1})$  with   $|\al_i| \equiv  \al_{i,0} +  \al_{i,1} \leq 3 $,
^ =   ^_1_x_1  ⋯^_ℓ_  x_ℓ     ^_i_x_i = ^_i,0 _x_i,0 ^_i,1 _x_i,1
Then  $\cC' $ is all continuous linear functionals on $\cC$ with norm
F _'  = sup_ f_  ≤1   |< F , f > |
Now we show that this satisfies the conditions of lemma 1.

Alt(  F_n ⊗G_m))_'   ≤F_n_'  G_m_'

We need to estimate
Alt(  F_n ⊗G_m)_'   = sup_ f_  ≤1|  Alt(  F_n ⊗G_m), f |
We have

Alt(  F_n ⊗G_m), f =    ( F_n ⊗G_m),Alt f

=      ∫F_n(ξ_1, …ξ_n)G_m(  η_1, …, η_m )  Alt f(ξ_1, …ξ_n, η_1, …, η_m ) d ξd η

=      ∫F_n(ξ_1, …ξ_n)G^*_m( ξ_1, …ξ_n) d ξ

G^*( ξ_1, …ξ_n) =∫G_m(  η_1, …, η_m )  Alt  f(ξ_1, …ξ_n, η_1, …, η_m )d ηThis gives
| Alt(  F_n ⊗G_m), f |  ≤F_n   G^* _
All this   makes sense because  $G^*$ is in fact  an element of $\cC(\bbT_k^n)$ with derivatives acting on the test function. To establish this we would need to show, say for $n=1$ and with $\textrm{Alt} f$ replaced by $f$,  the convergence of  difference quotients
h^-1 (  ^ _η f( ξ+ he _μ, η) -   ^ _ηf(ξ, η) )    →
_ξ_μ   ^ _ηf (ξ, η)
uniformly in $\eta$.  This follows by the uniform continuity of $ \pa_{\xi_{\mu} }  \pa^{\al} _{\eta}G^*(\xi, \eta  )$.
Now we have

^ _ξ G^*_m( ξ_1, …ξ_n) =     ∫G_m(  η_1, …, η_m ) (^ _ξ  Alt   f)  (ξ_1, …ξ_n, η_1, …, η_m )d η

=    G_m, ( ^ _ξ  Alt   f)  (ξ_1, …ξ_n,  ·)

which gives for $\|f\|_{\cC} \leq 1$

| ^ _ξ G^*_m( ξ_1, …ξ_n)|  ≤
G_m  _'     sup_ β, η |  ( ^β _η ^ _ξ  Alt f)  (ξ_1, …ξ_n, η_1, …, η_m ) |

≤        G_m   _'    Alt f   _   ≤  G_m  _'     f   _     ≤G_m_'

Hence    $  \| G^* \|_{\cC}   \leq \|  G_m  \|_{\cC'}      $ and so
\|  \textrm{Alt}(  F_n \otimes G_m))\|_{\cC'}   \leq  \|F_n\| _{\cC'} \|G_m\|_{\cC'}

Suppose the $F$ is presented in the form (<ref>), but with kernels $F_{\ell}(\xi_1, \dots , \xi_{\ell} ) $  which are not necessarily anti-symmetric.
In this case, as in the discussion of  products,  the interpretation is that it is the element with kernels   $\textrm{Alt} F_{\ell} $
and  the norm is
F_h =  ∑_ℓ h^ℓ /ℓ!  Alt F_ℓ _'
| < Alt F_ℓ, f>| = | <  F_ℓ,   Altf>|  ≤ F_ℓ _'  Alt f _  ≤ F_ℓ _'   f _
says that   $\|\textrm{Alt} F_{\ell} \|_{\cC'}  \leq    \|  F_{\ell} \|_{\cC'} $.  Thus we can estimate the norm with the original kernel as
F_h ≤∑_ℓ h^ℓ /ℓ!    F_ℓ _'

It will also  be convenient to use a representation which distinguishes $\psi, \bpsi$.  In the following
Dirac and internal indices  are suppressed.

Every element of $\cG_h$ can be written in the form

F =   ∑_n, m  1/n! m!   ∫F_nm (  x_1,…, x_n ,  y_1 ,  …,   y_m  )
ψ( x_1) , ⋯ψ( x_n)         ( y_1) , ⋯( y _m)                   d x dy
with $ F_{nm}$ anti-symmetric in  the $x_i$ and $y_i$ separately.
The norm is

F _h  =   ∑_n, m  h^n+m /n! m!     F_nm _'

Start with the representation (<ref>) and  write it as

F =  ∑_ℓ=0^∞ 1/ℓ!∑__1, …, _ℓ   ∫F_ℓ (_1, x_1,  ⋯, _ℓ ,  x_ℓ     )
ψ( _1, x_1) , ⋯ψ(   _ℓ,  x_ℓ  )  d x
Classify the terms in the sum over the $\om_i$ by the number of $0$'s that occur.  Then move all the zeros to the left both in the kernel and in the fields.  There is no net change in sign.   A term with $n$ zeros and $m$ ones
can occur in  $\ell!/n!m!$  different way corresponding to the $m$ different positions out of the original $\ell$ .     Thus we have

F =       ∑_ℓ=0^∞ 1/ℓ!∑_n+m = ℓ  ℓ!/n! m!   ∫F_ℓ ( (0, x_1),…, (0,x_n), ( 1,y_1) ,  ⋯,  (1,  y_m )     )

ψ( x_1) , ⋯ψ( x_n)         ( y_1) , ⋯( y _m)                   d x dy

If we now define
F_nm (  x_1,…, x_n ,  y_1 ,  …,   y_m  )
= F_n+m ( (0, x_1),…, (0,x_n), ( 1,y_1) ,  ⋯,  (1,  y_m )     )
and   collapse the sum over $\ell,m,n$ to just a sum over $m,n$ we have the expression (<ref>).
The analysis of the norm is entirely similiar.

§.§ localized  functions and norms

We cover  any torus $\bbT_j = \bbR^2/ L^j \bbZ^2$  into closed  unit blocks  (squares) $\sq$  centered on the points of  $ \bbZ^2/ L^j \bbZ^2$.      We consider paved sets   $X$  defined to be  unions of such  blocks.   They are not necessarily connected so we refrain from calling them polymers,  although the terminology is sometimes used.

We  consider elements  of the Grassmann algebra  $\cG_h$  which have the

E(ψ)  =  ∑_X E(X, ψ)
where  $E(X, \psi)$ only depends on $\psi$  only  in $X$.
With a weight function $\Ga(X)$ we   define a norm on the family  $E = \{  E(X)  \} $    as in  [9],  [12], [6],  by
E _h,    = ∑_X ⊃    E(X ) _h   (X)
Both $  E(X )$  and  $  \Ga(X) $ will be invariant under translations by elements of  $\bbZ^2$,  so this is independent of the unit block $\sq$.
The space of all $E(X, \psi)$ such that $\|E\|_{h, \Ga}< \infty $ is a Banach space denoted $\cG_{h, \Ga} $.

The weight   factor $\Ga(X)$  is take to have the form

(X) = A^|X|  Θ(X)
Here where $|X| $ is the number of  unit  squares in  the paved set  $X$, so $|X|$ is the volume of $X$.
In $A^{|X|} $ the $A$ is a sufficiently large constant depending on $L$,  say $A = L^{d+2} =L^4$.  Finally
$\Theta(X)$ has the form
Θ(X)  = inf_T ∏_ℓ∈T  θ(| ℓ| )
where the infimum  is over all tree graphs  $T$ on the centers of the blocks in $X$.  The  $\ell$ are the lines in the graph
and   $|\ell |$ is the length in an $\ell^{\infty}$ norm.
The factors $\theta(|\ell|) $ have a power law rate of increase.    Furthermore $\theta$ can be chosen so that $\theta(s/L)  \leq L^{-d-1} \theta(s) = L^{-3} \theta(s)$
and $\theta(0) = \theta(1) =1$.

The weight factor $\Ga(X) $ satisfies  $\Ga(X) \geq 1$ and

(X ∪Y) ≤(X) (Y) θ( d(X,Y) )
We also will use  a modification defining for positive integer $n$
_n(X)  = e^n|X| (X)
This also satisfies (<ref>).
[ $\Ga_n(X) $ was originally defined with $2^{n|X|} $ rather than  $e^{n|X|}$, an insignifacant difference]

A paved set $X$ is defined to be   small  if it is connected and satisfies  $|X| \leq  2^d = 4$.  Otherwise it is a large set.   let $\bar X$ be the smallest union of
$L$-blocks containing $X$.    Then  one   can show that for $p,q>0$ there are constants $c_{p,q}= \one $ such that

_q(L^-1 X̅ )
c_p,q    _q-p  (X)       X  small

c_p,q   L^-d-1 _q-p  (X)       X  large

Here are some elementary examples of localized functions

* If   $ E =  \int (\bpsi \psi)^2 $, then $ E= \sum_X E(X) $ with  $\| E\|_{h, \Ga}  \leq \frac14 Ah^4$.

* If    $ E =  \int  \bpsi \slpa \psi $  then $ E= \sum_X E(X) $ with  $\| E\|_{h, \Ga}  \leq   \frac12 Ah^2$.

Define  $E(X) = E(\sq) = \int_{\sq}  (\bpsi \psi)^2 $ if $X$ is a single block $\sq$,  and $E(X) =0$ otherwise.   Then taking account the remark after lemma <ref> we have
E_2,2 ()_ ' =  sup_f_  ≤1 |  ∫_   f(x,x,x,x) dx |   ≤1
This is the only contribution to $\|E(\sq) \|_h = \frac14 h^4 \| E_{2,2} (\sq)\|_{ \cC'} = \frac14 h^4$,
E _h,   = E() _h ()    =  1/4 h^4 A
The second case is similar,  but now
E_1,1 ()_ ' =  sup_f_  ≤1 |  ∫_  (_x  f) (x,x) dx |   ≤1

§.§ symmetries

Our localized functions $E(X, \psi)= E(X, \bpsi, \psi)$  expressed in the form (<ref>) will always have equal numbers
of $\psi$'s and $\bpsi$'s.   In addition they will have the following symmetries which are all symmetries of the massless Dirac operator.

* (torus symmetries) Let   $R$ is a    rotation by a multiple of $\frac{\pi}{2}$ or a   reflection and  let $a$ be  a  point in the torus.
Let  ${a,R} $ be  the   symmetry   $x \to Rx + a$.    Then there is  a matrix  $S$ in the group $Pin(2)$ such that    $S^{-1}  \ga_{ \mu} S = \sum_{\nu}  R_{\mu \nu} \ga_{\nu} $.  The fields   $\psi, \bpsi$ transform respectively  to    $ \psi_{R,a}  (x)  = S  \psi(R^{-1} ( x-a ) ) $  and $\bpsi_{R,a} (x)  =  \bpsi(R^{-1} ( x-a ) ) S^{-1} $.   We require  that
E(X, , ψ) = E(RX+a, _R,a, ψ_R,a )

* (internal rotations)    For $u \in U(n) $  and $(u\psi)_i = \sum_j u_{ij} \psi_j$
E(X,  , ψ) = E(X,   u^* , u  ψ)

* (chiral transformations)
For $\ga_5 = i \ga_0 \ga_1$
E(X,  , ψ) = E(X,   (i_5) , (i_5) ψ)

§.§ scaling

We  recall some general terminology about   the scaling  operation $\psi(x) \to \psi_L(x) = L^{-\frac12} \psi(x/L)$.    Terms  that increase like   $\int \bpsi \psi $
are called relevant.   Terms  that are invariant like   $\int (\bpsi  \psi)^2 $ and   $\int \bpsi \pa \psi $ are called marginal.    Terms that decrease like $\int  (\bpsi \psi )^6$
and $\int \pa \bpsi \pa \psi $  are called

We want to develop a similar classification for our localized functions.
If   $E(\psi)  = \sum_X E(X, \psi )$ is expressed as   a sum over paved sets  on   a torus  $\bbT_j=\bbR^2/ L^j \bbZ^2$,
we first reblock to  $L $-paved sets  defined to be unions of $L$-blocks centered on points of $L\bbZ^2/ L^j \bbZ^2$.  If $X'$ is
an $L$-block then
(E) (X', ψ)  = ∑_X: X̅ =  X' E(X, ψ)
Here  the sum is over unit paved sets  $X$ in $\bbT_j$
such that the smallest  $L$-paved set  $\bar  X$ containing $X$  is equal to $X' $.
Then we scale down to the torus $\bbT_{j-1}$  defining for unit  paved sets   $Y$

(E)(Y, ψ)  =   (E)( LY, ψ_L)  =   ∑_X: X̅ =LY    E(X, ψ_L )

We    want to remove local relevant and marginal parts from our localized functions and  leave irrelevant parts.
To do this we will need a workable criterion to identify irrelevant pieces.
We use the following condition  introduced in [7].

$E(X, \psi)$ is  irrelevant (normalized)  if

E_0(X)  =     0

E_1,1(X,1) = ∫E_1,1(X,x,y)   dx dy  =     0

E_1,1(X,x-x_0)=  ∫E_1,1(X, x, y) (x-x_0)  dx dy   =     0

E_2,2(X,1) =  ∫E_2,2(X,x_1,x_2,y_1,y_2 ) dx dy     =     0

The third condition is independent of the point $x_0$ which we take to  be in $X$.

For $\psi$ on $\bbT_{j-1} $ and $\psi_L $  on $\bbT_j$, let $E'( X, \psi) = E(X, \psi_L)$.
E'(X) _h ≤ E(X) _L^-1/2h  ≤ E(X) _h
Furthermore if  $X$ is  small    and $E(X) $ is normalized then

E'(X) _h ≤L^-3  E(X) _h

(as in [7])
For $f \in \cC(\bbT^{n+m} _{j-1}) $ we have
<  E'_nm (X ), f > = <E_nm (X) , f_L>
f_L (x_1, …, x_n , y_1, …, y_m ) = L^-1/2(n+m)  f ( x_1/L,  …,  x_n/L,  y_1/L,  …,  y_m/L  )
Then  $ \| f _L\|_{\cC}  \leq  L^{-\frac12(n+m) }  \|f  \|_{\cC} $ and so

E'_nm(X ) _'    =       sup_ f _   ≤1  |E_nm (X ), f_L  |

≤      sup_ f _    ≤1   E_nm (X )  _'     f_L_

≤       L^-1/2(n+m)  E_nm (X ) _'

Now multiply by  $h^{n+m}/n!m! $ and sum over $n,m$ to get the first result.

For the second result    note that for $n+m \geq 6$ we have  $ L^{-\frac12(n+m) }  \leq L^{-3}$ so (<ref>) is sufficient for these terms.
It suffices to consider  $n+m = 2,4$.   In analyzing these terms we are going to make use of identities like

ψ(x) =     ψ(  x_0 )  + (R_1,x_0  ψ) (  x  )
(R_1,x_0  ψ) ( x)  =   ∫_0^1  (x-x_0) ·( ψ)   ( x_0 + t( x-x_0) ) dt
The meaning of this manipulation of formal symbols  is the corresponding operation  on test functions.
For example if  $E(\psi)  = \int E(x) \psi(x) dx $  then making the replacement (<ref>) means we
are writing the distribution $<E,f>$ as the sum
<E,f> = <E, f(x_0) > + <E,  R_1,x_0 f>

For $n+m = 4$ we consider

E'_2,2 (X, ψ) =   E_2,2 (X, ψ_L ) = ∫E_4(X, x_1, x_2,y_1,y_2)  ψ_L(x_1)   ψ_L(x_2)   _L(y_1)   _L(y_2)
Pick a point $x_0$ in $X$ and make the replacement $  \psi_L(x_i) =    \psi_L (  x_0 )  + (R_{1,x_0}  \psi_L) (  x_i  ) $ and similarly
for $\bpsi$.
The term with all   the fields at $x_0$  is proportional to   $ \int E_{2,2}(X)$ and so vanishes by the normalization condition.
The other terms have at least one remainder term.

Consider for example

∫E_2,2 (X,x_1,x_2,y_1,y_2 )   (R_1,x_0  ψ_L) (  x_1)         ψ_L(x_0)   _L(x_0)   _L(x_0)
This refers to the distribution   $<E_{2,2} (X), f^{\star}   > $  with norm

sup_  f_   ≤1 |<E_2,2 (X), f^⋆   > |

f^⋆ (x_1, x_2,y_1,y_2 )  =     ( R_1,x_0 f_L ) (x_1, x_0,x_0 x_0)

=      ∫_0^1  (x_1-x_0)_μ   / x_1, μ   f_L( x_0 + t( x_1-x_0)  , x_0 ,  x_0,  x_0 ) dt

=    L^-3   ∫_0^1  (x_1-x_0)_μ    f/ x_1, μ  ( x_0 + t( x_1-x_0) /L , x_0/L ,  x_0/L, x_0/L ) dt

Here $f_L$ supplies the $L^{-2} $ and the derivative gives the extra $L^{-1} $.
But $|x_1-x_0| \leq 4$ thanks to our restriction to $X$ small and built from unit squares.  Thus we have
$| f^{\star}|   \leq \one L^{-3} \| f\|_{\cC} $.
Similarly we claim  that  for $| \al | \leq 3$
|^_ x_1  f^⋆  | ≤L^-3  f_
We use the identity $f_L (  x_1)  = f_L  ( x_0 )  + (R_{1,x_0}f_L) (  x_1  ) $
to change
$ \pa^{\al}_ {x_1} ( R_{1,x_0} f_L ) ( x_1, ...) $ to just   $\pa^{\al}_{x_1} f_L ( x_1, \dots)$.
Again the derivatives supply extra factors of $L^{-1}$.
Altogether then we have
f^⋆ _  ≤L^-3  f_
and hence  the norm  (<ref>) is bounded by
$\one  L^{-3} \| E_{2,2} (X) \|_{\cC'} $.  The other contributions to $E'_{2,2} (X)$ are similar and so
E'_2,2 (X) _ '   ≤L^-3  E_2,2 (X) _'
which suffices for (<ref>)

For $n+m =2$ we consider
E_1,1(X, ψ_L ) = ∫E_1,1(X, x,y  )  ψ_L(x)     _L(y)
Again  pick a point $x_0$ in $X$ and  now  make the replacement

ψ_L ( x)  =  ψ_L ( x_0 )  +  (x -x_0)_μ _μ    ψ_L ( x_0 )   +    (R_2,x_0  ψ_L) (  x )
and similarly for $\bpsi$.

(R_2,x_0  ψ_L) (  x_i )   =        ∫_0^1  (1-t) (x-x_0)_μ (x-x_0)_ν ( _μ _ν ψ_L )   ( x_0 + t( x-x_0) ) dt

The two leading terms here generate expressions which are seen to be proportional to $\int E_{2}(X,x,y) dxdy $ and
$\int E_{2}(X,x,y) (x-x_0) dxdy $ and so vanish by our normalization condition.    The surviving terms have at least one factor
$(R_{2,x_0}  \psi_L) (  x_i )  $.

Let us consider for example
∫E_1,1(X, x, y)  (R_2,x_0  ψ_L) (  x )        _L(x_0)
This has norm

sup_  f_   ≤1 |<E_1,1(X), f^⋆   > |
where now

f^⋆ (x, y )  =     ( R_2,x_0 f_L ) (x, x_0 )

=    ∫_0^1  (1-t) (x-x_0)_μ (x-x_0)_ν _μ _ν  f_L  ( x_0 + t( x-x_0),x_0 ) dt

=    L^-3 ∫_0^1  (1-t) (x-x_0)_μ (x-x_0)_ν_μ _ν  f  ( x_0 + t( x-x_0)/L, x_0/L  ) dt

Again we have
$| f^{\star}|   \leq \one L^{-3} \| f\|_{\cC} $ since the $f_L$ supplies a factor $L^{-1}$ and the derivatives give a factor $L^{-2}$.
Similarly we claim  that  for $ 1 \leq | \al| \leq 3$
|^_x f^⋆  | ≤L^-3  f_
This follows for $|\al | =1$ from the above representation.  For $|\al| = 2,3$  we use
(_ ν R_2,x_0 f_L) (  x, x_0 )  =   _ν   f_L ( x,x_0  )  - _ν  f_L  ( x_0,x_0 )  = ( R_1,x_0 _ν f_L ) (x, x_0 )
and so
(_μ _ν R_2,x_0  f_L ) (  x,x_0  )  =   ( _μ  _ν f_L) (x,x_0 )
Again the extra derivatives supply the $L^{-2}$.
Altogether  we have   $\| f^{\star} \|_{\cC}  \leq   \one L^{-3} \| f\|_{\cC} $
and hence  the norm  (<ref>) is bounded by   $\one  L^{-3} \| E_{1,1}(X) \|_{\cC'} $.  The other contributions to $E'_{1,1} (X)$ are similar and so
E'_1,1 (X) _'    ≤L^-3  E_1,1(X) _'
which suffices for (<ref>), and  completes the proof.
For  $p,q  \geq 0 $ and any family of localized functions $E(X)$
E _h, _q   ≤L^2 E_L^-1/2  h, _q-p
If  $E(X) $ is normalized for small sets this improves to
E _h, _q  ≤L^-1 E_h, _q-p

With $  E'(X, \psi )  = E(X, \psi_L ) $ we have
$ (\cL E)(Y) =  \sum_{X: \bar X =LY }   E' (X) $ and
E _h ≤∑_Y ⊃  ∑_X: X̅ =LY    E' (X) _h  (Y)

∑_Y⊃    ∑_X: X̅ =LY   = ∑_X: X̅ ⊃L   ≤∑__0 ⊂L ∑_X ⊃_0  =L^2  ∑_X ⊃_0

E _h, _q  ≤L^2  ∑_X⊃_0    E' (X) _h  _q(L^-1 X̅)
For general  $E(X)$ we have $\| E' (X) \|_h \leq \| E (X) \|_{L^{-1/2} h}$ from lemma <ref>,
and  $\Ga_q (L^{-1} \bar X)  \leq \one  \Ga_{q-p}( X) $ from (<ref>),  so the right side is bounded by $\one  L^2\|E\|_{L^{-1/2} h, \Ga_{q-p} } $
as announced.

In the second case  if $X$ is small and $E(X)$ is normalized we use  $\| E'(X) \|_h \leq   \one L^{-3} \| E(X) \|_{h}$ from lemma <ref>
and  again   $\Ga_q (L^{-1} \bar X)  \leq \one  \Ga_{q-p}( X) $.
If $X$ is large we use  $\| E'(X) \|_h \leq    \| E(X) \|_h$ and   $\Ga_q (L^{-1} \bar X)  \leq \one L^{-3}  \Ga_{q-p}( X) $  from (<ref>).
In either case we have
E' (X) _h  _q(L^-1 X̅) ≤L^-3  E(X) _h    _q-p (X)
Insert this bound in (<ref>) and get the announced  $\one L^{-1} \|E\|_{h, \Ga_{q-p} } $

§.§ Gaussian integrals

We study Gaussian integrals on our  Grassmann algebra.    We study particularly  the fluctuation integrals after $k$ renormalization group steps,  although
much of what follows is more general.   The torus is   $\bbT_{N+M-k} $  and the covariance is
C_k (x-y)  = ∑' _  p ∈^*_N+M-k   C̃_k ( p)    C̃_k ( p)  =   -i/ p^2(  e^-p^2 - e^L^2p^2 )
The   Gaussian integral with   covariance $C_k(x-y)$ is a linear function on the Grassman algebra   defined by

∫  ψ_a_1 (x_1) _b_1(y_1)  ⋯ψ_a_n(  x_n )  _b_n(y_n )    d μ_C  (ψ, )
=         { (C_k)_a_i b_j (x_i - y_j)}         n=m
0     n≠m

where  $ \{ ( C_k)_{a_i b_j} (x_i - y_j)\  \} $ is  a  $2n \times 2n$ matrix.   More precisely this is the integral if all internal indices are
the same.   In the general case the integral factors into a product of this type.  This definition is consistent with the more familiar
finite dimensional case.

We also have the identity
ψ_a_1 (x_1) _b_1(y_1)  ⋯ψ_a_n(  x_n )  _b_n(y_n )= _n ψ_a_1 (x_1)  ⋯ψ_a_n(  x_n )     _b_1(y_1) ⋯_b_n(y_n )
where $\ep_n = \pm 1$  is defined by $\ep_2= -1$ and $\ep_{n+1} = (-1)^n \ep_n$.   Thus in terms of
our standard representation for $n=m$

∫ ψ_a_1 (x_1)  ⋯ψ_a_n(  x_n )     _b_1(y_1) ⋯_b_n(y_n )    d μ_C  (ψ, )
=   _n   { (C_k)_a_i b_j (x_i - y_j)}
In the following let $C=C_k$ and suppress Dirac indices.

We   split the covariance  as $\tilde C =\tilde  C_1 \tilde C_2$ with
C̃_1(p) = e^-1/2 p^2   C̃_2(p) =  -i/ p^2 (  e^- 1/2  p^2 - e^-(L^2- 1/2)p^2 )
Then we have

C(x-y) =∫C_1(x- z) C_2(z-y) dz  = (C_1( x- ·) , C_2(·- y))
It follows by Gram's inequality [15]  that
| { C(x_i- y_j)}    | ≤∏_i=1^n    C_1( x_i- ·)_2   C_2(·- y_j )_2 =   C_1_2^n   C_2^n_2
Here $  \|C_1\|_2, \|C_2\|_2 $ are both finite and $\cO(L^2)$.
The same is true for derivatives and we define

h(C)  = sup_ || ≤3{   ^ C_1 _2,   ^ C_2 _2 }

For $F \in  \cG_{ h(C)} $
| ∫F      d μ_C    |
≤  F _h(C)

Take  $F$ of  the form  (<ref>), and then by (<ref>) we have
∫F      d μ_C
=∑_n=m 1/n!m!   ∫F_nm (x_1,   …, x_n, y_1,  …,   y_m )   _n { C(x_i- y_j) }
The integral here  has the form  $<F_{nm}, f > $  with $n=m$ and
f(x_1,   …, x_n,y_1, ⋯y_m)  = _n   { C(x_i-  y_j) }
Then  with $\pa^{\al}_x = \pa^{\al_1}_{x_1}  \cdots   \pa^{\al_n}_{x_n} $

^_x ^β_y f(x_1,   …, x_n,y_1, ⋯y_n) =      {  ^_i_x_i  ^β_j_y_j    C(x_i-  y_j) }

=     {  ( ^_i C_1( x_i- ·) , (-) ^β_jC_2(·- y_j)) }

Then by Gram's inequality

f _=      sup_, β, x,y | ( ^_x ^β_y f)  (x_1,   …, x_n,y_1, ⋯y_m) |

≤     sup_ |_i|, |β_j| ≤3     ∏_i=1^n    ^_i  C_1 _2   ∏_j=1^m     ^β_jC_2_2   ≤h(C) ^n+m

$|<F_{nm}, f >| \leq   \|F_{nm} \|_{\cC'}  h(C)  ^{n+m}   $  and the result follows.

We also  need to consider integrals in which only some of the  generators are integrated out.
Suppose the Grassmann algebra has two sets of generators $ \psi, \bpsi $ and $ \eta, \bar \eta $.
We consider elements  of the form

F(  ψ, , η, η̅)

=  ∑_n= m,n' =m'    1/n!m!n'!m' !  ∫F_nm,n'm' ( x, y,  x', y' )
ψ(x)  ( y   )   η(x')  η̅( y'   )     d  x d y  d  x' d y'

where $x =(x_1, \dots, x_n)$, $\psi(x)  =\psi(x_1) \cdots \psi(x_n) $, etc.
The associated norm is
F _h,h'

=  ∑_n= m,n' =m'    h^n+m h'^n'+m' /n!m!n'!m' !    F_nm,n'm'   _'

The integral
F^*(   ψ, )  = ∫F (  ψ, , η, η̅)  d μ_C(η, η̅)
F^*_h  ≤ F_h, h(C)

The integral is evaluated as
F^*( ψ) =    ∑_n=  m    1/n!m!   ∫F^*_nm ( x, y )  ψ(x)  ( y   )      d  x d y
F^*_ nm ( x, y )   =   ∑_n'=m'    1/ n'!m' !  ∫F_nm,n'm' ( x, y,  x', y' )  _n (   { C( x'_ i  - y'_j)  }  )     d  x' d y'
Then with $g( x', y' ) =  \ep_n \det \B   \{ C( x'_ i  - y'_j) \B \} $
F^*_ nm , f =  ∑_n'=m'    1/ n'!m' !   F_nm, n'm' ,  f ⊗g It is straightforward that  $  \| f \otimes g \|_{ \cC}  \leq \| f \|_{ \cC}  \|   g \|_{ \cC}  $ and
again by Gram's inequality    we have  $\|g \|_{\cC}  \leq h(C) ^{n'+m'} $.
F^*_'  =  sup_ f_ ≤1  |<   F^*_ nm , f >   |    ≤∑_n'=m'     h(C) ^n'+m' / n'!m' !      F_nm,n'm'  _'
Use this in
F^*_h =   ∑_n= m    h^n+m/n!m!       F^*_nm _'
and we have the result.

The following result  is also useful

If  $ F^+ ( \psi, \eta)  = F( \psi + \eta ) $
F^+  _h, h'  =  F _h+h'

We use the representation (<ref>) which does  not distinguish $\psi, \bpsi$.  Then we have

F^+ (ψ,  η)  =   ∑_ℓ 1/ℓ! ∫F_ℓ (  ξ_1, …, ξ_ℓ  ) (ψ(ξ_1) + η( ξ_1 ) )  ⋯(ψ(ξ_ℓ ) + η( ξ_ℓ ) )d ξ

=   ∑_ℓ 1/ℓ! ∑_n+m = ℓ ℓ!/n!m! ∫F_ℓ (  ξ_1, …, ξ_n,  ζ_1, …, ζ_m  )
ψ(ξ_1)  ⋯ψ(ξ_n)) η( ζ_1 )  ⋯η( ζ_m ) d ξd η

= ∑_n, m 1/n!m! ∫F_n+m (  ξ_1, …, ξ_n, ζ_1, …, ζ_m )
ψ(ξ_1)  ⋯ψ(ξ_n) η( ζ_1 )  ⋯η( ζ_m )   d ξd η

Here in the first line we multiply out the product  and in each of the  $2^{\ell}$ terms  and move all the $\psi(\xi_i) $ to the left.   Make the same permutation in the  kernel and the signs cancel.  Relabel the $\eta(\xi) $ as $\eta(\zeta) $. Terms with $n$   $\psi$-fields and $m$  $\eta$-fields all give the same contribution. There are $\ell !/ n! m!$ of these
corresponding to the number of start positions.  This  gives the second line.
From the third line we identify
F^+ _n,m  (  ξ_1, …, ξ_n, ζ_1, …, ζ_m ) =F_n+m (  ξ_1, …, ξ_n, ζ_1, …, ζ_m )

F^+  _h,h'  =        ∑_n, m h^n(h') ^m/n!m!    F^+_n,m  = ∑_n, m h^n(h') ^m/n!m!     F_n+m

= ∑_ℓ 1/ℓ! ∑_n+m = ℓ  ℓ!/n!m! h^n(h') ^m    F_ℓ    = ∑_ℓ 1/ℓ! (h+h')^ℓ    F_ℓ  =  F _h+h'

§ A SINGLE RG TRANSFORMATION

§.§ some definitions

After $k$  RG transformations we will have an action which contains a term
V_k = g_k ∫( ψ)^2
with a new coupling constant $g_k$,  as well as many other terms.    If this were the only term in the next step we would want to compute
the fluctuation integral
e^V_k  _C_k  (ψ)   ≡∫e^V_k( ψ+ η)  d μ_C_k (η)
We would like to write this as the exponential of something.   To find the leading term of this something.  one might    look at the logarithm and expand in $V_k$
loge^V_k  =    V_k+ 1/2  V_k,V_k ^T + …where $\blan V_k,V_k \bran^T= \blan V_k^2 \bran -\blan V_k \bran^2$.   This is just formal because  $V_k$ is not small (due to the large volume)
and so the definition of the logarithm is problematic.    Nevertheless  $ \blan V_k\bran $ and  $ \frac 12\blan V_k,V_k \bran^T $
are the correct leading terms and will play an important roll in the more detailed analysis.

In computing  these integrals it is useful to use the following fact.   If
F  _C  (ψ)   ≡∫F( ψ+ η)  d μ_C (η)
\blan F  \bran_{C}     = e^{\De_{C} } F
where $\De_C$ is the
formal differential operator
_C  =  ∫dx dy  C_ab (x-y)    /  _b  (y )   /  ψ_a  (x  )
In the case at hand   $ \blan V_k\bran   = e^{\De_{C} }V_k $.  Any derivatives give terms proportions to $C_k(0) $ which vanishes
since $C_k$ is an odd function.  (This is special to the massless model.)  Thus we

V_k_C_k    =     V_k

V_k, V_k ^T_C_k  =        V_k^2 _C_k  - V_k^2 = ( e^_C_k -1) V_k^2

Terms like this accumulate after $k$ steps and we will find that the quadratic term has the form
Q_k ≡1/2   V_k, V_k ^T_w_k   = 1/2 ( e^_w_k -1) V_k^2

w_k(x) =  ∑_j=0^k-1 C_j,L^-(k-j   (x) =  ∑'_p e^ipx  -i /p^2 (  e^-p^2/L^2k  - e^-p^2 )

In subsequent  steps  we  cannot avoid allowing  local  terms not present in the original action.
With $\ga_5 = i \ga_0 \ga_1$  there  may  be pseudo-scalar terms
of the form
V^p _k = p _k ∫( _5 ψ)^2
with $p_k = \cO(g_k^2)$ and $p_0 =0$.
Again we have $\blan V^p_k \bran = V^p_k$  and
there are associated quadratic terms   $<V_k, V^p _k>^T = \cO(g_k^3) $ and  $\frac12<V^p _k, V^p _k>^T= \cO(g_k^4)$.
It is only the first that we need to track in  any detail.
After $k$ steps these will accumulate to
Q^p_k ≡V_k, V^p_k ^T_w'_k   = ( e^_w'_k -1) V_kV^p_k
where $w_k'$ is $w_k$ with the $j=0$ term removed.

In addition there may be vector terms
of the form  (sum over $\mu$ implied)
V^v _k = v _k ∫( _μ  ψ)^2
with $v_k = \cO(g_k^2)$ and $v_0 =0$.  Again we have $\blan V^v_k \bran = V^v_k$  and
there are associated quadratic terms   $<V_k, V^v _k>^T = \cO(g_k^3) $ and  $\frac12<V^v _k, V^v _k>^T= \cO(g_k^4)$.
After $k$ steps the first of  these  accumulates  to
Q^v_k ≡V_k, V^v_k ^T_w'_k   = ( e^_w'_k -1) V_kV^v_k

In fact it is not   $Q_k,  Q^p_k,  Q^v_k $ which occur,  but normalized versions   $  Q^{\reg}_k,  Q^{p, \reg}_k,  Q^{v, \reg} _k $
which have relevant terms removed and have good bounds.  The precise definition is yet to come.
In the following we group together the rogue terms by
V_k' = V^p_k +V^v_k Q_k' = Q^p_k +Q^v_k  Q_k'^ =  Q^p,  _k + Q^v, _k

§.§ statement of the theorem

We going to claim that after $k$ renormalization group transformation the partition function can be written
as an integral over fields  $\psi$  on  $\bbT_{N+M-k} $     of the form    $\sZ = \int e^{  S_k(\psi) } d \mu_{G_k}(\psi) $
where $G_k$ is defined in (<ref>)  and

S_k(ψ)  =       ∫(   _k   - z_k  ψ+  g_k  (   ψ)^2  +  p_k  ( _5 ψ)^2
+  v_k  ( _μ ψ)^2   ) +   Q_k^  +   Q_k'^   +  E_k

=       ∫(   _k   - z_k  ψ)
+  V_k +  V_k' +  Q_k^  +   Q_k'^   +   E_k

and where   $E_k$  has a local expansion
$E_k = \sum_X  E_k(X  ) $

Let $L,h,C_Z,C_E$ be sufficiently large and chosen in that order (i.e. $h$ is sufficiently large depending on $L$, etc.), and  let
$g_{\max} $ be sufficiently small  depending on these constants.
Suppose after $k$ renormalization steps  $\sZ = \int e^{ S_k} d \mu_{G_k}$ where $S_k$ has the form   (<ref>)
with $0< g_k   <   g_{\max} $ and

E_k _h, _4   ≤    C_E g_k^3

|z_k| ≤     C_Z g_k

|p_k|, |v_k|    ≤    C_E g_k^2

And suppose  $E_k(X)$ has all the symmetries of section <ref>.
Then  $\sZ = \int e^{  S_{k+1}} d \mu_{G_{k+1}}$ with  $S_{k+1} $ of the same form, with the same symmetries,  but with  new parameters.
There are  positive constants   constants  $\beta_k, \beta'_k$  bounded above and below and  also constants  $ \theta_k, \theta'_k$
bounded above  such that

E_k+1 =     ( E_k + E_k^*)

g_k+1   =    g_k   +   β_k  g^2_k -2 β' _kg_kp_k - 4β'_k g_kv_k  +  g^*_k

z_k+1  =     z_k +  θ_k  g_k^2 -  θ^p_k g_kp_k - θ^v_k g_kv_k+  z^* _k

_k+1 =     L^2 (_k + _k^*)

as well as

p_k+1   =    p_k -   2  β_k' g_kv_k+ p_k^*

v_k+1   =     v_k-    β'_k g_kp_k+  v_k^*

The  starred quantities  are functions of    $E_k,  g_k, z_k, p_k, v_k$  in   the  domain (<ref>)    and  satisfy there

E_k^* _1/2 h,    ≤     C_E h^-1/2 g_k^3

|g^*_k|, h^-2 |z^*_k|, |p^*_k|, |v^*_k|  ≤      C_E h^-4   g_k ^3

* The linear  operator $\cR$ is defined in lemma <ref> to follow.     For small sets $X$ it   removes the relevant parts from $E_k(X) $  and these   show up in  the higher order corrections  $\vep^*_k,  z^*_k,  g^*_k,v_k^*,p_k^*$.   Then $\cR E_k$ is normalized for small sets in the sense of definition <ref>.

* The operator
$\cL$ defined in (<ref>) reblocks and rescales.
By lemma <ref>  and lemma <ref>

E_k  _h, _4   ≤(L^-1 )  E_k _h, _4   ≤(L^-1)   E_k _h, _4  ≤( L^-1 )  C_E g_k^3

E_k^* _h, _4 ≤(L^2)   E_k^* _L^-1/2 h,   ≤(L^2)   E_k^* _1/2  h,     ≤(L^2)  C_E   h^-1/2g_k^3
It follows that

E_k+1 _h, _4  ≤((L^-1) + (L^2)  h^-1/2  )   C_Eg_k^3  <    C_E g_k^3   ≤C_E g_k+1^3
which advances the bound we started with.
So  the  theorem is   enough to control the growth of    $E_k$  as long as $ g_k$ stays small.

But the growth of  $g_k$ itself  (and  $z_k$)  is not necessarily controlled.   In fact the initial value
must be carefully chosen to avoid problems.   This is  renormalization and is the content of   section <ref>.

* The following proof has many points of similarity with  the analysis of Brydges, Dimock, and Hurd  [6], [7] who consider the infrared problem for
a scalar $\phi^4$ model  in $4- \ep$ dimensions.

§.§ start of the proof
Starting  with $\sZ = \int e^{ S_k} d \mu_{G_k}$
we first absorb the incremental field strength into the measure   by

e^ -  z_k ∫ψ d μ_G_k  = (I + z_kG_k )    d μ_G^z_k  (ψ)
= e^    '_k    | _N+M-k|     d μ_G^z_k  (ψ)

G^z_k(x,  y)  =   ∑'_p    e^ip(x-y)  -i/p^2     1/ z_k + e^p^2
This is established in  Appendix <ref>  where we also show that $|\vep'_k|  \leq   \one |z_k|  $.
=   e^ ( _k +  '_k )   | _N+M-k|    ∫exp( V_k +V'_k  + Q̂_k + Q̂'_k +  E_k  )   d μ_G^z_k
Next  we split  $G^z_k = G^z _{k+1,L}  + C^z_k $ with

G^z_k+1,L (x, y)  =        ∑'_p    e^ip(x-y)  -i/p^2      1/ z_k + e^L^2p^2

C^z_k(x, y)  =        ∑'_p    e^ip(x-y)  -i/p^2  [   1/ z_k + e^p^2    -    1/ z_k + e^L^2p^2   ]

=   e^_k +  '_k   | _N+M-k|     ∫[    exp(  V_k + V_k' +Q̂_k+ Q̂'_k+ E_k) ( ψ+ η)  )  d μ_C^z_k (η) ] d μ_G^z_k+1,L    (ψ)
But again as in (<ref>)
d μ_G^z_k+ 1,L
= e^- ”_k   | _N+M-k|  -  z_k∫ψ d μ_G_k+1,L
We also take out the leading term in $E_k$ defining $\de E_k$ by
E_k(ψ+ η) = E_k(ψ)   + E_k ( ψ, η)
Then we have

=   e^  ( _k +  '_k -_k”)   | _N+M-k|     ∫exp(   -    z_k∫ψ+ E_k(ψ) )          Ξ_k (ψ)            d μ_G_k+1,L    (ψ)
The  fluctuation integral is

Ξ_k(ψ)        =     ∫exp(  V_k + V_k' +  Q_k^  +   Q_k'^  ) ( ψ+ η) + E_k( ψ, η)  )   d μ_C^z_k (η)
which we  abbreviate  as
Ξ_k(ψ)   = e^W_k   _ C^z_k   (ψ)
W_k =   V_k   +  V_k' +  Q_k^  +   Q_k'^   + E_k

§.§ extraction

Consider the leading  term $E_k(\psi) = \sum_X E_k(X, \psi) $.
We   isolate relevant and marginal terms and leave an irrelevant remainder.

$E_k $ can be written
E_k  =  E_k^   +E_k

E^ _k ( ψ)  =        ∫(    ^E_k  -    z^* _k       ψ+  g^*_k    (ψ)^2 +  p_k^* ( _5 ψ)^2
+  v_k^*   (_μ ψ)^2  )
and $   \cR E_k ( \psi )= \sum_X \cR E_k(X, \psi) $ is irrelevant in the sense of  (<ref>). For fixed $n$
we have the  $\cO(g_k^3) $ bounds with constants $\one$ independent of all parameters

|_k^E| ≤A^-1  E_k_h, _n   |z_k^*| ≤A^-1  h^-2 E_k_h, _n

|g_k^*|   |v^*_k|, |p^*_k|  ≤A^-1  h^-4 E_k_h, _n

R E_k _h, _n  ≤ E_k_h, _n

Suppress the index $k$, let $|X|$ =  volume of $X$,   and at first  choose  $ E^{\loc}  =\sum_X  E^{\loc}  (X) $ as  follows (cf.  [7]).
E^  (X)  =  α_0 (X) |X|  +  ∫_X  _2(X) ψ+ ∫_X    _2,μ (X)   _μ  ψ+ ∫_X   _4(X)   ψψHere if  $X$ is a small set in the sense of section  <ref>  then

α_0(X)  =      1/|X| E_00 (X)

α_2 (X)  =      1/|X| E_11 (X,1)

α_2,μ (X)  =      1/|X| ( E_11 (X, x_μ - x^0_μ  )   - ∫_X (x_μ - x^0_μ)  / |X|   E_1,1 (X, 1 ) )

α_4  (X)  =      1/|X|   E_22 (X,1)

The  definition of   $\alpha_{1,\mu} (X) $ is independent of  the  reference  point $x^0$, but we  take $x^0 \in X$.    If $X$ is not a small set   then  all the $\al(X)=0 $ and
$ E^{\loc}  (X)  = 0$,   We define $\cR E(X) = E(X) - E^{\loc} (X) $

Then       $(\cR E)_{11} (X,1 ) =0$ for $X$ small  since
E^ _11( X, 1)  = ∫_X _2  = _2 |X| = E_11(X,1)
Further    $(\cR E)_{11} (X,x_{ \mu} - x^0_{\mu}  ) =0$ for $X$ small  since

E^_11  (X,x_ μ - x^0_μ  ) =      ∫_X _2, μ (X) dx +  ∫_X _2 (X) (x_μ -x^0_μ ) dx

=      |X|  _2, μ (X) +  |X|  _2 (X)  ∫(x_μ -x^0_μ ) dx

=     E_11 (X,x_ μ - x^0_μ )

Similarly $(\cR E)_{00}(X) =0$ and  $(\cR E)_{22} (X,1 ) =0$.  Thus  for $X$ small $\cR E(X) $ is irrelevant (normalized)  in the sense of (<ref>).

Now we sum  the various terms in $E^{\loc}(X) $ over (small)  paved sets $X$.   We have
∑_X α_0(X) |X|   =  ∑_X α_0(X) ∑_⊂X  ||
=  ∑_  ( ∑_X ⊃   α_0(X)  )  ||
= _k^E | _M+N-k |
Here we defined
_k^E =  ∑_X ⊃   α_0(X)  = ∑_X ⊃   1/|X| E_00 (X)
which is independent of $\sq$ by translation invariance.   We have  $|E_{00} (X)| \leq \| E(X) \|_{h} $
and so   since $1 \leq A^{-1}  \Ga_n(X)  $
| ^E | ≤∑_X ⊃  E(X)  _h     ≤A^-1  ∑_X ⊃  E(X)  _h  _n(X)  = A^-1    E _h, _n

For the quadratic terms we have

∑_X  ∫_X  _2  (X) ψ=   ∑_X   ∑_⊂X  ∫_   _2  (X) ψ=    ∑_  ∫_   (  ∑_X ⊃ _2(X) ) ψ=    ∫_2 ψwhere
$ \al_2   =  \sum_{X \supset \sq} \al_2(X) $ is independent of $\sq$.

∑_X  ∫_X  _2, μ (X)_μ  ψ=   ∫_2,μ   ψ

where $\al_{2,\mu}   =  \sum_{X \supset \sq} \al_{2,\mu} (X) $ is independent of $\sq$.

Now we claim that if $S^{-1} \ga_{\mu} S = \sum_{\nu} R_{\mu \nu}\ga_{\nu} $, then

S^-1  _2 S =    _2

S^-1  _2, μ  S =    ∑_ν R_μν_2,ν

To see this start with $E_{1,1}(X, \bpsi, \psi)  = S^{-1} E_{1,1}(RX, \bpsi_R, \psi_R) S$.
Then  $E_{1,1}(X, x,y)  =S^{-1}  E_{1,1}(RX, Rx,Ry) S$,
and hence   $E_{1,1} (X,1) =S^{-1}  E_{1,1} (RX,1) S$.  Then  $\al_2(X) = S^{-1} \al_2(RX) S$ and summing over
$X \supset \sq$ gives the first identity.   The other is similar.

From these transformation properties it follows that   $\al_2 $ is a multiple of the identity and $\al_{2, \mu} $ is a multiple of $\ga_{\mu}$.
For details on this argument see the Appendix <ref>.
Thus we can define scalars $m_k^*, z_k^*$ by.

m_k^*   =     _2    =  ∑_X ⊃ _2  (X)

z_k^*  _μ  =     -  _2,μ  =     -       ∑_X ⊃ _2,μ (X)

and then the quadratic terms  are
m^*_k ∫ψ-         z^* _k    ∫ψHowever  chiral invariance of $E$ gives  that  $i \ga_5[\al_{2,\mu}] i\ga_5 =  \al_{2,\mu}$  and hence that  $i\ga_5 [ m_k^*] i\ga_5 = m_k^*$.
If the scalar $m_k^* \neq 0$ this says   $(i\ga_5 )^2 = 1$ which is false;   it is $-1$.  Thus $m_k^* =0$.

For a bound  on $z_k^*$ we note that    $|x_{\mu} - x^0_{\mu}| \leq |X| $    and so
1/|X|  | E_11 (X,  x_μ - x^0_μ ) | ≤ E_11(X)    ≤1/h^2  E(X)_h
the same holds for $  |X|^{-1}  \int (x_{\mu} - x^0_{\mu})  E_{11} (X, 1 )  $   and so twice this bound for $\al_{2,\mu} (X)$.  Then we have
|z_k^*|  = |1/2  ∑_X ⊃ (_2,μ (X)_μ )  |
≤h^-2   ∑_X ⊃    E(X)_h   = A^-1   h^-2    E_h, _n

For the quartic terms   we have  as in (<ref>)

∑_X  ∫_X  _4 (X) ψ=    ∫_4 ψψ

$ \al_{4}   =  \sum_{X \supset \sq} \al_{4} (X) $ is independent of $\sq$.

Now $\al_{4} $  can be regarded as a  linear operator on $\bbC^2 \otimes  \bbC^2 $.   A basis for this space   is  the  sixteen operators

I ⊗I      I    ⊗_μ     I  ⊗_5

_μ ⊗I          _μ  ⊗_ν      _μ ⊗_5

_5 ⊗I        _5 ⊗_μ      _5 ⊗_5

However from the  covariance  of $E_4(X) $   we have that.
( S^-1  ⊗S^-1 )  _4    (S  ⊗S )  = _4
This is violated by all off-diagonal element in the above array for some $S$.   Furthermore one can show that $\ga_0 \otimes \ga_0$ and $\ga_1 \otimes \ga_1$
have the same coefficient in this basis.
Thus  there are constants  $g_k^*,  p_k^*, v_k^* $.
_4  = g_k^* ( I  ⊗I )+  p_k^*  ( _5 ⊗_5)   + v_k^*∑_μ  ( _μ  ⊗_μ)
For details of this argument see Appendix <ref>.

The quartic term is then

g^*_k  ∫( ψ)^2
+  p_k^*  ∫( _5 ψ)^2  +   v_k^*  ∫∑_μ  (_μ ψ)^2

g^* _k   =
_4  = ∑_ X ⊃   _4(X)
We  have   $| \al_4(X) | \leq    \|  E_{22}(X)  \| \leq  4h^{-4}  \|  E(X)  \|_{h } $ and so

| g^*_k  |   ≤A^-1  h^-4 E_h, _n
We get the same bound for  $v_k^*, p_k^*$.

Combining all the above if we sum  $ E(X)  =  E^{\loc} (X) + \cR E(X) $ over $X$ we get
$ E  =  E^{\loc}  + \cR E$ with $E^{\loc}$ given by (<ref>) .
We still need the bound on  $ \cR E(X) $.     We  give a new polymer expansion for the term in $E^{\loc} $ based on unit blocks.
From Lemma <ref>  we have that  $\int (\bpsi \psi)^2 $ has a local expansion $\sum_X E' (X) $         with  $\| E' \|_{h, \Ga_n}  \leq \one Ah^4$.     Combine
this with the bound  $| g^*_k  |   \leq  \one A^{-1}  h^{-4} \|E\|_{h, \Ga_n} $    and  conclude the     $g_k^* \int (\bpsi \psi)^2 $ has a local expansion $\sum_X E'' (X) $
with  $\| E'' \|_{h, \Ga_n}  \leq \one \|E \|_{h, \Ga_n} $. The other terms in  $E^{loc} $ are similar and  so  $\| E ^{\loc} \|_{h, \Ga_n}  \leq \one \|E \| _{h, \Ga_n} $ and hence
$\|\cR E  \|_{h, \Ga_n}  \leq \one \|E \| _{h, \Ga_n} $.

§.§ fluctuation integral

As we will see the fluctuation integral can be written as an exponential

Ξ_k ≡e^ W_k   _C^z_k   = exp( W_k^# )
where    $W_k^\# $ has a local expansion.   More generally we will find
e^t W_k  _C^z_k   = exp( W^#_k (t ) )
where   $ W^\#_k (t )  $ is analytic   in complex  $t$ for   $|t|  <  r g_k^{-1} $ with  $r$ sufficiently small.
Matching derivatives at $t=0$  gives

1 =      exp( W^#_k (0 ) )

<W_k> =     (W^#_k)' (0 )

<W_k,W_k>^T =      (W^#_k)” (0 )

Then   expressing  $W_k^\# = W_k^\#(1)$   as an expansion around $t=0$ we have

exp( W^#_k  ) =exp(   W_k  + 1/2 W_k, W_k ^T +  1/2 πi  ∫_ |t| =rg_k^-1   dt/t^3(t-1)  W^#_k(t)    )
Since  $ W_k =   V_k   +   V_k' +\  Q_k^{\reg}  +   {Q_k'}^{\reg}   + \de E_k$ and since $<V_k> = V_k$ and $<V'_k> = V'_k$

W_k  =      V_k   +   V' _k  +  Q_k^      +   Q_k'^    +  E_k

1/2   W_k, W_k ^T =       1/2    V_k, V_k ^T  +     V_k, V' _k ^T    +   [ ⋯]

Here in the second line we have isolated the terms we want to analyze in detail in the next section.
In subsequent sections we  give   the exact definition  of $W_k^\#(t) $ and   the   remainder term in (<ref>).  Finally  the omitted terms in (<ref>) are
The omitted terms are

[ ⋯] =        V_k,  Q_k^  ^T + V_k ,  Q_k'^ ^T +  V_k,  E_k  ^T

+ 1/2 V'_k, V_k'  ^T  + V'_k,   Q_k^  ^T   + V'_k,  Q_k'^  ^T +
V'_k,  E_k  ^T

+ 1/2   Q_k^ ,   Q_k^  ^T   +   Q_k^ ,    Q_k'^  ^T  + Q_k^ ,    E_k ^T

+ 1/2   Q_k'^ ,    Q_k'^  ^T    + Q_k'^ ,    E_k ^T

+1/2 E_k,    E_k ^T

§.§ the quadratic terms

We study the $\cO(g_k^2) $ terms  in  (<ref>)  which are

1/2    < V_k, V_k> ^T_C^z_k  +  < Q_k^ > _C^z_k
We start with a computation of $Q_k$,  then discuss the modifications to generate $Q_k^{\reg}$, and
then  analyze the above  expression.
By evaluating the formal derivatives we compute
Q_k = 1/2  (e^ _w_k  -1 ) V_k^2     ≡g_k^2   (w_k)
(w_k)  =  _6 (w_k) +  _4 (w_k) + _2 (w_k)  +  _0(w_k)

_6(w_k)  =   4 ∫(ψ)(x) ( (x)   w_k (x-y)    ψ(y)  )  (ψ)(y) dx dy

_4( w_k  )
-2n  ∫(ψ)(x)    (  w_k(y-x)   w_k(x-y)  )
(ψ)(y)dx dy

+   4  ∫(ψ)(x) (  (y) w_k(y-x)w_k(x-y) ψ(y ) )  dx dy

+ 2 ∫((x)  w_k(x-y)    ψ(y) )    ((x)  w_k(x-y)    ψ(y)+     (y)   w_k(y-x)   ψ(x)          )  dx dy

_2(w_k)   =
4  ∫(x) (w_k(x-y) w_k(y-x) w_k(x-y)    )  ψ(y)   dx dy

-4n ∫(x) w_k(x-y)   ψ(y)      ( w_k(x-y) w_k(y-x)   )   dx dy

=  - n  ∫(w_k(x-y)  w_k(y-x)   w_k(x-y) w_k(y-x) )     dx dy

+    n^2 ∫(   ( w_k(x-y) w_k(y-x) ) ) ^2 dx dy

Here we used

_w[  ψ^i_ a(x)  ^j_b(y)]  =      _ij w_ab(x-y)

_w[∑_i  ψ^i_ a(x)  ^i_b(y)]  =      n  w_ab(x-y)

This expression will not have good estimates as $k$ increases.     The trouble is that  $w_k(x-y) $ is develops  a short distance singularity of the form $\cO( |x-y|^{-1} )$.
This is  summable (i.e has bounds independent of $N, k$)   but powers $w_k(x-y)^2$ and higher are not.    Thus the  $\cH_6(w_k) $  term  will have good estimates,  but this is  not so for the  $\cH_4(w_k)  , \cH_2(w_k) , \cH_0(w_k) $
terms.       To fix it we have to remove relevant terms, and this will give us the definition of $Q_k^{\reg}$.

First in
$ \cH_4 (w_k)  $ we    replace  $\psi(y) , \bpsi(y) $ by   $\psi(x),   \bpsi (x) $   and let $\hat  \cH_4( w_k  )$
be the difference.
There is no contribution from the  last term in  $\cH_4( w_k  )$  since $w_k$ is an odd function.   We have

_4( w_k  )
=       ∫(ψ)(x)   (x)  β(w_k)  ψ(x )   dx dy   + _4( w_k  )

β(w_k)  =  ∫[ -2n(  w_k(y-x)   w_k(x-y) ) + 4 w_k(y-x)w_k(x-y)] dy
is independent of $x$.  The integral    $\int w_k(y-x)   w_k(x -y) )dy$  is a scalar in spin space so

β(w_k)  = - 4(n-1)   ∫w_k(y-x)   w_k( x-y) )dy = 4(n-1)  ∫w_k(x-y)^2   dy

_4( w_k  )
=      β(w_k)  ∫(ψ)^2    + _4( w_k  )

In $\cH_2(w_k) $ we replace  $\psi(y) $
by  $\psi (x)   +\sum_{\mu}  (y-x)_{\mu}   \pa_{\mu}  \psi (x) $  and  let $\hat \cH_2(w_k)  $ be the difference.
The new  $\bpsi \psi  $   terms vanish since we are integrating an odd function  and we find
_2(w_k) =    ∫(x) θ_ μ (w_k)     _μ ψ(x)    dx   +_2 (w_k)

θ_μ (w_k)     =   4    ∫(w_k(x-y) w_k(y-x) w_k(x-y)  (y_μ  - x_μ )  )  dy

-4n   ∫( w_k(x-y)w_k(y-x) dy   )   w_k(x-y)   (y_μ  - x_μ )  dy

is independent of $x$.
\theta_{ \mu} (w_k)   = \ga_{\mu}  \theta (w_k) $   with   $ \theta  (w_k)   = \frac12 \tr ( \ga_{\mu} \theta_{\mu}(w_k) )

_2  (w_k) =  θ(w_k)  ∫ψ+ _2 (w_k)
where with  $w(y-x) = -w(x-y)$

θ(w_k)=     2    ∫( w_k^3(x-y)  ( -  )  dy    -2n ∫(   w_k^2(x-y)  )  (  w_k(x-y) (  - )  )  dy

We also define  $ \hat \cH_6 (w_k)     = \cH_6(w_k)    $  and $\hat \cH_0 (w_k)  =0$.  Then $Q_k^{\reg}$ is defined by
Q_k^  =  g_k^2 (w_k)
= g_k^2 (  _6 (w_k) + _4 (w_k) + _2 (w_k)  + _0(w_k) )
This will have good estimates.     If we define $\vep(w_k) $  by $ \vep(w_k) |\mathbb{T}_{N+M-k}| = \cH_0(w_k) $
then we have

(w_k)  = ∫( (w_k) + θ(w_k)  ψ+ β(w_k) (ψ)^2 ) +

Now consider the  expression  (<ref>).
At  first suppose   it is  $ <   Q_k >_{C^z_k}  $ rather than   $ < Q_k^{\reg} >_{C^z_k} $.
Then we have

1/2   < V_k  ,   V_k  >^T_C^z_k    +  <   Q_k >_C^z_k

=    1/2 (e^ _C^z_k  -1) V_k^2  +  1/2 e^ _C^z_k (e^ _w_k  -1 ) V_k^2

=      1/2 (e^ _C^z_ke^ _w_k   -1) V_k^2
=  1/2 ( e^ _w_k + C^z_k - 1) V_k^2  = g_k^2(w_k + C^z_k)

Now consider the
actual expression     (<ref>) which we write as

1/2   < V_k  ,   V_k  >^T_C^z_k    +  < Q_k^   >_C^z_k

=     1/2   < V_k  ,   V_k  >^T_C^z_k    +  <   Q_k   >_C^z_k   +     g_k^2<   (w_k)  -    (w_k)  >_C^z_k

=       g_k^2 (w_k + C^z_k)       +     g_k^2  ((w_k)  -    (w_k) )

=     g_k^2    ((w_k + C_k) - ( w_k))  + g_k^2 (w_k)         + E^z_k

Here in the second step  we used  (<ref>)  and the fact  that  $ \hat  \cH  (w_k)  -    \cH  (w_k)  $ is strictly local so the fluctuation integral has no effect.
In the last step we defined

E^z_k    =  g_k^2    ((w_k + C^z_k) -  (w_k + C_k))
Also    from (<ref>)
( w_k + C_k)   -( w_k  )   =  ∫( ^Q_k  +  β_k    (ψ)^2  +   θ_k     ψ)     +     ( w_k + C_k)   - ( w_k  )

^Q_k =      ( w_k + C_k) - ( w_k )

β_k =      β( w_k + C_k) -  β( w_k )

θ_k =        θ( w_k + C_k) -  θ(  w_k  )

Insert   this into (<ref>) and  cancel the terms $ \hat  \cH (w_k) $.  Then     we have

1/2   < V_k  ,   V_k  >^T_C^z_k    +  <Q_k^  >_C^z_k

g_k^2 ∫(  ^Q_k   +  θ_k        ψ+ β_k    (ψ)^2 ) +  g_k^2    (w_k+ C_k)    + E^z_k

This is the expression we were looking for.  It exhibits the corrections to the coupling constants and  preserves the general form since
$g_k^2    \hat  \cH(w_k+ C_k) ) $  scales to  $g_k^2    \hat  \cH(w_{k+1} ) $  which after a change of coupling constant is $ Q_{k+1}^{\reg} =g_{k+1}^2    \hat  \cH(w_{k+1} )   $.

§.§ the lowest order rogue terms

We study the  terms  $ \frac12    < V_k, V'_k> ^T  +  <  {Q_k'}^{\reg}   > $ in (<ref>) which are $\cO(g_k^3)$.
This expression breaks into a pseudo-scalar part and a vector part.

We start with the pseudo-scalar part which is
$ \frac12    < V_k, V^p_k> ^T  +  < Q^{p, \reg} _k >  $.
First we  define  $\cH^p(w'_k)$  by
Q_k^p  = (  e^_w'_k -1 ) V_kV_k^p  ≡g_k p_k  ^p(w'_k)
and we find that
$ \cH^p =  \cH^p_6 + \dots +  \cH^p_0 $ where

^p_6(w'_k)  =   8 ∫(ψ)(x) ( (x)   w'_k (x-y) _5   ψ(y)  )  (_5 ψ)(y) dx dy

^p_4( w'_k  )
-4n  ∫(ψ)(x)    (  w'_k(y-x)   w'_k(x-y)_5  )
(_5 ψ)(y)dx dy

+   8  ∫(ψ)(x) (  (y)_5  w'_k(y-x)w'_k(x-y)_5 ψ(y ) )  dx dy

+ 4  ∫((x)  w'_k(x-y) _5   ψ(y) )    ((x)  w'_k(x-y) _5   ψ(y)+     (y) _5  w'_k(y-x)   ψ(x)          )  dx dy

^p_2(w'_k)   =
8   ∫(x) (w'_k(x-y)_5 w'_k(y-x) w'_k(x-y) _5   )  ψ(y)   dx dy

-8n ∫(x) w'_k(x-y) _5  ψ(y)      ( w'_k(x-y)_5 w'_k(y-x)   )   dx dy

^p_0 (w'_k)
=     -2 n  ∫(w'_k(x-y)_5  w'_k(y-x)   w'_k(x-y)_5 w'_k(y-x) )     dx dy

+       2n^2 ∫(   (w'_k(x-y)_5 w'_k(y-x) ) ) ^2 dx dy

In $\cH^p_4(w'_k) $ we again  replace $\bpsi(y) , \psi(y) $ by $\bpsi(x), \psi(x) $ and let  $\cH^p_4(w'_k) $ be the difference.    The first
term  in $ \cH^p_4( w'_k  )$ is zero since  $\int w'_k(y-x)   w'_k(x-y) dy $ is a multiple of
the identity in spin space and $\tr \ga_5 =0$. In the second term we identify  $\ga_5^2=1$.   In the third term we use
$w'_k(x-y) \ga_5 = \ga_5 w'_k( y-x)$.
Then we have

^p_4( w'_k  )
=        8  ∫(ψ)(x) (  (x)  [ ∫w'_k(y-x)w'_k(x-y) dy ] ψ(x ) )  dx

+ 8  ∫((x)  w'_k(x-y) _5   ψ(x) )    (      (x) _5  w'_k(y-x)   ψ(x)          )  dx dy    + ^p_4( w'_k  )

The bracketed expression is identified as $- \frac14 (n-1)^{-1}  \beta(w'_k) $.
Also we write    $w'_k = \ga_{\mu} w_{k, \mu} $
and then
∫w'_k, μ (y) w'_k,ν(y)  dy   = 1/2   _μν  ∫∑_μ ( w'_μ ( y) )^2dy =1/2  _μν  ∫w'_k(y) ^2
and the latter is identified as $ \frac18 (n-1)^{-1}  \beta(w'_k) \de_{\mu \nu} $.
We  also use
∑_μν   _μν  (   _μ_5  ⊗_5 _ ν    )
= ∑_μ    (  _μ_5  ⊗_5 _μ )   =  - ∑_μ    (  _μ  ⊗_μ )
Then we have with $\beta'(w'_k) =   (n-1)^{-1} \beta(w'_k) $  (sum over $\mu$ implied)

^p_4( w'_k  )
=         - 2  β'(w'_k) ∫(ψ) ^2
-      β' (w'_k)   ∫∑_μ  (_μ      ψ)^2         + ^p_4( w'_k  )

Now consider the term   $ \cH^p_2 (w'_k)$.  We  replace  each  $\psi(y)$ by $\psi (x) + (y_{\mu}  - x_{\mu} ) \pa_{\mu} \psi (x) $
The leading terms vanish since we are integrating an odd function.   Then we have

^p_2(w'_k) =       ∫(x) θ^p_ μ (w'_k)     _μ ψ(x)    dx   + ^p_k(w'_k)
where (dropping some  $\ga_5$'s)

θ^p_ μ (w'_k)
=       8 ∫(w'_k(x-y) w'_k(y-x) w'_k(x-y) (y_μ  - x_μ )  )  dy

- 8n   ∫w'_k(x-y)  _5  (y_μ - x_μ)      ( w'_k(x-y)_5 w'_k(y-x)   )   dx dy

But  $\tr ( \theta^p_{ \mu}) = 0$ and $\tr ( \ga_5\theta^p_{ \mu}) = 0$ and  $\theta^p_{\mu} $ is independent of $\mu$.  Thus
\theta^p_{ \mu} (w'_k)   = \ga_{\mu}  \theta^p (w'_k) $  where  $ \theta^p_{\mu}  (w'_k)   = \frac12 \tr (  \ga_{\mu} \theta_{\mu} (w'_k) )$.
(w'_k) =   θ^p(w'_k)  ∫ψ+^p_2(w'_k)

θ^p (w'_k)
=       4∫(w'_k(x-y) w'_k(y-x) w'_k(x-y) (-)  )  dy

- 4n   ∫(   w'_k(x-y)  _5  (-)  )     ( w'_k(x-y)_5 w'_k(y-x)   )    dy

We also define  $ \hat   \cH^p_ 6(w'_k)  =\cH^p_ 6(w'_k) $ and $\hat \cH^p_0 =0$ and
Q_k^p,   =g_kp_k  ^p(w'_k)   = g_kp_k (  ^p_6(w'_k) + …+    ^p_0(w'_k) )
Also with   $\int \vep^{p,Q}(w'_k)   = \cH^p_0(w'_k)$
we have

^p (w'_k)  =   ∫( ^p,Q(w'_k)  +  θ^p (w'_k)      ψ- 2  β'(w'_k)      (ψ) ^2
-      β' (w'_k)     (_μ      ψ)^2  )
+ ^p (w'_k)

Now as in (<ref>)  and (<ref>) (without the factor $\frac12$)

< V_k, V^p_k> ^T_C^z_k    +  <  Q_k^p,  >_C^z_k
=     g_kp_k    (^p_k(w'_k + C^z_k)  +( ^p  (w'_k) -    ^p (w'_k))  )

=     g_kp_k    (^p_k(w'_k + C_k) -    ^p  (w'_k) ) +g_kp_k  ^p  (w'_k)   + E^z,p_k

where  $E^{z,p}_k =g_kp_k( \cH^p_k(w'_k + C^z_k)-\cH^p_k(w'_k + C_k)) $.
But from (<ref>)

^p(w'_k + C_k  ) -    ^p(  w'_k)    =       ∫( ^p,Q _k   +  θ^p_k      ψ- 2  β'_k      (ψ) ^2
-      β'_k     (_μ      ψ)^2  )

+      ^p(w'_k + C_k  )  -^p (w'_k)

^p,Q _k =      ^p,Q   ( w'_k + C_k) - ^p,Q ( w'_k )

β'_k =      β'  ( w'_k + C_k) -  β' ( w'_k )

θ^p_k =        θ^p    ( w'_k + C_k) -  θ(  w'_k  )

Now insert this into (<ref>), cancel the $g_kp_k  \hat  \cH^p  (w'_k) $     and find

< V_k, V^p_k> ^T_C^z_k    +  <  Q_k^p,  >_C^z_k

g_kp_k  ∫( ^p,Q_k
+  θ^p_k     ψ-2  β'_k         (ψ)^2
-  β'_k    (_ μ    ψ) ^2    )  +     g_kp_k ^p(w'_k + C_k  )  + E^z,p_k

Now consider the vector part which is $     < V_k, V^v_k> ^T    +  <  Q_k^{v,\reg}  >  $.
We compute
Q_k^v  = (  e^_w'_k -1 ) V_kV_k^v  ≡g_k v_k  ^v(w'_k)
The  $ \cH^v(w'_k) $ is the same as $ \cH^p(w'_k) $ in (<ref>) - (<ref>) except that we replace each $\ga_5$ by $\ga_{\si}$
and sum over $\si $.   Thus  instead of  (<ref>)  we have

^v_4( w'_k  )
=        8  ∫(ψ)(x) (  (x)  [ ∫w'_k(y-x)w'_k(x-y) dy ] ψ(x ) )  dx

+ 8  ∫∑_  ((x)  w'_k(x-y) _    ψ(x) )    (      (x) _   w'_k(y-x)   ψ(x)          )  dx dy    + ^v_4( w'_k  )

The bracketed expression is again identified as $- \frac14 \beta'(w'_k)$.
Again we  write  $w'_k =\sum_{\mu} \ga_{\mu} w_{k, \mu} $  and identify  $ \int w_{k, \mu}(y-x)  w_{k, \nu}(x-y) dy  = -\frac18 \beta'(w'_k) \de_{\mu \nu}  $.
We also use

∑_μν   _μν  (   _μ_   ⊗_  _ ν    )
=      ∑_μ    (  _μ_   ⊗_  _μ )

=      ∑_= μ    (  _μ_   ⊗_  _μ ) +  ∑_≠μ    (  _μ_   ⊗_  _μ )

=      2( I ⊗I )  + 2 (_5 ⊗_5)

^v_4( w'_k  )
=       -4 β'(w'_k)   ∫(ψ)^2 -2  β'(w'_k)    ∫(_5 ψ)^2      + ^v_4( w'_k  )

For the quadratic terms we have  instead of (<ref>) we have

^v_2(w'_k) =        ∫(x) θ^v_ μ (w'_k)     _μ ψ(x)    dx   + ^v_2(w'_k)

θ^v_ μ (w'_k)
=       16 ∫(w'_k(x-y) w'_k(y-x) w'_k(x-y) (x_μ  - y_μ )  )  dy

The second term is absent here since it involves the trace of three gammas.
But  $\tr ( \theta^v_{ \mu}) = 0$ and $\tr ( \ga_5\theta^v_{ \mu}) = 0$ and  $\theta^v_{\mu} $ is independent of $\mu$.  Thus
\theta^v_{ \mu} (w'_k)   = \ga_{\mu}  \theta^v (w'_k) $  where  $ \theta^v_{\mu}  (w'_k)   = \frac12 \tr (  \ga_{\mu} \theta^v_{\mu} (w'_k) )$.
(w'_k) =   θ^v(w'_k)  ∫ψ+^v_2(w'_k)

θ^v (w'_k)
=       8∫(w'_k(x-y) w'_k(y-x) w'_k(x-y) (-)  )  dy

We also define  $ \hat   \cH^v_ 6(w'_k)  =\cH^v_ 6(w'_k) $ and $\hat \cH^v_0 =0$ and
Q_k^v,  =g_kv_k  ^v(w'_k)   = g_kv_k (  ^v_6(w'_k) + …+    ^v_0(w'_k) )
Also with   $\int \vep^{ v,Q}(w'_k)   = \cH^v_0(w'_k)$
we have instead of (<ref>)

^v (w'_k)  =   ∫( ^v,Q(w'_k)  +  θ^v  (w'_k)      ψ- 4  β'(w'_k)      (ψ) ^2
-   2   β' (w'_k)   (_5     ψ)^2  )
+ ^v (w'_k)
and so

^v (w'_k+C_k ) -   ^v (w'_k)  =        ∫( ^v,Q_k  +  θ^v_k      ψ- 4  β'_k       (ψ) ^2
-   2   β'_k   (_5     ψ)^2  )

+      ^v (w'_k+C_k ) -  ^v (w'_k)

where  $ \vep^{v,Q} _k ,\beta'_k,  \theta^v_k $ are defined as in (<ref>).

The rest of the analysis follows the pseudoscalar  case and we have

< V_k, V^v_k> ^T_C^z_k    +  < Q_k^v,  >_C^z_k

g_kv_k ∫( ^v,Q_k  +  θ^v_k      ψ- 4  β'_k       (ψ) ^2
-   2   β'_k  (_5     ψ)^2  )+    g_kv_k ^v(w'_k + C_k  )  + E^z,v_k

Finally  $   < V_k, V'_k> ^T   +  < {Q'_k}^{\reg}  >$ is the sum of (<ref>) and (<ref>).  When summing we define   $ {E^z_k}'  = E^{z,p}_k  + E^{z,v}_k$.

§.§ estimates

Before  proceeding we develop estimates on some of the objects we have created.
Here and throughout the paper $\cO(1)$ stands for a constant independent of all parameters.   We regard the number of internal components $n$ as fixed and $\one$.
The letter  $C$ stands for a constant that may depend on $L$, and it  may change from line to line.

We start with estimates on  $\beta_k, \theta_k$.  ($\vep_k^Q$ is finite but is not bounded uniform in $k$.)

* There exist  positive constants $C_{\pm}$ such that
( n  -1) C_-    ≤   β_k     ≤( n  -1)C_+    and $\beta'_k$ satisfies the same bound without the $(n-1)$.

* There exists a constant $C$  such that $ |\theta_k|,   |\theta^p_k|,  |\theta^v_k| \leq C  $

The coefficient $\beta_{k} =  \beta( w_k+ C_k ) - \beta(w_k) $  is from (<ref>)
=  4 (n-1)  ∫[(w_k +C_k)^2   -    w_k^2  ]
=  4 (n-1)  ∫[2w_kC_k  + C_k^2   ]
We first consider $\int     C_k^2  $
and compute
∫C_k^2  =   ∑'_p  ∈^*_M+N-k      1 /p^2   ( e^-p^2 -  e^-L^2p^2 )^2=   ∑'_p  ∈^*_M+N-k    e^-2p^2  /p^2   ( 1-  e^-( L^2-1) p^2 )^2
We split this  into  two pieces.  First consider  the region
$(L^2-1) p^2 \leq 1$. We use the bound
x e^{-1}      \leq   1-e^{-x}   \leq   x  $ for $0 \leq  x \leq  1   $.
e^-1  ( L^2-1)^2   ∑'_(L^2-1) p^2 ≤1   p^2 e^-2p^2       ≤∫C_k^2   ≤( L^2-1)^2     ∑'_(L^2-1) p^2 ≤1  p^2 e^-2p^2
which suffices.
In the region $(L^2-1) p^2 \geq 1$ we use the   bound
1-    e^{-1}  \leq   1-e^{-x}  \leq  1$ for    $x \geq 1$
(1-e^-1 )^2   ∑'_(L^2-1) p^2 ≥1     e^-2p^2  /p^2    ≤∫C_k^2   ≤∑'_(L^2-1) p^2 ≥1    e^-2p^2  /p^2
which also suffices.

Now let's look at the other term
which is
∫2 w_k C_k   =   2 ∑'_p  ∈^*_N+M     1 /p^2   ( e^-p^2/L^2k -  e^- p^2 ) ( e^-p^2 -  e^-L^2p^2 )
Since this is positive and since we already have a lower bound from the first term, it suffice to get an upper bound on this.
But    $( e^{-p^2- } - e^{-L^2p^2}) $ is $ \cO(p^2)$ as $p \to 0$ and $\cO(e^{-p^2})$ as $p \to \infty$.  So this term is bounded by a constant.
This completes the  bound on $\beta_k$

The coefficient $\theta_k$ is  $    \theta( w_k+ C_k ) - \theta(w_k)$ where   $\theta(w_k)$ is defined in  (<ref>).
We  use the bounds from Appendix <ref> which are

| C_k(x-y) |  ≤e^ - |x-y|/L    |w_k(x- y) | ≤|x-y| ^- 1 e^-|x-y|
Referring to (<ref>)  we see that the integral defining $\theta  (w_k) $   is exponentially decaying as $x-y \to \infty$
and has an $\cO(|x-y|^{- 2})$  singularity as $x \to y$.  The latter is not summable.
But in $ \theta_k =     \theta( w_k+ C_k ) - \theta(w_k)$  at least one of the factors $w_k $  is replaced by   $C_k$.
This reduces the short distance singularity to   $\cO(|x- y|^{- 1})$ which is summable.  The bound on $\theta_k$ follows and
the others are similar.  This completes the proof.

Next we   develop local expansions for  $V_k, V'_k,  Q_k^{\reg} ,  {Q'_k}^{\reg} $ and give associated estimates.
In fact as in  Lemma <ref>  we already have $V_k = \sum_X V_k(X)$ and $V' _k = \sum_X V' _k(X)$

V_k _h, _n     ≤C  h^4  g_k     V'_k  _h, _n     ≤C h^4 (|p_k|+|v_k| )
For the  quadratic terms we have:

$ Q_k^{\reg} =  \sum_X   Q_k^{\reg}(X) $ and ${Q'_k}^{\reg}  =\sum_X  {Q'_k}^{\reg}(X) $  with

Q_k^ _h, _n     ≤    C   h^6    g^2_k    Q'_k^ _h, _n     ≤C  h^6  (|p_k|+|v_k| )   g_k

We have    $Q_k^{\reg}  = g_k^2   \hat  \cH(w_k) $, but  before extracting the local pieces we have $Q_k =  g_k^2 \cH(w_k)$.    We first consider the quartic contribution  $ g_k^2 \cH_4 (w_k)$
which from (<ref> )   is
_4( w_k  )
2n  ∫(ψ)(x) n   (   w^2_k(x-y) )
(ψ)(y)dx dy  +  2 more terms
We study this first term in detail.  The    $\hat  \cH _4 (w_k) $ is the remainder when we   replace each
$\psi(y) , \bpsi(y) $ by    $\psi (x), \bpsi(x)$. It is given by

_4( w_k  )
=   2 n ∫(ψ)(x)    (  w^2_k(x-y))

( (x)  (ψ(y)- ψ(x))    + ((y)- (x))  ψ(x) +    ((y)- (x))  (ψ(y)- ψ(x))    )
dx dy + …

We consider the first term here in detail,  call it $K $.   We localize the integral  writing
$K = \sum_{\sq, \sq'}  K(\sq \times \sq')  $ where $\sq, \sq'$ are unit blocks and
=  2n  ∫_×'     (  w^2_k(x-y))     (ψ)(x)    (x)    (ψ(y)- ψ(x))   dx dy
This has norm

K(×')  _h
≤h^4     sup_f _ ≤1 | ∫_×'   (  w^2_k(x-y)) (   f(x,x,x,y) - f(x,x,x,x) )   dx dy |
We again use the bound  (<ref>)
|w_k( x-y) | \leq \one |x-y|^{-1} e^{-|x-y|}
If $x \in \sq, y\in \sq'$ and                     $d(\sq,\sq') \geq 1$ this is less than  $e^{- d(\sq,\sq') }$  and (<ref>) is bounded by $\one h^4e^{- d(\sq,\sq')/2 }$.
On the other hand if $d(\sq, \sq') =0$  we  use
f(x,x,x,y) - f(x,x,x,x)
=   ∫^1_0 (y-x) ·f  (x,x,x, x + t(y-x) ) dt
This gives the estimate $|  f(x,x,x,y) - f(x,x,x,x) | \leq |x-y| \|f \|_{\cC} $.   The $ |x-y|^{-2}$ singularity in $ \tr  (  w^2_k(x-y)) $
is reduced to an integrable $ |x-y|^{-1}$ singularity.   So in this case  (<ref>) is bounded by $\one h^4$.   Altogether
then we have
K(×')   _h ≤h^4e^- d(,')
The other terms in $\hat \cH_4( w_k  )$ are estimated similarly.

Now we consider the quadratic term which has the form $g_k^2 \hat  \cH_2(w_k)$.
First without the extraction we have from (<ref>)
_2(w_k )   =   -4  ∫(x)  w_k^3  (x-y)     ψ(y)   dx dy   + …This modified to  $\hat   \cH_2(w_k )   $ by  replacing   $\psi(y)$   by      $ \psi (x)   + (y-x)\cdot  \pa \psi (x)$.
The difference is
_2(w_k )   =   -4  ∫(x)   w^3_k  (x-y)   ( ψ(y) - ψ(x)   - (y-x)·ψ(x))    dx dy   + …Call this first term $H$  and write
$H = \sum_{\sq, \sq'} H(\sq \times \sq')  $
where now
=   -4  ∫_×' (x) w^3_k  (x-y)  ( ψ(y) - ψ(x)   - (y-x)·ψ(x))    dx dy
This has  norm

H(×') _h

≤h^2 sup_ f_  ≤1|∫_×'  w^3_k  (x-y)
(  f(x,y)  -f(x,x) -  (y-x) ·( _yf)(x,x)  ) dx dy |

By the bound on $w_k$, if  $d(\sq,\sq') \geq 1$ this is bounded by $\one h^2 e^{-d(\sq,\sq')} $.   For $d(\sq,\sq') =0$ we use the identity
f(x,y)  -f(x,x) -  (y-x) ·( _yf)(x,x)   = ∫_0^1 (1-t)  (y-x) (y-x)(  f) (x,   x + t(y-x) )  dt
This is $\cO( |x-y|^2) \|f\|_{\cC} $.
So the  $ |x-y|^{-3}$ singularity in $  w^3_k(x-y)$
is reduced to an integrable $ |x-y|^{-1}$ singularity, and  the norm is bounded by  $ \one h^2$.  Altogether then
and we have
H(×')   _h ≤h^2e^- d(,')
The  other term in  $\hat \cH_2( w_k  )$ is estimated similarly.

The same bound   holds for  $\hat \cH_6( w_k  )= \cH_6( w_k  )$  (now with $h^6$ )  since we have only one $w_k(x-y) $ which is integrable.

Altogether we have
Q_k^  =  ∑_, ' Q_k^ (×' )
Q_k^  (×) _h    ≤g^2_k   h^6 e^- d(, ' )
Now define   $Q_k^{\reg} (X)  =  Q_k^{\reg}  (\sq \times \sq)$ if $X= \sq \cup \sq'$ and zero otherwise.
Then we have

Q_k^  _h, _n    =      ∑_X⊃   Q_k^ (X)  _h  _n(X)  = ∑_ '    Q_k^ (×' )  _h  _n(∪')

≤g^2_k   h^6    ∑_'   e^- d(, ' )   _n(∪')
≤C g^2_kh^6

The last line follows since the exponential decay $ e^{- d(\sq, \sq' )  } $ beats the polynomial growth of $ \Ga_n(\sq \cup  \sq')  $.
This completes the proof.

$E^z_k    = \vep_k^z|\mathbb{T}_{N+M-k} | +
\sum_X  E^z_k(X) $   and    ${E'}^z _k   = { \vep'} _k^z| \mathbb{T}_{N+M-k} | +  \sum_X {E^z_k}'(X)  $ with

E^z_k  _h, _n     ≤C   h^6 g^2_k  |z_k|      E^z_k' _h, _n     ≤C h^6 g_k  (|p_k|+|v_k| )|z_k|

Recall  $E^z_k    =  g_k^2    \B(\cH(w_k + C^z_k) -  \cH(w_k + C_k)\B)$.  The term $ \vep_k^z|\mathbb{T}_{N+M-k} |$ is the contribution of the constant term coming from $\cH_0$.    For the
rest  we follow the analysis of the previous lemma,  but now   we use the estimate from  Appendix <ref>
| C^z_k(x-y) - C_k(x-y) |  ≤|z_k|  e^-|x-y|/L

In the quartic term $\cH_4$ we have terms of the form
(w_k + C^z_k)^2 - (w_k + C_k)^2   =   2w_k(C^z_k-C_k)  +  ((C^z_k)^2 - C_k^2  )
The only short distance singularity is the integrable   $|x-y|^{-1} $ from the $w_k$.   Thus we
can estimate the term as in the previous lemma and
get a bound  $C   g^2_k  |z_k|  h^4 $.

In the quadratic term   $\cH_2$ we have terms of the form
(w_k + C^z_k)^3 - (w_k + C_k)^3   =  3w_k^2(C^z_k-C_k)  + 3 w_k ((C^z_k)^2 - C_k^2  )  +  ((C^z_k)^3 - C_k^3  )
Here the $w_k^2(x-y)  = \cO( |x-y|^{-2} )  $ term is not integrable and is potentially worrisome.
This occurs in a term like
∫(x)  w_k^2 (x-y) (C^z_k(x-y) -C_k(x-y) )  ψ(y)  dx dy
However we again replace $\psi(y)$
by $\psi(x)   + (\psi(y) - \psi (x) ) $.  The $\psi(x)$ term gives zero since the integrand is odd.      The $ \psi(y) - \psi(x)  $  term  has an estimate with an extra   $|x-y|$  and we are
again reduced to an integrable singularity and a bound  $ C   g^2_k  |z_k|  h^2$.

The terms in $\cH_6$ only have $C_k^z - C_k$  and an easy   $C   g^2_k  |z_k|  h^6$  estimate.
This completes the proof for $  E^z_k(X) $ and the proof for  $  {E^z_k}'(X) $ is the same.

$\de E_k(X, \psi,  \eta) = E_k( \psi + \eta ) - E_k(\psi) $

E_k _1/2 h, 1/4 h , _n    ≤ E_k_h, _n
Furthermore if   $\frac14 h>   h(C) $
< E_k >_C _ 1/2  h, _n  ≤h(C) h^-1    E_k_h, _n

Let $
E^+_k (X, u,  \psi, \eta )  \equiv   E_k (X,  \psi + u\eta )
Then we have the representation  for $r >  1$

E_k (X, ψ, η) = 1/2 πi ∫_|u|  = r  du/ u(u-1)   E^+ _k (X, u,  ψ, η)
Indeed $  E^+_k (X, u,  \psi, \eta ) $ is analytic in $u $ (i.e. the kernel is an analytic Banach space valued function)
and  for   $|u|  \leq r$ it  satisfies by lemma  <ref>
E^+ _k (X, u) _1/2h,h'  ≤ E_k(X)  _1/2 h + rh'
which is finite provided  $\frac12 h + rh' \leq   h$.
This gives the  bound
E_k(X) _1/2 h  ,h '    ≤r^-1   E_k(X) _1/2  h +rh'
For  the first result we take $h'    = \frac14 h $ and  $r$ slightly less than 2.

For  second result  integrate   $\int [\cdots] d\mu_{C} (\eta)$  and

E_k (X) _C(ψ)      = 1/2 πi ∫_|u|  = r  du/ u(u-1)     E^+ _k (X, u )  _C (ψ)
We have from  lemma <ref>   for $|u| \leq  r$
E^+ _k (X, u )  _C  _1/2 h  ≤   E^+ _k (X, u )  _1/2 h, r h(C)   ≤E_k(X) _1/2 h + rh(C)
E_k (X) _C _1/2 h   ≤r^ - 1   E_k(X) _1/2 h + rh(C)
The result follows by taking  $r$ slightly less than $ \frac12h h(C)^{-1}$.

§.§ cluster expansion

To complete the analysis of  the terms in the fluctuation integral (<ref>) we need a cluster expansion.   First we present it  in a more general context.
We still work on the torus $\mathbb{T}_{N+M-k}$, but consider general localized functions $E(\psi) = \sum_X E(X, \psi) $.     The covariance $C$ is either $C_k$  or  $C_k^z$.

The following   is a  version of the cluster expansion was   developed by  Gawedski and Kupiainen [17] for bosons, and
extended to fermions by    Feldman, Magnen, Rivasseau, Seneor  [13].
Technical details related to the norm we are using  go back to   Brydges and Yau [9] (although they have a rather different  treatment of fluctuation integrals).
Earlier references can be found in the cited papers.

Let   $h$ be sufficiently large depending on $L$,  and   $ \| E \|_{h,\Ga} \leq c_0    $ with $c_0 = \one$   sufficiently small. Then there are complex numbers
$E^{\#} (X) $ such that
∫exp( ∑_X E(X, ψ) ) d μ_C (ψ)   = exp( ∑_X  E^#(X) )
with the estimate
|E^#|_   ≡∑_X |  E^#(X) |(X)   ≤E _h, _4

part I:
First we make a Mayer expansion.  The    $\exp ( \sum_XE(X) ) $  is written as     $\prod_X\B(  ( e^{E(X) } -1)  +1\B) $.
The product is  expanded and intersecting terms    are grouped together.
We have
exp( ∑_X E(X)  ) = ∑_ { Y_β}   ∏_β  K(Y_β )
where the sum is over disjoint paved sets $\{ Y_{\beta} \}$, and where we  have   defined

K(Y) =  ∑_ {X_i}  →Y    ∏_i   ( e^ E(X_i)  - 1 ) )
Here  the sum is over distinct   overlap connected paved sets  $ \{X_i\} $ whose union is $Y$.
[overlap connected means the graph on the  $ \{X_i\} $ consisting of all pairs $ \{ X_i, X_j \} $ such that
$X_i \cap X_j  \neq \emptyset$ is connected.  The $X_i $ themselves may not be connected. ]
To estimate this we use
e^ E(X_i)  - 1 _h  ≤2 E(X_i) _h
and so
K(Y) _h   ≤∑_ {X_i  }       ∏_i  2  E(X_i)   _h
We also have from (<ref>) $\Ga_3 (Y) \leq  \prod_i  \Ga_3(X_i) $ and then
K _h, _3 =
∑_Y  ⊃  K(Y) _h _3(Y)   ≤∑_  {X_i} : ∪_i X_i  ⊃  ∏_i  2  E(X_i)   _h   _3(X_i)
We replace the sum over unordered paved sets  by a sum over ordered paved sets  and have
K _h, _3  ≤∑_n=1^∞  1/n! ∑_ (X_1, …, X_n)       ∏_i  2  E(X_i)   _h   _3(X_i)
with the restrictions that  $ (X_1, \dots, X_n)   $ are overlap connected  and that  $\cup_i X_i  \supset \sq  $.

1( ∪_i X_i ⊃) ≤∑_j=1^n   1(  X_j ⊃)
and when summed over $X_i$ this gives a contribution independent of $j$.  Thus we can assume $j=1$.  Then $\sum_{j=1}^n = n$  and we have
K _h, _3   ≤∑_n=1^∞  1/(n-1) ! ∑_ (X_1, …, X_n)      1(  X_1 ⊃)  ∏_i  2  E(X_i)   _h        _3(X_i)
still with the overlap connected condition.   But if $ (X_1, \dots, X_n)  $ are overlap connected there is also a tree graph connecting  these elements.
Thus we have
1(  (X_1, …, X_n)   overlap connected  ) ≤∑_τ 1(  τ connects  (X_1, …, X_n) )
We can regard the tree graph $\tau$ as rooted on $X_1$ and specified by
a mapping on $(1,2, \dots, n)$ satisfying  $\tau(j) < j $.  The  statement  that $\tau$ connects means $X_j \cap X_{\tau(j) } \neq \emptyset$
for $j =2, \dots,  n$.
Now we have.
K _h, _3   ≤∑_n=1^∞  2^n/(n-1) ! ∑_τ ∑_  (X_1, …, X_n) : X_j ∩X_τ(j)  ≠∅  1(  X_1 ⊃)  ∏^n_i=1     E(X_i)   _h      _3(X_i)

We estimate successively the sums over $X_n, X_{n-1} , \dots$.  In the first step we have
∑_  X_n ∩X_τ(n)  ≠∅     E(X_n)   _h      _3(X_n)  ≤∑_∈X_ τ(n)   ∑_X_n ⊃     E(X_n)   _h      _3(X_n)
=    |X_τ(n) | E _h, _3
In  the $j^{th} $ step   we will have  built up a factor $|X_j|^{d_j-1} $ where the incidence number  $d_j$ is  the number of elements in $   \tau^{-1} (j)  $.    We estimate this
by   $ |X_j|^{d_j-1} \leq (d_j-1)! e^{ |X_j| }  $ and  use    $ e^{ |X_j| } \Ga_3(X) = \Ga_4(X) $     to obtain

∑_  X_j ∩X_τ(j)  ≠∅  |X_j|^d_j-1    E(X_j)   _h      _3(X_j)  ≤
(d_j-1)!   ∑_  X_j ∩X_τ(j)  ≠∅    E(X_j)   _h    _4 (X_j)

≤      (d_j-1)!     |X_τ(j) | E _h, _4

The last step is just

∑_  X_1 ∩≠∅  |X_1|^d_1-1    E(X_1)   _h      _3(X_1)  ≤
(d_1-1)!   ∑_  X_1 ∩≠∅    E(X_1)   _h  _4(X_1)

=        (d_1-1)!   E _h, _4

Altogether then we have
K _h, _3   ≤∑_n=1^∞  2^n E ^n_h, _4 /(n-1) ! ∑_τ  ∏_j=1 ^n-1   (d_j-1)!
By Cayley's theorem the number of trees on $n$ vertices  with incidence numbers $d_j$ is
| τ| ≤ (n-2)!/  ∏_j=1^n  (d_j-1)!
From  this  the sum over $\tau$ is bounded by $(n-2)! 4^{n-1} $ (for details see [9] or  [11]).   Thus

K _h, _3   ≤∑_n=1^∞  8^n E ^n_h, _4  ≤E _h, _4

Part II:
Now we have

∫exp( ∑_X E(X, ψ) ) d μ_C(ψ)
= ∑_ { Y_β}  [  ∫∏_β  K(Y_β )        d μ_C(ψ)  ]
We are assuming that $C (x,y) =(\overline{   C_1(x,\cdot)},  C_2(\cdot, y) ) $ as in (<ref>).
Let  $\{U_{\al} \} $ be a partition of the torus  consisting of the $Y_{\beta} $ and all the unit blocks $\sq $ not in any $Y_{\beta} $.
Correspondingly we  introduce weakening parameters $0 \leq  s_{\al,\al'} \leq 1$  in each factor separately  and  we define
for  $i = 1,2$
C_i(s,x,y )  =   ∑_  1_U_ (x) C_i(x,y)  1_U_ (y)   + ∑_≠'    s_,'   1_U_ (x)  C_i(x,y)  1_U_'(y)
C (s, x,y) = (  C_1(s, x,·) ,  C_2(s, ·, y) )

If all the $ s_{\al,\al'}  =1 $  this is equal to $C$.  In the bracketed expression we express the value  $s_{\al,\al'}  =1 $
by expanding around $  s_{\al,\al'}  =0 $.  Then  (<ref>) can be written

∑_ { Y_β}   ∑_  ∫ds_   / s_   [  ∫∏_β  K(Y_β )        d μ_ C(s )    ] _s_^c =0
Here    $\ga$ be  a  subset of the set   of all  pairs  $\{ U_{\al} , U_{\al'} \}$,   that is a graph on vertices
$\{ U_{\al} \}$.
Also   $\int ds_{\ga} = \prod_{{\al,\al'} \in \ga} \int_0^1ds_{\al,\al'} $ and $\pa/ \pa s_{\ga} = \prod_{{\al,\al'} \in \ga} \pa /\pa s_{\al,\al'} $.
The sum includes $\ga = \emptyset$ which is the totally decoupled term $ \int    \prod_{\beta}  K(Y_{\beta} )        d \mu_{ C(0 ) } $.

Note   that not all $\ga$ contribute in  (<ref>).   Indeed  we can replace the covariance   $C(s)$ by  $ 1_Y C(s) 1_Y$ where $Y = \cup_{\beta} Y_{\beta} $.  Equivalently we can
replace $C_1(s)$ by $1_Y C_1(s) $ and    $C_2(s)$ by $C_2(s)1_Y $.    Then $C(s)$ is independent of $s_{\al, \al'} $ if      $\{ U_{\al} , U_{\al'} \}= \{\sq , \sq' \}  $   or if
$\{ U_{\al} , U_{\al'} \}= \{ Y_{\beta}, \sq\} $ and there is no matching pair $ \{\sq,  Y_{\beta'} \}   $.   In  either case if $\ga$ is a graph with such a line then  $\pa/ \pa s_{\ga} $ gives zero  and $\ga$  does not contribute.

The  graph   $\ga$     splits into connected components $\{ \ga_j \}$.     The covariance   $C(s)$
only connects points in the same  component  since  lines $\{ U_{\al} , U_{\al'} \}$  connecting different components are not in $\ga$ and so  have $s_{\al, \al'} =0$.
Thus we have the factorization
[  ∫∏_β  K(Y_β )        d μ_ C(s )    ] _s_^c =0
= ∏_j  [  ∫∏_Y_β   ∈_j   K(Y_β )        d μ_ C(s )  ]_s_^c =0
The sum over $\ga$ is regarded as a sum  over disjoint $\{\ga_j\} $
we   can write (<ref>)
∑_ { Y_β}     ∑_ { _j  }    ∏_j   ∫ds_ _j  / s__j     [  ∫∏_Y_β   ∈_j   K(Y_β )        d μ_ C(s )  ]_s_^c =0
In this expression   classify the terms in the sum over $ \{ Y_{\beta} \}  , \{ \ga_j  \} $ by the paved sets they generate and get

∫exp( ∑_X E(X, ψ) ) d μ_C(ψ)
= ∑_ { X_j }
∏_j  K^# (X_j  )
Here the sum is over disjoint paved sets  $\{ X_j \}$
and for   a paved set $X$
K^# (X)  =   ∑_ { Y_β}, →X   ∫ds_   / s_   [  ∫∏_β  K(Y_β )
d μ_C(s)    ] _s_^c =0
The sum is over  disjoint  $  \{ Y_{\beta}\}$ and additional disjoint  blocks $\sq$   whose union is $X$,   and  a graph $\ga$
connecting these sets.
In the following we make a slight change in notation explicitly recording the extra blocks $\{\sq_i\} $ which were implicit in $\ga$.
Thus we write
∑_ { Y_β}, →X  =  ∑_ { Y_β}, {_i}  →X    ∑_
where $X$ is the union of the indicated sets and $\ga$ connects them.

Part III
We want to estimate $K^{\#} (X)$ and we start with the bracketed integrals in the last equation.    We write
K(Y_β,   ψ, )
=  ∑_n_β = m_β     1/n_β!m_β!  ∫K_n_βm_β (Y_β, _β,  _β     )
ψ(_β)  (  _β  ) d  d where $ \bx_{\beta} =  (x_{{\beta},1},  \cdots ,  x_{{\beta},n_{\beta}} )$ and $\psi(\bx_{\beta})  =  \psi(x_{{\beta},1} ) , \cdots  \psi( x_{{\beta},n_{\beta}} ) $, etc.
Inserting this and carrying out the integral we have

∫∏_β  K(Y_β )        d μ_ C(s )

=     ∑_n_β = m_β    ∏_β  h^n_β + m_β /n_β ! m_β!  ∫∏_β  K_n_βm_β (Y_β, _β,  _β     )
(  { h^-1 C(s, x_β, i , y_β,j)  }   )    d  d

Here  $  \{ C(s, x_{\beta, i} , y_{\beta,j})  \}   $ are the entries of an $2N \times 2N$ matrix where $N = \sum_{\beta}  n_{\beta} =\sum_{\beta} m_{\beta}  $.  Since
$\det ( h^{-1} C)  =
h^{-2N} \det C$   the factors of $h$ in the above expression
cancel out.

We have

{ C(s, x_β, i , y_β,j)   =(   C_1(s,  x_β, i , ·),      C(s,  ·,  y_β,j) )
and then by Gram's inequality

|   (   { h^-1 C(s, x_β, i , y_β,j)  )  |     ≤∏_β,ih^-1/2    C_1(s, x_β,i , ·)_2
∏_ β,j h^-1/2     C_2(s,  ·,y_β,j)_2

From Appendix <ref> we have  $|C(x,y)| \leq \one e^{-|x-y|/L}$.    The same bound holds separately for $C_1,C_2$.
Hence    $\| C_i(s,x,\cdot) \|_2 \leq  \one L^2 $ and  we   chose $h$ large   so  $h^{-\frac12}  \| C_i(s, x, \cdot) \|_2 \leq  \one h^{-\frac12}   L^2 \leq 1 $.
Thus the determinant is less than one. Similarly  for  spatial derivatives

| ^   (   { h^-1 C(s, x_β, i , y_β,j) }  )  | ≤1
Thus as in lemma <ref>

|  ∫∏_β  K(Y_β )        d μ_ C(s )   | ≤∑_ { n_β, m_β }   ∏_β  h^n_β + m_β /n_β!m_β!      ∏_β   K_n_βm_β (Y_β) _'
= ∏_βK(Y_β)  _h
The  covariance $C(s)$  is actually  analytic  in  complex  $s_{\al,\al'}$ (it is the complex inner product in (<ref>)).  If  we take   $| s_{\al,\al'}|  \leq h^{\frac14}
\exp( d(U_{\al} ,U_{\al'} )/2L ) $
we have  $|  C_i(s,x,y)|  \leq  \one  h^{\frac14}   e^{-  |x-y| /2L}$  and therefore  for $h$ large   $  h ^{-\frac12} \| C_i(s,x,\cdot) \|_2 \leq  \one h^{-\frac14 }  L^2 \leq 1$.
The bounds (<ref>)  still hold  in this complex domain.
The integral $ \int    \prod_{\beta}  K(Y_{\beta} )  d \mu_{ C(s ) } $ is analytic in the same domain and satisfies the bound (<ref>).
By  Cauchy bounds for $|s_{\al, \al'}| \leq 1$
/ s_   [  ∫∏_β  K(Y_β )     d μ_C(s)    ] _s_^c =0
≤∏_ {,'}  ∈ h^-1/4  e^ -  d(U_,U_' )/2L   ∏_βK(Y_β)  _h
where $ \{\al,\al'\}  \in \ga$ means $ \{ U_{\al},U_{\al'} \}  \in \ga$.

|  K^# (X) |   ≤∑_ { Y_β},{ _i} →X  ∑_   ∏_{ ,'} ∈ h^-1/4  e^ -   d(U_,U_' )/2L   ∏_βK(Y_β)  _h
where $\{U_{\al} \} =  \{\{ Y_{\beta}\},\{ \sq_i\} \}$ and $\cup_{\al} U_{\al} =X$.
Part IV:
Next we estimate the norm  of $K^{\#}$.
First note that every connected graph $\ga$ has a  tree subgraph $\tau$ spanning the same vertices.  So  $\sum_{\ga} \leq \sum_{\tau } \sum_{\ga>\tau} $.
We split the factor $h^{-\frac14} e^{ -   d(U_{\al},U_{\al'} )/2L } $ into two factors $h^{-\frac18}   e^{ -   d(U_{\al},U_{\al'} )/4L } $ and in one factor estimate the
contribution of lines not in the tree $\tau$ by 1.
This gives

|  K^# (X)  |  ≤      ∑_ { Y_β},{ _i} →X   ∏_βK(Y_β)  _h  ∑_τ
∏_{,'} ∈τ  h^-1/8  e^ -   d(U_,U_' )/4L

[ ∑_> τ     ∏_ {,'} ∈  Ch^-1/8  e^ -   d(U_,U_' )/4L  ]

We estimate the bracketed expression by enlarging the  sum over connected $\ga > \tau$  to a sum over all graphs $\ga$ connected or not.
We have

∑_     ∏_{,'} ∈  Ch^-1/8  e^ -   d(U_,U_' )/4L
≤       ∏_{,'}   ( 1 +C h^-1/8
e^ -   d(U_,U_' )/4L  )

≤     exp(Ch^-1/8 ∑_{,'}
e^ -   d(U_,U_' )/4L )

≤    exp(C h^-1/8 ∑_| U_ |)   ≤∏_ e^|U_ |

Here we used
∑_{,'}  e^- d(U_,U_' )/4L
≤∑_{,'}    ∑_∈U_' e^- d(U_,)/4L
≤∑_   ∑_  e^- d(U_,)/4L
≤C ∑_ |U_ |
Now  we have
|  K^# (X)  |  ≤∑_ { Y_β},{ _i} →X   ∏_βK(Y_β)  _h ∏_ e^|U_ |
∑_τ  ∏_{,'} ∈τ  h^-1/8  e^ -   d(U_,U_' )/4L

Next we pass to an estimate on
$ |  K^{\#}  |_{\Ga_{1}  }  = \sum_{X\supset   \sq}  |  K^{\#} (X) |\Ga_1(X) $.
Repeatedly using the inequality (<ref>) we have
_1(X) ≤∏_β _1(Y_β ) ∏_i _1(_i)
∏_ { ,'} ∈τ θ( d(U_ , U_' )   )
The factor  $\prod_{\al} e^{|U_{\al} |}  $ changes the $\Ga_1$ here to $\Ga_2$.
We can kill the factor $\Ga_2(\sq_i) $ if we borrow a factor $h^{-1/16} $ from the product over lines.
Since $\theta$ is polynomially bounded      $ \theta( d( U_{\al}, U_{\al'} ) ) \leq C\exp(  d(U_{\al}, U_{\al'})/4L  )$
for some constant $C$.  Also we identify   the sum over  $X \supset \sq$ and $\{ Y_{\beta}\},\{ \sq_i\} \to X$ as just a sum over  $\{ Y_{\beta}\},\{ \sq_i\} $
restricted only  by the condition that one of them contain $\sq$, a fact which we temporarily leave out of the notation.
Thus we have

|  K^#  |__1      ≤     ∑_ { Y_β},{ _i}        ∑_τ
∏_ {,'} ∈τ  h^-1/16  e^ -   d(U_,U_' )/8L
∏_βK(Y_β)  _h _2(Y_β)

The sum over  $\{ Y_{\beta}\},\{ \sq_i\} $ is a sum over $\{U_{\al} \}$.
We classify the terms in the sum by the number of $Y_{\beta}$ and the number of $\sq_i$. .
So now

|  K^#  |__1       ≤∑_ℓ ∑_n+m = ℓ   [   ∑_ { U_ }      ∑_τ
∏_ {,'} ∈τ  h^-1/16  e^ -   d(U_,U_' )/8L
∏_βK(Y_β)  _h _2(Y_β)    ]

where the sum over $ \{ U_{\al} \}$ is now restricted  to have   $n$ elements $Y_{\beta} $ and $m$  elements $\sq_i$.

We work on the bracketed expression.     The sum over unordered   $ \{ U_{\al} \}$ with at least one element containing $\sq$  is written as a sum over ordered sets
$( U_1, \dots,  U_{\ell})  $ with $U_1 \supset \sq$.
The   tree is identified as a map $\tau$ on $(1, \dots,  \ell)$ with $\tau(j) <j$.  Still with the   restriction on the numbers of $Y_{\beta}$ and $\sq_i$  we have
[ ⋯] = 1/ℓ! ∑_ ( U_1, …U_ℓ )
∏_j=2^ℓ    h^-1/16  e^ -   d(U_j,U_τ(j)  )/8L  ∏_βK(Y_β)  _h _2(Y_β)
We successively do the sums over $U_{\ell},   U_{\ell -1}, \dots$.  In the first step if   $U_{\ell} $  is some $Y_{\beta} $ then the sum over $U_{\ell} $ is

∑_Y_β   e^ -  d(Y_β ,U_τ(ℓ)  ) /8L   K(Y_β ) _h _2 (Y_β )

≤      ∑_Y_β  ∑_⊂Y_β     e^ -  d(,U_τ(ℓ)  ) /8L    K(Y_β ) _h _2 (Y_β )

≤    ∑_    e^ -  d(,U_τ(ℓ)  ) /8L  K _h, _2
≤C | U_τ(ℓ)|  K _h, _2

Here in the second step we used $ \sum_{Y_{\beta} } \sum_{\sq \subset Y_{\beta}  } = \sum_{\sq}  \sum_{Y_{\beta} \supset \sq  }$.
If $U_{\ell} $ is some $\sq_i $  then just
∑__i    e^ -  d(_i  ,U_τ(ℓ)  ) /8L     ≤C  |U_τ(ℓ)|

The proof now proceeds as in part I.   In the $j^{th}$ step we again accumulate  a factor $|U_j|^{d_j-1} $.    If
$U_j = Y_{\beta}$ then  this is $|Y_{\beta}|^{d_j-1} \leq  (d_j-1)! e^{|Y_{\beta}|}$ and $e^{|Y_{\beta}|}\Ga_2(Y_{\beta}) =\Ga_3(Y_{\beta})$.
If $U_j = \sq_i$ then this is  $|\sq_i|^{d_j-1}=1$ and can be ignored.  Except for these changes the $j^{th} $ step is just like the first step.
The last step uses the pin $U_1 \supset \sq$.

We   find the bracketed expression in (<ref>) is bounded by $   \|K   \| ^n _{h, \Ga_3 }   (  h^{-1/16})^{\ell-1}   C ^{\ell}     $.  If  $\ell \geq 2$ then
$ (  h^{-1/16})^{\ell-1}  C ^{\ell}   \leq  (  h^{-1/32}C)^{\ell}  \leq (\frac12)^{\ell} $. Now the sum in (<ref>) for $\ell \geq 2$  is bounded by  a sum of      $ \|K   \| ^n _{h, \Ga_3 }  (\frac 12 ) ^{n+m} $
with the restrictions $n+m \geq 2$ and $n \geq 1$.    This gives has a  bound    $\one    \|K   \| _{h, \Ga_3 } $.  The case $\ell =1$ is special.  In this case
$K^{\#}(X)  = \int K(X) d \mu_{C} $ and by lemma <ref> if $h \geq h(C)$ then   $|K^{\#} |_{\Ga_1} \leq \|K \|_{h, \Ga_1}  \leq \|K \|_{h, \Ga_3} $. Altogether then

|  K^#  |__1   ≤K    _h, _3

Part V
Finally we claim that
∑_ { X_}
∏_  K^# (X_ )   =  exp( ∑_Y E^# (Y)  )
E^# (Y)  =
∑_n=1^∞  1/n!  ∑_(X_1, …X_n) : ∪_iX_i = Y  ρ^T(X_1, …, X_n) ∏_i K^# (X_i)
Here $ \rho^T(X_1, \dots, X_n) $  is a certain function which. vanishes unless $(X_1, \dots, X_n)$ are overlap connected,  and is bounded by the number of
tree graphs on $(1,2, \dots, n)$.     This is a standard combinatoric argument, see for example [11].
This is again estimated by a spanning tree argument as in Part I.   The result is an estimate   $|E^\#|_{ \Ga}    \leq \one | K^{\#} | _{ \Ga_1} $.
Combining this with    (<ref>) and (<ref>) we have the desired result
|E^#|_    ≤| K^# | __1   ≤ K  _h, _3   ≤E  _h, _4
This completes the proof

$E^\#$  is actually an  analytic function of  $E$  on the ball $\|E\|_{h,\Ga} \leq c_0$  in  the complex Banach space $\cG_{h, \Ga_4}$, i.e.  it is continuously differentiable.
To see this it is sufficient to establish the analyticity  for each of the steps  $E \to K \to K^\# \to E^\#$.    We discuss the first step in more detail.   First define
$F(X) = e^{E(X) } -1$.  This is analytic since it is the composition of two analytic functions.   Then as in (<ref>)

K(Y) =  ∑_ {X_i}  →Y    ∏_i F(X_i)    = ∑_n=0 ^∞ 1/n!  ∑_(X_1, …X_n) →Y ∏^n_i=1 F(X_i)
and it suffices to show this is an analytic function of $F$.

Define a   bounded  multi-linear function $K_n$  from $ \cG_{h, \Ga_4} \times \cdots \times \cG_{h, \Ga_4}$ to complex-valued localized functions
[K_n(F_1, …F_n) ](Y)  =  ∑_(X_1, …X_n) →Y  F_1(X_1) ⋯F_n(X_n)
and let $[K_n(F)](Y)   = [K_n(F, \dots,  F) ](Y) $ be the associated homogeneous polynomial.    Then we have

K(Y)  = ∑_n=0 ^∞ 1/n!  [ K_n(F) ](Y )
The  proof of the theorem shows  this is an absolutely convergent power series.  As such it  is an     analytic function.  The other
steps are similar.     See  Appendix A in  [24] for more on
analytic functions on a complex Banach space.

We now quote  a variation of this  cluster expansion.   We suppose  that there are additional fermi fields present that are not being integrated out.
So our function $E(X,\psi, \eta) $ is assumed to has the form (<ref>)

Let $h$ be sufficiently large depending on $L$,  and let $\|E \|_{h',h, \Ga} \leq c_0$ sufficiently small.   Then  there are paved functions
$E^{\#} (X) $ such that
∫exp( ∑_X E(X, ψ, η) ) d μ_C (η)   = exp( ∑_X  E^#(X, ψ) )
E^#  _h',    ≤E _h',h, _4

The proof follows the proof of theorem <ref>  with the fields $\psi$ as spectators.   The analyticity remarks hold in this case as well.   We omit the details.

§.§ more estimates

With the cluster expansion as a tool, we are now in a position to complete our analysis of the expansion of the fluctuation integral  starting with the remainder term
in (<ref>).

Let  $h$ be  sufficiently large and  $g_k$ sufficiently small.

* $W_k(X, \psi, \eta)   =  \B( V_k+V_k'+ Q_k^{\reg}  + {Q'_k}^{\reg} \B) (X,  \psi + \eta)    + \de E_k(X, \psi, \eta) $ satisfies

W_k _ 1/2 h, 1/4  h, _4      ≤C h^4 g_k

* For $t$ complex with $|t| \leq c_0 (Ch^4 g_k) ^{-1} $ there is $W_k^{\#}(t)$
such that
e^-tW_k    _C^z_k   =   exp( - ∑_XW^#_k(t,X)  )
which is analytic in $t$ and satisfies

W^#_k(t)  _1/2h ,   ≤
* The remainder term

E^#_k(X)  ≡1/2 πi   ∫_ |t| = c_0 (Ch^4 g_k) ^-1   dt/t^3(t-1)   W^#_k(t,X)

E_k^# _1/2 h,    ≤C   h^12    g_k^3

* By   lemma <ref> and (<ref>),
$ V_k^+ (X,  \psi,  \eta)    \equiv V_k (X,  \psi + \eta) $ and $ {V'}^+  _k (X,  \psi,  \eta) $

V^+ _k _ h, h,_4     ≤       V_k _2h,_4    ≤C  h^4 g_k

V'^+  _k  _ h,h, _4       ≤       V'_k  _2h, _4     ≤Ch^4 (|p_k| + |v_k|)     ≤C C_E h^4 g_k^2 ≤Ch^4g_k

The bounds on the other terms in $W_k$ are even smaller.   However the bound
$ \| \de E_k \|_{\frac 12 h, \frac 14  h, \Ga_4   }   \leq   \one  \|E_k\|_{h, \Ga_4 } \leq \one C_Eg_k^3$ from lemma <ref>
requires that we weaken the $h$ parameter.
Altogether we have  the announced (<ref>).

* For  $|t| \leq    c_0 (Ch^4 g_k) ^{-1}  $  we have
\| t  W_k \|_{\frac12 ,\frac 14 h, \Ga_4  }  \leq   c_0
which is  small.   The result follows by the cluster expansion theorem <ref>.

* This   follows from     $\| W^\#_k(t)  \|_{ \frac 12 h, \Ga }  \leq   \one  $ and  $|t|^{-3}=c_0^{-3} (Ch^4 g_k )^3  $.

Another   $\cO(g_k^3) $  term is   $ < V_k,  Q_k^{\reg} >^T _{C_k^z} $  which we now estimate:

$< V_k,   Q_k^{\reg} >^T_{C_k^z}$ has a polymer expansion which satisfies
< V_k,   Q_k^ >_C_k^z ^T _1/2 h,   ≤C h^10  g_k^3

We could compute the integral and give a direct bound.   But since we have the cluster expansion at our disposal
it is quicker to use it.
W(t,s ) = tV_k + s  Q_k^
We have the bounds   $\|V_k\|_{h, \Ga_4} \leq Ch^4g_k$ and $\| Q_k^{\reg} \|_{h, \Ga_4}  \leq Ch^6g^2_k$ (lemma  <ref>).
So if we take   $|t| \leq  \frac 12  c_0 ( Ch^4g_k)^{- 1} $ and $|s| \leq  \frac 12  c_0 ( Ch^6g^2_k)^{- 1} $
W^+ (t,s) _1/2h, 1/2h,    ≤W(t,s) _h, _4 ≤c_0
By the cluster expansion (theorem <ref>)  there  is a $W^{ \#}(t,s, \psi ) = \sum_X W^{ \#}(t,s,X, \psi ) $ such that
e^ W(t,s)   _C (ψ)
=    ∫(e^  W^+ (t,s)    )(ψ, η) dμ_C  (η)             = e^  W^ #(t,s,X, ψ)
with  $\|  W^{ \#}(t,s) \|_{\frac12 h, \Ga}  \leq \one   $.
Taking derivatives at $s=t=0$ we find that
< V_k,   Q_k^ >_C ^T =      ^2 W^#  / t s (0,0)
So   $< V_k,  Q_k^{\reg}>^T  = \sum_X E(X) $ with  $E(X) = ( \pa^2 W^{\# } /\pa t \pa s ) (0,0,X)$.  Since $W^\# (t,s, X ) $ is analytic in $(t,s)$
we can write this as
E(X) =    1/ ( 2 πi)^2  ∫_ |t| =  1/2  c_0 ( Ch^4g_k)^- 1  ∫_|s| =  1/2  c_0 ( Ch^6g^2_k)^- 1   dt ds/ t^2 s^2W^# (t,s, X )
This  has the bound    $\| E\|_{\frac 12h, \Ga}  \leq   C h^{10} g_k^3$ as claimed.
This completes the proof.

The  higher order terms in (<ref>)  can be handled in the same way and are even smaller.  For example  the fourth order terms are

< V_k, E_k  >^T_1/2h,   ≤      Ch^4g_k  E_k_h, _4  ≤CC_E h^4 g_k^4

< V'_k, Q_k^ >^T_1/2h, ,    < V_k,  Q'_k^ >^T_1/2h,    ≤      C h^10 (|p_k|+|v_k| )g^2_k   ≤CC_E h^10 g_k^4

< V'_k, V'_k >^T_1/2h,   ≤      C h^8 (|p_k|+|v_k| )^2   ≤CC^2_E h^8 g_k^4

§.§ completion of the proof

We collect the results and finish the proof of theorem <ref>.
From (<ref>)     the partition function is
$ \sZ =   \int    \exp (   \tilde S_{k+1} )    \        d \mu_{G_{k+1,L } }    $
S̃_k+1  =    ( _k +   '_k -   ”_k )|_N+M-k  | -z_k    ψ+ E_k  + W_k^#
The $E_k$ is inherited from $S_k$, but
in  lemma   <ref>  the relevant parts   are extracted  by $E_k = E_k^{\loc} +  \cR E_k $.
The $ W_k^\#$ is the contribution of the fluctuation integral  $\Xi_k$ as defined in section    <ref>, and further  developed in subsequent sections.
Collecting terms from   (<ref>), (<ref>)-(<ref>),  (<ref>), (<ref>), (<ref>), (<ref>),  (<ref>) we find

S̃_k+1  (ψ)     =
(  _k   + ^*_k) | _M+N -k |

+ (g_k   +   β_k  g^2_k - 2β' _kg_kp_k - 4β'_k g_kv_k  +  g^*_k )  ∫(ψ)^2

- ( z_k +  θ_k  g_k^2  -θ_k^p g_kp_k- θ_k^v g_kv_k  +  z^* _k  )   ∫ψ

+  (p_k -  2β_k' g_kv_k+ p_k^*)∫(_5 ψ)^2 + (v_k -  β'_k g_kp_k+  v_k^*)∫(_μ ψ)^2

+  g_k^2 (w_k + C_k  )   +   g_kp_k ^p(w'_k + C_k  ) +  g_kv_k ^v(w'_k + C_k  )   +   E_k   + E^*_k

E_k^* =       E_k^#+  E_k^z+ E^z_k' + < E_k >    +  the terms (<ref>)

_k^*  =       '_k -   ”_k  +   ^E_k +  g_k^2 ^Q_k +  g_kp_k_k^p,Q  +   g_kv_k_k^v,Q         +   ^z_k +     ^z_k'

We now feed in the bounds we have established.  We use   (<ref>),  (<ref>),  the bound  $\|< \de E_k> \|_{\frac12  h,  \Ga} \leq CC_E h^{-1} g_k^3 $ from lemma <ref>,  and (<ref>)
and find

E_k^* _1/2 h ,   ≤C(  h^12   + C_Z h^6   + C_Eh^-1+C_Eh^10 g_k) g_k^3
≤C_E h^-1/2   g_k^3
Here we take $h$ large enough so $Ch^{-\frac12} \leq 1$ and   $C_E$ sufficiently large  depending on $h, C_Z$.
We also have from lemma <ref>
|g_k^*|, h^-2  | z_k^*|, | p_k^*|, | v_k^*|  ≤C_E h^-4 g_k^3
These  are the claimed bounds.

Now we scale replacing $\psi$ on $\bbT_{M+N-k}$  by $\psi_L$ with $\psi$ on $\bbT_{M+N-k-1}$.  Then   $d \mu_{G_{k+1,L } }$ scales to
$ d \mu_{G_{k+1 }}$,
and  $\tilde   S_{k+1} $ scales to   $S_{k+1} (\psi) = \tilde S_{k+1} ( \psi_L) $ and $\sZ = \int e^{S_{k+1} } d \mu_{G_{k+1}}$.

We identify the new values  $\vep_{k+1},   z_{k+1}, g_{k+1}$ as given by (<ref>),(<ref>).
Once $g_{k+1}$ is identified we can replace  $ g_{k}^2  \hat \cH(w_k + C_k  ) $ by  $g_{k+1}^2  \hat \cH( w_k + C_k   )$ with
a small $\cO(g_k^3)$ error which can be absorbed  into $E_k^*$ and $\vep_k^*$.
g^2_k+1 ( w_k + C_k  , ψ_L  ) =g^2_k+1 ( (w_k + C_k)_L^-1   , ψ) =g^2_k+1( w_k+1   , ψ) =  Q_k+1^(ψ)
Similarly we identify  $ {Q'_{k+1}}^{\reg} $.
Reblocking and scaling changes $ \cR E_k  + E^*_k $ to  $  E_{k+1} =    \cL (  \cR E_k  +   E^*_k )$.   (We  estimated $E_{k+1} $ in section <ref>.)

Altogether we have    the announced

S_k+1(ψ)    =      ∫( _k+1  -  z_k+1 ψ+   g_k+1   ( ψ̅ψ)^2 +   p_k+1 (_5 ψ)^2    + v_k+1  (_μ ψ)^2   )

+ Q_k^ + Q'_k+1^  +  E_k+ 1

This completes the proof.

§.§ some derivatives

As a corollary to theorem <ref>  we   estimate some derivatives of $E^*( E_k, g_k, z_k, p_k, v_k) $ as defined in (<ref>).   The theorem
gives estimates  in the domain  (<ref>). For our estimates we shrink the domain by a factor $\frac12$.   The new domain is

|g_k|   <  1/2 g_max  E_k _h, _4   ≤1/2 C_E g_k^3 |z_k| ≤1/2 C_Z g_k  |p_k|, |v_k|,   ≤1/2 C_E g_k ^2

In the domain (<ref>)

D_E_k  E_k^* _( _h, _4, _ h/2, )  ≤    h^-1/2

E_k^* / g_k _h/2, ,   E_k^* / z_k _h/2,   ≤     C_E g_k^2

E_k^* / p_k _h/2, ,   E_k^* / v_k _h/2,    ≤
CC_E h^8 g_k^2

$ E_k^* $ is an analytic function of $E_k$   in  the full domain  $\| E_k \|_{h, \Ga_4}   \leq   C_E g_k ^3$.  Indeed in the sequence of maps  $E_k \to W_k \to W_k^\# \to E_k^\#$ in lemma  <ref>,  the maps $ W_k \to W_k^\# $ is analytic by the remarks following theorem <ref>.  The other steps are linear and hence analytic.  So $E_k^\#$ is analytic  and the other contributions to $E_k^*$ are
either linear or quadratic and hence analytic.

For     $\| E_k \|_{h, \Ga_4}   \leq  \frac12 C_E g_k^3$
and $\| \dot{E_k}\|_{h, \Ga_4}  =  1$  and $|t| \leq    \frac12 C_E g_k^3$       we have   $\| E_k+ t \dot{E_k} \|_{h, \Ga_4}   \leq   C_E g_k^3$
and we are in the domain of analyticity.
The derivative $D_{E_k}  E_k^*$ is a linear function on $\cG_{h, \Ga_4}$
and we can write
( D_E_k E_k^* ) ( E_k),   Ė_̇k̇ = d/dt  E_k^*( E_k + t Ė_̇k̇) |_t=0
= 1/2 πi  ∫_|t| = 1/2 C_E g_k^3 dt/t^2     E_k^*(E_k + t Ė_̇k̇ )
The theorem says   $\| E_k^*\|_{h/2, \Ga}  \leq C_Eh^{-\frac12} g_k^3$ which yields
( D_E_k E_k^* ) ( E_k),   Ė_̇k̇ _h/2,   ≤( 1/2 C_E g_k^3)^-1   C_Eh^-1/2  g_k ^3 ≤2h^-1/2
Since $\|D_{E_k}  E_k^* \|_{\cL( \cG_{h, \Ga_4}, \cG_{ h/2, \Ga}) } $ is the supremum of this over $\| \dot{E_k}\|_{h, \Ga_4}  =  1$ we have the stated bound.

For the second bound   note that  $ E_k^* $ is analytic  in  $|g_k| \leq g_{\max} $. Indeed inspection of the sequence of maps   $W_k \to W_k^\# \to E_k^\# $     shows that each
preserves analyticity and the other contributions to   $ E_k^* $ are analytic as well.   With complex $g_k$ we have the basic bound $\|E_k^*\|_{h/2, \Ga} \leq C_E h^{- \frac12}  |g_k|^3$ as in the theorem.

Now  for real $g_k < \frac12 g_{\max} $ consider complex $g'_k$ with $|g_k'-g_k| < g_k $.  In this domain $|g'_k| < 2 g_k < g_{\max} $.  So $E_k^*$ is
analytic here with the bound   $\|E_k^*\|_{h/2, \Ga} \leq C_E h^{- \frac12}  |g'_k|^3   \leq 8 C_E h^{- \frac12}  g_k^3 $.  Then by a Cauchy inequality
the derivative at $g_k$ satisfies
E_k^* / g_k _h/2,   ≤g_k^-1  ( 8C_E h^- 1/2  g_k^3 )  ≤C_Eg_k^2

Similarly for   the third bound  $E^*_k$    is analytic  in  $|z_k| \leq C_Z g_k  $.  Then in the smaller domain
$|z_k| \leq \frac12  C_Z g_k $   by a Cauchy inequality
E_k^* / z_k _h/2,   ≤( 1/2  C_Z g_k  )^-1 ( C_E h^- 1/2  g_k ^3)   ≤C_E   g_k^2
For the derivative in $p_k$   we  look at  $E_k^*$ more carefully.      It has the form  $ E_k^* =    E_k^\#+ {E^z_k}' $  +  the terms (<ref>)  plus terms
that do not depend on $p_k$.     The first  term  $E_k^\#$  is analytic  in  $|p_k| \leq   g_k $   (just as  for  $g_k$) and satisfies the bound (<ref>) there.  Then by a Cauchy inequality
in the smaller domain $ |p_k| \leq   \frac12  g_k $  (and hence for $|p_k| \leq  \frac12 C_E g_k ^2 $)
E_k^# / p_k _h/2,   ≤( 1/2   g_k  ) ^-1   ( C   h^12    g_k^3 )
≤C h^12   g_k^2 ≤CC_Eg_k^2
For the other terms are either linear or quadratic in $p_k$, and it is better to evaluate derivatives directly.    The term    ${E^z_k}'  $ is proportional to $g_k p_k z_k$,  the derivative reduces this to $g_k z_k$, and as in lemma <ref> it is bounded by $CC_Vh^6g_k^2 \leq CC_Eg_k^2$.
Fourth order terms in   (<ref>) are estimated in (<ref>).   The derivative in $p_k$ reduces the  bounds by $C_Eg_k^2$
and they are all less than $CC_Eh^8g_k^2$.
Altogether the   bound is the announced  $\| \pa E_k^* / \pa p_k \|_{h/2, \Ga} \leq CC_Eh^8g^2_k$.   The bound on   $ \pa E_k^* / \pa v_k  $ is just the same.

§ THE  FLOW

We  study the flow of the  renormalization group equations (<ref>),(<ref>).
Our basic structural stability  result is the following.
Let $ L,h,C_v, C_E$ be chosen as in theorem <ref>  and then let   $g_f<  \frac12g_{\max} $ be sufficiently small.  Then the flow equations

E_k+1 =     ( E_k + E_k^*)

g_k+1   =    g_k   +   β_k  g^2_k - 2β' _kg_kp_k - 4β'_k g_kv_k  +  g^*_k

z_k+1  =     z_k +  θ_k  g_k^2- θ^p_k g_kp_k - θ^v_k g_kv_k+  z^* _k

p_k+1   =    p_k -    2 β_k' g_kv_k+ p_k^*

v_k+1   =     v_k-     β'_k g_kp_k+  v_k^*

_k+1 =     L^2 (_k + _k^*)

have a unique solution   $\{E_k, g_k,  z_k, p_k, v_k, \vep_k  \}_{0  \leq k \leq N} $
with  boundary conditions

g_N = g_f,        _N =0      and      E_0 =0,      z_0=0,    p_0=0,   v_0 =0

The problem is an analysis of the flow around
the origin.  As noted the condition  $\beta_k>0$ means   the coupling constant grows which it what we want for an ultraviolet problem.
Our analysis follows the work of Bauerschmidt, Brydges, and Slade [5]  who study the case   $\beta_k  <0$
for an infrared problem.

The proof will occupy the remainder of this section, and includes bounds of the various quantities.
For most of the analysis we can ignore  the energy density $\vep_k$  since it does not affect the other variables.  The final condition $\vep_N =0 $
on the energy density is quite arbitrary.    One might contemplate tuning the  field strength  so the final  $z_N$   takes some pre-assigned value,  but for now it
is simpler to work with the initial  condition $z_0 =0$.

§.§ the quadratic flow

We start by analyzing the quadratic flow of $g_k,z_k$, dropping terms of order $\cO(g_k^3)$.
Recall that from  lemma <ref> there are positive  constants $C_{\pm}, C $ so that $C_- \leq \beta_k \leq C_+ $ and $|\theta_k| \leq C$.
(We have absorbed the factors $n-1$ into the constants.)

* Define   $\bar g_k$  for $0 \leq k \leq N$  by
\bg_{k+1}  =    \bg_k  + \beta_{k}  \bg_k^2
and   $ \bg_N = g_f>0 $.   For $g_f$ sufficiently small it is bounded by

g_f/ 1  +    C_+ g_f(N-k)  ≤g̅_k ≤ g_f/  1 + 1/2 C_- g_f(N-k)

* Define    $\bar z_k$ for  $ 0 \leq k \leq N$  by  $\bar  z_{k+1}  =\bar  z_k + \theta_k    \bg_k^2  $
and  $z_0 = 0$.    Then   $|\bar z_k| \leq  C \bg_k$

$ \bar g_{k+1}>0 $  determines $ \bar g_k> 0 $   as the unique positive root of  $\beta_k   \bar g_k^2 +   \bar g_k -  \bar g_{k+1}=0 $
which  is
g̅_k   = 1/2 β_k  ( -1 + √( 1 + 4 β_k  g̅_k+1  ) )
So  starting with $\bar g_N = g_f$ we get $\bar g_{N-1} $,then $\bar g_{N-2}$, etc. and
$ \bar g_N >\bar g_{N-1}  >  \cdots > \bar g_0>0$.

For the bound  it is convenient to work with $\bar g_k^{-1} $ rather than $\bar g_k$.    We

g̅_k^-1   - g̅_k+1^-1  =       g̅_k^-1  - (  g̅_k  +  β_k g̅_k^2)^-1
=      β_k g̅_k^2/ g̅_k(g̅_k + β_k g̅_k^2)
=      β_k /1 + β_k g̅_k

It follows that
C_-/ 1 + C_+ g̅_k    ≤( g̅_k^-1   - g̅_k+1^-1 )  ≤C_+/ 1 + C_- g̅_k
We  can assume $C_+ \bar g_k  \leq C_+  g_f \leq 1$ and so
1/2 C_-
≤(g̅_k^-1   - g̅_k+1^-1 ) ≤C_+    Then since
g̅_k^-1  = g_f^-1   +  ∑_j = k ^N-1   ( g̅_j^-1   - g̅_j+1^-1 )
we have
g_f^-1  +  1/2 C_-(N-k)   ≤g̅_k^-1   ≤g_f^-1  +  C_+(N-k)
and so
g_f/ 1  +    C_+  g_f(N-k)  ≤g̅_k  ≤  g_f/ 1  +  1/2 C_- g_f(N-k)

For the field strength  the solution is
z̅_k =  ∑_ℓ=0^k-1 θ_ℓ _ℓ^2
We know  $|\theta_{\ell}|  \leq C$  and
by  the following remark the sum over    $\bg_{\ell}^2$ is  bounded by $C \bg_k$.    Thus     $|\bar z_k| \leq C \bg_{k}$.

We will use the following estimates similar to those of  Lemma 2.1 in [5].
One can deduce directly from the equation $\bg_{k+1} = \bg_k + \beta_k \bg_k^2$  that for $0 \leq j \leq k \leq N $

∑_ℓ= j^k β_ℓ   g̅_ℓ   ≤      |log_j |

∑_ℓ= j^k β_ℓ   g̅_ ℓ^n | log_ℓ|^m   ≤     _k+1 ^n-1   | log_k+1 |^m       n>1, m ≥0

Or since $C_- \leq     \beta_{\ell} \leq C_+$ we have for $C = \one C_-^{-1} $

∑_ℓ= j^k    g̅_ℓ   ≤     C |log_j |

∑_ℓ= j^k      g̅_ ℓ^n | log_ℓ|^m   ≤     C _k+1 ^n-1   | log_k+1 |^m       n>1, m ≥0

§.§ reduction of the problem

Now consider the general case.
We make some changes in notation defining

_k =   ( E_k +  E_k^*)  ρ_k  =  (g_k^*, z_k^*,p_k^*,v_k^*)

and also the explicit part of the flow

ϕ̅( g_k, z_k,  p_k,v_k)
( g_k +   β_k  g^2_k - 2β' _kg_kp_k - 4β'_k g_kv_k,     z_k +  θ_k  g_k^2- θ^p_k g_k p_k - θ^v_k g_kv_k,

p_k -    2 β_k' g_kv_k,   v_k-     β'_k g_kp_k)

Then  the RG flow is given by

( E_k+1,  g_k+1,   z_k+1,p_k+1,  v_k+1  )  =    Φ_k ( E_k, g_k,  z_k, p_k,v_k)

≡     ( _k(E_k, g_k, z_k, p_k,v_k),        ϕ̅_k( g_k, z_k,p_k,v_k )   +    ρ_k(E_k, g_k, z_k, p_k,v_k )  )

We look for solutions with the boundary conditions (<ref>).
We introduce the notation
x_k = (E_k, g_k, z_k,p_k,v_k)
Then (<ref>)   is also written

x_ k +1  = Φ_k (x_k)  ≡( _k(x_k ), ϕ̅_k (x_k)   +     ρ_k(x_k) )

We introduce an approximate flow  with no feedback from the $E_k$ to the coupling constants.
So we drop the  $\rho_k $   and study

x_k+1  =  Φ̅_k (x_k )  ≡(_k (x_k),  ϕ̅( x_k)    )
with the boundary conditions  (<ref>).  This has a unique solution  of the form
x̅_k  =( E̅_k ,  g̅_k, z̅_k,  p̅_k,  v̅_k       )
with   $\bar p_k=0 ,  \bar v_k=0  $. The $( \bar   g_k, \bar z_k) $ is the solution  of  $( \bar   g_{k+1} , \bar z_{k+1} )   = \bar \phi( \bar   g_k, \bar z_k ,0,0)$
defined in the previous section.
The  $\bar E_k$
is the solution of
$ \bar  E_{k+1}   =  \si_k   (\bar  E_k,  \bar   g_k, \bar z_k,0,0) $  starting with $\bar E_0=0$.   (Note that this   involves   $ E^*_k(\bar E_k, \bg_k,  \bar z_k, 0,0)$).        By the remark after theorem <ref>, $\bar E_k$    is well-defined for all  $k$ and it satisfies

E̅_k _h, _4   ≤( L^-1  +  L^2 h^-1/2 ) C_E g̅_k^3 ≤C_E g̅_k^3

Next define   for $0 \leq t \leq 1$

x_k+1 = Φ^t_k(x_k  )  =     ( _k(x_k  ), ϕ̅_k (x_k )   +   t   ρ_k(x_k ) )
This interpolates between $\Phi^0_k(x_k)  = \bar \Phi_k(x_k) $ and $\Phi^1_k (x_k)= \Phi_k(x_k) $.
We are looking for a  family of sequences  $x_k(t) $,   each in the domain of theorem <ref>,     satisfying this recursion with the  boundary condition (<ref>).     At  $t=0$ we have such a sequence namely  $x_k(0) = \bar x_k$,

If $x_k =x_k(t)$ is a such a  solution of (<ref>),  then differentiating
$ x_{k+1}(t) = \Phi^t_k(x_k(t)  )  \equiv  \Phi_k(t,x_k(t)  ) $
we have a solution to the ODE

ẋ_k+1(t)  = (  D_x   Φ^t_k (x_k(t) ) ẋ_k (t)   + (0, ρ_k(x_k(t) )    x_k(0) = x̅_k
Conversely a solution of (<ref>)  gives a solution of (<ref>).

As an aid to studying the last equation we study a related equation.   Fix sequences $x_k$ in the domain of theorem 1   and $r_k$ and consider
the  linear  equation for $y_k=(  E_k,  g_k, z_k, p_k,v_k   )$

y_k+1  =     (  D_x   Φ^t_k (x_k ) )   y_k    + r_k
For  this equation null boundary conditions

g_N =  0       and     E_0 = 0, z_0 = 0,  p_0=0, v_0=0
are appropriate as we will see.

Furthermore we define $\bar \Phi^0_k$ by setting $t=0$ and also dropping the terms $\sigma_k(x_k) $ so that

Φ̅^0_k(x_k) = (0, ϕ̅(x_k) )=  (0, ϕ̅(g_k,z_k , p_k, v_k ) )
We replace     (<ref>) by the simpler   low order equation.

y_k+1  =  (    D_x   Φ̅^0_k  (x̅_k )  )  y_k    + r_k
also with null  boundary conditions.  We will  consider solutions of all  these equations in reverse order.

In studying these it will be convenient to introduce the following norms on spaces of sequences $x=(x_0,x_1, \dots,  x_N)$.

x _X_w
=      sup_k  {  g̅_k^-3 E_k _h,_4 ,  g̅_k^-2  |logg̅_k| ^-1|g_k|,  g̅_k^-2  |logg̅_k| ^-1|z_k|,   g̅_k^-2 |p_k|,  g̅_k^-2 |v_k|   }

x _X_r
=      sup_k  {   g̅_k^-3 E_k _h, _4 ,   g̅_k^-3 |g_k|,  g̅_k^-3  |z_k|,  ^-1 g̅_k^-3 |p_k|, g̅_k^-3 |v_k|  }

$X_w, X_r$  are the  Banach spaces for which these norms are finite.
The norms are chosen to accommodate the next lemma.

§.§ the first  linear equation

We study the linear equation (<ref>)  with the notation $L_k =   D_x   \bar \Phi^0_{k } (\bar   x_k )$

For  $r_k  \in X_r$ there exists
a unique solution  $y_k =(  E_k, g_k, z_k , p_k, v_k)  $  of

y_k+1  =   L_k   y_k    + r_k
with null boundary conditions (<ref>) .
It   satisfies for some constant $C$

y   _X_w  ≤C  r_X_r
The solution is a linear function of $r$ and  we write it as $y = S^0r$ so
the estimate  says
S^0_(X_r,X_w)  ≤C

Let $r_k = (r_k^E, r_k^g, r_k^z,  r_k^p, r_k^v )$.  The operator $L_k$ is zero on the first component so we just have
$E_{k+1}  = r^E_{k} $.   Since $ \|E_{k+1} \|_{h, \Ga_4}   = \| r^E_{k}  \|_{h, \Ga_4}   \leq  \bg_k^3 \| r \|_{X_r} $
we have for $( \pi_E y)_{k+1} = E_{k+1} $
π_E y _X_w   = sup_k   g̅_k+1^-3 E_k+1 _h, _4
≤ r _X_r

In the remaining variables $(  g_k, z_k ,p_k, v_k ) $  we compute from  (<ref>):

D_x   Φ̅^0_k  (   x_k )  = (
1+ 2 β_k  g_k -2 β_k' p_k  -4β'_k v_k     0       - 2 β_k' g_k     -4β_k' g_k

2 θ_k  g_k      1    - θ^p_k g_k      - θ^v_k g_k

-2 β_k'v_k     0    1      -2β_k' g_k

-  β_k' p_k     0     - β_k' g_k   1

We evaluate at
$\bg_k, \bar z_k$ and $ \bar v_k =0, \bar p_k=0$  and get

L_k =  D_x   Φ̅^0_k  (x̅_k )  = (
1+ 2 β_k g̅_k    0        -2 β_k' _k     -4 β_k' _k

2 θ_k g̅_k      1   -  θ^p_k _k     -  θ^v_k _k

0     0    1     -2 β_k' _k

0     0    -   β_k' _k   1

Our strategy is to first   solve (<ref>)  for $p_k,v_k$ with zero initial condition.  Then solve for $g_k$  with  final condition $g_N = g_f$.   Then solve for $z_k$
with zero initial condition.

The equations for $p_k,v_k$ are

p_ k+1  =      p_k  - 2 β_k' _k v_k +r^p_k

v_ k+1  =      v_k  - β_k' _k v_k + r^v_k

The matrix of coefficients on the right side  (i.e. the lower right corner of (<ref>) ) has eigenvalues $1 \pm \sqrt{2} \beta_k' \bg_k$ and a matrix
of eigenvectors can be taken of the form $I + \cO( \bg_k)$.  Thus we can change  to new variables  $\tilde p_k,\tilde v_k$ and  new $\tilde  r_k^p, \tilde r_k^v  $ which satisfy

p̃_ k+1  =     (1 + √(2) β_k' _k)  p̃_k  + r̃_k^p

ṽ_ k+1  =        (1 -  √(2) β_k' _k) ṽ_k  + r̃_k^v

It suffices to prove bounds in these variables.

The solution for $\tilde p_k$ with $\tilde p_0 = 0$ is
p̃_k =   ∑_ℓ=0 ^k-1(  ∏_i = ℓ+1 ^k-1 (1 + √(2) β_i' _i) ) r̃_ℓ ^p
In the product we take $\beta'_i \leq \beta_i$ and then
by lemma <ref> to follow with $\gamma = \sqrt 2$
∏_i = ℓ+1 ^k-1 (1 + β_i' _i)  ≤(  _k/_ ℓ+1  )^ ≤(  _k/_ ℓ  )^
We also have $ |\tilde r_{\ell} ^p| \leq  \bg^3_{\ell} \|r\|_{X_r} $.  Then we can estimate by (<ref>)

| p̃_k | ≤^ _k ∑_ℓ=0 ^k-1^3-  _ℓ    r_X_r  ≤C      ^2_k   r_X_r
The $\tilde v_k$ trivially satisfies the same bound and hence so do $p_k,v_k$.  Then the projection $ (\pi_p y)_k = p_k$ satisfies
π_py _X_w   ≤sup_k _k ^- 2    | p_k |  ≤C   r_X_r
The equation for  $g_k$ is
g_k+1 =  ( 1 +2  β_k  _k ) g_k +r̃^g_ k
r̃^g_k  = - 2 β'_k _k p_k  - 4  β'_k _k p_k + r^g_k
This is solved starting with $g_N = g_f$  and counting down by
g_N-j  =  A_N-j  ( g_N-j+1  - r^g_N-j  )      where     A_k   = ( 1 + 2 β_k  g̅_k ) ^-1
The solution is
g_N-j = -    ∑_ℓ=1^j [  ∏_i=ℓ^j  A_N-i ] r̃^g_N-ℓ
By lemma <ref> to follow
∏_i= ℓ^j  A_N-i    ≤(g̅_N-j /g̅_N - ℓ+ 1 ) ^2
We also  have     $|r^g_k | \leq  \bg_k^3  \|  r \|_{X_r}   $.  Also  since $|p_k|, |v_k| \leq C \bg_k^2 \|r\|_{X_r} $  both  $  \beta'_k \bg_k |p_k| $
and  $\beta'_k \bg_k |v_k| $  are bounded by  $C \bg_k^3 \|r\|_{X_r}$.  Therefore
$|\tilde r^g_k| \leq C   \bg_k^3  \|  r \|_{X_r} $ and $|\tilde r^g_{N-\ell} | \leq C   \bg_{N-\ell} ^3  \|  r \|_{X_r} $.  Then again   by (<ref>)

| g_N-j |   ≤C  ^2_N-j  ∑_ℓ=1^j   g̅_N- ℓ
r _X_r    ≤C ^2_N-j  |log_N-j |    r _X_r
So for  the projection  $(\pi_g y)_{N-j}  = g_{N-j} $ we have

π_g y_X_w   ≤sup_j     g̅_N-j ^-2| log_N-j |^-1  |g_N-j|      ≤C    r _X_r

Finally  consider the $z_k$ equation which is
z_k+1 = z_k  +2 θ_k g̅_k  g_k +r̃^z_k
r̃^z_k = -  θ^p_k _k  p_k  -  θ^v_k _k v_k +   r^z_k
With $z_0 =0$  this has the solution
z_k  =  ∑_ℓ=0 ^k-1 ( 2θ_ℓ    g̅_ℓ    g _ℓ   + r̃^z_ℓ   )
Taking account the bounds on $|p_k|, |v_k| $    and $|\theta^p_k|, |\theta^v_k| \leq C$
we have    $  |\tilde r^z_{\ell}|  \leq C  \bar g_{ \ell}  ^3 \|r \|_{X_r}  $.   Also
$|g _{\ell}|
\leq   C   \bar g^{2}_{\ell}  |\log \bar g_{\ell} | \   \| r \|_{X_r} $ from  (<ref>)
and $|\theta_{\ell}| \leq C$.
Hence again by (<ref>)
|z_k|   ≤C   ∑_ℓ=0 ^k-1     _ℓ ^3  | log_ℓ  |   r _X_r
≤C _k ^2  | log_k |   r _X_r
Then we have  for $( \pi_z y)_k  =z_k$
π_zy  _X_w   =   sup_j  g̅^-2_k| logg̅_k |^-1   | z_k |  ≤C   r _X_r
This completes the proof.

For $g_f$ sufficiently small and $\ga >0$
1/2   (  _k/_ ℓ  )^  ≤∏_i=ℓ^k-1(1 + β_i _i)    ≤3/2   (  _k/_ ℓ  )^

Define $\al_i$ by
(1 + β_i _i)   = (1 + β_i _i)^ ( 1 +  _i)
Then $\al_i = \cO(g^2_i)$ and since  $\sum_i \bg_i^2$ is small we have
1/2 ≤∏_i=ℓ^k-1 ( 1 +  _i)   ≤3/2
On the other hand from   $( 1+ \beta_i \bg_i)  = \bg_{i+1} / \bg_i$
∏_i=ℓ^k-1  (1 + β_i _i)^   = ∏_i=ℓ^k-1(  _i+1 /_i ) ^=   (  _k/_ ℓ  )^

§.§ the second linear equation

For the next result   we introduce  a new domain.
Let $\cB_C$ be the   ball of radius $C$ in $X_w$ and we consider the domain   $\bar x + \frac18\cB_{C_E} $.
If  $x \in \bar  x +\frac18 \cB_{C_E} $  then  $x - \bar x \in \frac18  \cB_{C_E}  $
and this says

E_k - E̅_k  _h, _4   ≤   1/8 C_E g̅_k^3

|g_k-g̅_k|,|z_k-z̅_k|       ≤    1/8  C_E  g̅_k^2  |logg̅_ k|

|p_k- p̅_k |,|v_k- v̅_k| ≤    1/8 C_E   g̅_k^2

Always assuming $\bg_k \leq g_f <\frac12  g_{\max} $ are sufficiently small,  the bound on $g_k$ implies say
4/5 _k ≤g_k ≤5/4_k

The next result shows that elements of $  \bar x +\frac18  \cB_{C_E} $ satisfy the conditions of theorem <ref> and corollary <ref>.  Hence
the mapping $ x_{k+1} =  \Phi_k( x_k )  $ is well-defined on  such sequences, as is   $\Phi^t_k$, and we have control over derivatives.

For $x \in   \bar x+ \frac18 \cB_{C_E} $

E_k  _h, _4   ≤   1/2 C_E g_k^3

|z_k|       ≤    1/2 C_Z g_k

|p_k |,|v_k | ≤     1/2  C_E  g_k^2

We have     $\|  \bar E_k  \|_{h, \Ga_4}   \leq   \one (L^{-1} + L^2h^{-\frac12} )  C_E \bar g_k^3 $ from (<ref>)  which is less than $\frac18   C_E \bar g_k^3 $
for $L,h$ large.
Together with   (<ref>) this gives $ \|  E_k  \|_{h, \Ga_4}   \leq  \frac14 C_E \bar g_k^3 \leq  \frac12 C_E   g_k^3   $.
We have   $|\bar z_k|   \leq  C \bg_k $ from lemma <ref> which together with the bound on $|z_k-\bar z_k|$ gives  $  |z_k|       \leq  \frac12 C_Z \bg_k$.
We have $\bar p_k =0$ so    $|p_k |  \leq   \frac18 C_E   \bar g_k^2  \leq \frac 12 C_E g_k^2$ .   The same bound holds for $v_k$.  This completes the proof.

Now for  $x \in \bar x + \frac18  \cB_{C_E}$ we  study the  equation    $y_{k+1}  =      D_x  \Phi^t_{k}(  x_k )   y_k    + r_k $  from   (<ref>).  We regard it as  a perturbation
of the previous equation $  y_{k+1}  =   L_k y_k       + r_k $.  Since $ L_k = D_x \bar  \Phi^0_k  (\bar x_k )$ we
can write

y_k+1  =   L_k y_k  +     _k(t,x_k)  y_k     + r_k
_k(t,x_k)   = D_xΦ^t_k+1  (  x_k  )   - D_xΦ̅^0_k  (x̅_k )
We first need estimates on $ \cW_{k}(t,x_k)  $.

There  are five components  in
_k= (_E, _V) =  (_E, _g, _z, _p, _v)
and each is a function for the five variables
x_k =(E_k, V_k)  = (E_k,g_k, z_k,p_k,v_k)  We have
_k    = (
_EE      _EV

_VE      _VV

Here  the first index indicates the component  we are considering  and  the second index indicates which derivative we are considering
in the contribution of $D_x \Phi_k, D_x  \bar \Phi^0_k$ to $\cW_k$.

For    $x \in \bar x +  \frac18 \cB_{C_E} $

_EE  _(X_w, X_r)  ≤      (L^-2 + L^2h^-1/2)

_VE  _(X_w, X_r)  ≤    h^-2

_EV  _(X_w, X_r)  ≤      CC_E h^8   g_f | logg_f |

_VV  _(X_w, X_r)  ≤     C C_E g_f | logg_f |^2

We  write

_k(t,x_k)   =        ^(I)_k(t,x_k)   +   ^(II)_k(t,x_k)

^(I)_k(t,x_k)  =     D_x Φ^t_k  (  x_k  )    -D_x Φ̅^0_k  ( x_k )

=    ( D_x _k(x_k),    tD_xρ_k(x_k) )

=    ( D_x_k(x_k),    t D_x g^*_k(E_k)  ,      t D_xz^*_k(E_k), t D_xp_k^* (E_k), t D_xv^*_k(E_k)   )

^(II )_k(t,x_k)    =      D_x  Φ̅^0_k ( x_k  )   -   D_x Φ̅^0_k   ( x̅_k )

We begin by studying $\cW_{VV}$ which has the form
_VV     = (
_gg      _gz       _gp      _gv

_zg     _zz      _zp      _zv

_pg     _pz      _pp      _pv

_vg      _vz       _vp      _vv

There is only a  contribution from  $\cW^{(II)} $.
From (<ref>) and (<ref>) we have

_VV  =       D_x   Φ̅^0_k  (   x_k ) - D_x   Φ̅^0_k  ( x̅_k )

=    (
2 β_k  (g_k-_k)  -2β_k'p_k-4 β'_kv_k)     0        - 2β_k'  (g_k-_k)     - 4β_k'  (g_k-_k)

2 θ_k   (g_k-_k)       0     θ^p_k  (g_k-_k)       2 θ^v_k  (g_k-_k)

-2 β_k'v_k     0    0      -2β_k'  (g_k-_k)

-  β_k' p_k     0     - β_k'  (g_k-_k)    0

Since we are assuming  $|g_k- \bg_k | \leq \frac18 C_E  \bg^2_k | \log \bg_k|$   and $|p_k|,|v_k|   \leq \frac18 C_E \bg_k^2$
every entry  is bounded by  $  C  C_E \bg^2_k | \log \bg_k| $.  Taking account also that
$\Phi_k$ is a map from a $k^{th}$ component to $(k+1)^{th}$ component we have for any entry   in $\cW_{VV} $

_ VV  f _ X_r
≤    sup_k     g̅_k+1 ^-3| ( _ VV f)_k+1  |

≤    sup_k   g̅_k+1 ^-3[ CC_E  ^2_k | log_k|]   | f_k |

≤    sup_k   g̅_k+1 ^-3[ CC_E  ^2_k | log_k|]    g̅_k^2 |logg̅_k| f_X_w

≤     sup_k  CC_E g̅_k | logg̅_k|^2  f_X_w

≤    CC_E    g_f | logg_f  | ^2  f_X_w

which   says

_ VV _(X_w,X_r)  ≤CC_E    g_f | logg_f  | ^2 Now consider  $\cW_{EV}$.   Only  $\cW^{(I)} $ contributes  and
_EV  = D_V _k=     D_V ( E_k + E_k^*) =   D_V E_k^*
D_V E_k^*    =   ( E_k^*/ g_k,       E_k^* /z_k, E_k^*/ p_k,       E_k^* /v_k )
These derivatives are estimated in corollary <ref>.   We have
$  \| \pa E^*_k / \pa g_k \|_{\frac12h, \Ga}   \leq     C_E \bg^2_k  $ and
$  \|\cL \ \pa E^*_k / \pa g_k \|_{h, \Ga_4}   \leq  \one L^2 C_E \bg^2_k  $. The  derivative in  $ z_k $ has   the same bound,
and derivatives with respect to $p_k,v_k$ have the same bound but with an extra constant $Ch^8$.

_EV  _(X_w,X_r)  ≤    sup_k    _k+1 ^-3 [ C C_E h^8_k^2]  _k^2 |log_k|

≤    sup_k   CC_E h^8 _k  |log_k|  ≤C C _E  h^8   g_f  |logg_f|

Next  consider
_VE  = D_E ρ_ k=    t (D_Eg_k^*,  D_Ez_k^*, D_Ep_k^*, D_Ev_k^*)
Since $g^*_k(E_k) $ is a linear function of $E_k$
we have  $D_E g^*_k(E_k ) =  g^*_k(E_k ) $.  We have from lemma <ref> the estimate    $ | g^*_k(E_k)|  \leq \one  h^{-4} \|E_k\|_{h, \Ga_4}   $ and so
$ \| D_E g^*_k   \| \leq  \one  h^{-4} $ (norm in the dual space $\cG'_{h, \Ga_4}$).
D_E g^*_k   _(X_w, X_r)
≤sup_k     g̅_k+1 ^-3[h^-4 ]      g̅_k^3 ≤h^-4
The same bound holds for  $ D_Ep_k^*, D_Ev_k^*$ and for  $  D_Ez_k^*$ it holds with $h^{-2}$ instead of $h^{-4}$
Thus we have
_VE  _(X_w, X_r)   ≤h^-2

Finally consider
_EE  = D_E (  E_k  +  E_k^* )
From  (<ref>)    we have $\| \cL \cR E_k \|_{h, \Ga_4}  \leq  \cO(1)L^{-1} \| E_k\|_{h, \Ga_4}  $.       Since it is a linear function it is its own derivative and it follows that
$\| D_E\cL \cR  \| =  \cO(1) L^{-1}$  (norm in $\cL(\cG_{h, \Ga_4} )$).
This yields

D_E _(X_w, X_r)   =  sup_k     _k+1 ^-3  [  (1) L^-1 ]  _k ^3       ≤(1) L^-1
For the other term   we have
$ \| D_E  E^*_k  \|_{\cL(\cG_{h, \Ga_4},  \cG_{h/2, \Ga}) }  \leq \cO(1) h^{-\frac12}  $
from  corollary <ref>,  and $\| \cL \|_{\cL(\cG_{h/2, \Ga},\cG_{h, \Ga_4} ) } \leq \one L^2  $ as in (<ref>).
D_EE^*_k _( _h, _4)   = D_E E^*_k _( _h, _4)     ≤L^2 h^-1/2
and as in  (<ref>)  $ \| D_E\cL E^* \|_{\cL(X_w, X_r) }     \leq  \one L^2 h^{-\frac12} $.
_EE  _(X_w, X_r)   ≤(L^-1   +  L^2  h^-1/2 )
This completes the proof.

Let $g_f$ be sufficiently small and suppose   $ x \in \bar x +\frac18 \cB_{C_E}  $.    Then for $r \in X_r$ there exists
a unique solution of

y_k+1  =      D_x  Φ_k( t, x_k )   y_k    + r_k
in $X_w$  with  null boundary conditions (<ref>).   There is a  constant $C$ such that the linear solution operator $y  = S(t,x) r $ satisfies

S(t,x) _(X^r,X^w) ≤      C

We claim  (<ref>)  is equivalent to

y =    S^0  (t,x) y        +S^0 r
This follows since  $y=S^0r$ solves  $y_{k+1}  =   L_k   y_k    + r_k$.   So if  $y$ satsifies
(<ref>)  then

y _k+1 =      ( S^0  (t,x) y)_k+1    +  (S^0 r)_k+1

=     (  L_k ( S^0 (t,x) y)_k    +    _k(t,x)  y_k   )  +   (  L_k   (S^0 r) _k    + r_k )

=       L_k ( ( S^0  (t,x) y)_k  + (S^0 r) _k)     +    _k(t,x)  y_k   +     r_k

=       L_k y_k    +    _k(t,x_k)  y_k         + r_k

which is (<ref>) and hence (<ref>).  Conversely a solution of (<ref>) gives a solution of (<ref>).
The solution to (<ref>) is
y = (1 - S^0   (t,x) )^-1 S^0 r ≡S(t,x) r
provided the inverse exists in $\cL(X_w)$.   We argue below that  $ \|  S^0 \cW \|_{\cL(X_w)}   < \frac12$ so the inverse  does exist
with  $  \| ( I- S^0 \cW )^{-1} \|_{\cL(X_w)}   < 2$.  The we have the required
S _(X_r,X_w)   ≤  S^0 _(X_r,X_w)     (1 - S^0  (t,x) )^-1 _(X_w) ≤C

For the estimate we have
S^0 = (    1     0
0      S^0
_EE        _EV

_VE      _VV

) =
_EE      _EV

S^0  _VE      S^0 _VV

We argue that  each entry is small by  lemma  <ref> and lemma <ref>.
Start with
_EE _(X_w) =    _EE _(X_w,X_r) ≤( L^-1 + L^2h^-1/2  )  < 1/2
which holds for first   $L$  and  then $h$  sufficiently large.
Similarly  for $g_f$ sufficiently small  $\| \cW_{EV} \|_{\cL(X_w)}  \leq C C_Eh^8 g_f |\log g_f|  < \frac12$.
Next  for  $h$ sufficiently large
S^0 _VE _(X_w)     ≤ _VE _(X_w,X_r)  S^0_(X_r,X_w)   ≤Ch^-2   < 1/2
Finally for $g_f$ sufficiently small
$\|  S^0 \cW_{VV} \|_{\cL(X_w)}   \leq  CC_Eg_f| \log g_f|^2 < \frac12$.   This completes the proof.

§.§ proof of theorem <ref>

We  study $x_k = (E_k, g_k,  z_k, p_k, v_k,)$ for $x \in \bar x + \frac18  \cB_{C_E} $.    For any such  sequence $x$  the sequence
$y =  S(t,x)  \rho(x) $ is defined to  solve  the equation

( S(t,x)  ρ(x))_k+1    =   D_x   Φ_k (t,x_k   ) (S(t,x  )  ρ(x) )_k    + (0, ρ_k(x))
with  null boundary conditions (<ref>).
Suppose we can find  a solution $x_k = x_k(t) $  in $\bar x +  \frac18  \cB_{C_E}$ of the system of  ordinary differential equations

ẋ_k  = F_k(t,x) ≡(S(t,x)  ρ(x) )_k
for  $0 \leq t  \leq 1$,  with the initial condition  $x_k(0) =\bar x_k$.
The   null  boundary conditions for  $S(t,x)  \rho(x) $ imply  null boundary condition for  $\dot x_k(t)  $.   (That is $\dot g_N(t) =0, \dot E_0(t) =0, \dots $.)  This  means that
$x_k(t)$  has  constant boundary conditions, and   since $x_k(0) = \bar x_k $ satisfies the boundary conditions (<ref>), the same is true for  $x(t)$.
(That is $g_N(t) = g_N(0) = \bg_N = g_f$, and   $E_0(t) = E_0(0) = \bar E_0   =0$, etc.)

Now  taking the sequence  $x_k = x_k(t)$ in   (<ref>)
we have
ẋ_k+1 (t) =    D_x   Φ_k (t,x_k(t)  ) ẋ_k(t)     + (0, ρ_k(x_k(t) ) )
By uniqueness of such solutions   $x_{k+1}(t)  = \Phi_k ^t( x(t))$.   In  particular  $x_k \equiv x_k(1) $, still in  $\bar x +  \frac18  \cB_{C_E}$,  satisfies (<ref>)   $x_{k+1}  = \Phi_k ( x_k)$
and  hence (<ref>) with the boundary conditions (<ref>).

Thus the problem is reduced to  the existence of solutions to (<ref>) which  is an ordinary differential equation in the  Banach space  $X_w$.   We need a solution $x(t)$
in the domain $  \bar x + \frac18 \cB_{C_E} $with  $x(0) =\bar x$  and defined for at least  $0 \leq t \leq 1$.    By a fundamental theorem of ODE  there is a unique solution provided      $ \| F(t, x)  \|_{X_w}  < \frac18C_E  $  for   $x \in \bar x +  \frac18  \cB$ and $F$   satisfies Lipschitz condition on this domain.
(See for example [1]).

Elements of   $  \bar x + \frac18 \cB_{C_E} $ satisfy the hypotheses of theorem <ref>  by lemma <ref> and so we can conclude
that $|g_k^*|, |z_k^*|,|p_k^*|,|v_k^*|$ are all bounded by  $C_Eh^{-2} g_k^3 \leq   2 C_Eh^{-2} \bg_k^3 $.
ρ( x)  _X_r     =   sup_k  {   g̅_k^-3 |g^* _k|,   g̅_k^-3  |z^*_k|, g̅_k^-3 |v^* _k|,  g̅_k^-3 |p^*_k|    }  ≤2C_E h^-2
Then also by the bound on $S$ in lemma <ref>  we have for $h$ sufficiently large   the required
F(t,x ) _X_w  ≤ S(t,x) _( X_w,X_r)    ρ( x)  _X_r  ≤C C_E h^-2   <  1/8 C_E

For the Lipschitz continuity recall that  in the proof of corollary <ref>  we observed that $E_k^*$ is analytic in $x_k$.   Keeping this in
mind and inspecting the various terms in $\cW_k$ in the proof of lemma <ref> we see that $\cW_k$  is analytic in $x_k$.  Hence the same is
true for $ S(t,x) =  (1 - S^0   \cW(t,x) )^{-1} S^0 $.   Since also $\rho_k(x_k)$ is analytic we have that  $F_k(t,x_k)$ is  analytic and  $F(t,x_k)$ is analytic since every entry is analytic.    The analyticity implies  Lipschitz continuity  in $x_k$ and it is uniform  in $t$ which suffices for the existence theorem.
Thus theorem <ref> is established for all but the vacuum energy $\vep_k$.

Finally  consider $\vep_k$ which satisfies $\vep_{k+1}  = L^2 ( \vep_k + \vep_k^*)$.  The solution with initial condition $\vep_0$
is then
_k = L^2k _0 + ∑_j=0^k-1 L^2(k-j)  ^*_j
To get $\vep_N =0$ we choose
_0 =   - ∑_j=0^N-1 L^-2j ^*_j
This completes the proof.

§ ULTRAVIOLET STABILITY BOUND

We specialize now to unit volume  $M=0$, so we are studying the partition function $\sZ $ as   on  $\bbT_0 = \bbR^2/ \bbZ^2  $ with momenta cutoff at about $|p| = L^N$  as defined in (<ref>).  This
is  scaled up to an integral on the large torus   $\bbT_N =\bbR^2/ L^N \bbZ^2$ with  unit  momenta cutoff at about $|p| =1$.  Successive renormalization group transformations change it to an integral over smaller tori  $\bbT_{N-k} $ still  with unit momentum cutoff.    We choose the coupling constants as in theorem  <ref> so we can   continue all the way to $k=N$.  Then we are again on a  unit torus $\bbT_0 = \bbR^2/ \bbZ^2 \equiv \sq$, but now  with unit momentum cutoff.
The partition function is given by
=  ∫e^S_f(ψ)   d μ_G_f (ψ)
Here $G_f  \equiv  G_N$  is independent of $N$ and has the kernel

G_f(x-y) = ∑'_ p ∈_0^*  e^ip(x-y)  -i/p^2  e^ -|p|^2  = ∑_ p ∈2 π^2, p ≠0  e^ip(x-y)  -i/p^2  e^ -|p|^2
Since     $g_N = g_f$ and $\vep_N =0$  the expression for   $S_f \equiv S_N $ from
(<ref>)  is
S_f  =    ∫_   (     - z_N  ψ+  g_f  (   ψ)^2
+  p_N  ( _5 ψ)^2 +  v_N  ( _μ ψ)^2   ) +  Q̂ _N + Q̂_N'    +  E_N
(Stability bound) Under the hypotheses of theorem <ref> with parameters chosen as in  theorem <ref>  we have uniformly in $N$
S_f _h  ≤CC_V g_f
1/2    ≤≤3/2

Since our solution has   $x \in \bar x + \frac18 \cB_{C_E} $ the bounds (<ref>) hold for all $k$ and so for $k=N$.
Then as in   lemma <ref> with $|z_N| \leq C_Zg_f$
∫ψ_h ≤CC_Vg_f    g_f ∫(    ψ)^2  _h ≤Cg_f
All the other terms in $S_N$  are higher order and smaller.  See in particular  (<ref>), (<ref>), and  (<ref>)
and use  $\| F\|_h < \| F\|_{h, \Ga_4} $
The bound on $S_f$ follows.

For the partition function  note that the Fourier coefficients
$\tilde G_f(p) $ are rapidly decreasing and have no small divisors.   Hence $h(G_f)$ defined as in (<ref>)  is bounded by $\one$
and we can choose  $h$ so   $ h(G_f )   \leq  h$.
Then $  \| S_f \|_{h(G_f)  } \leq   \| S_f \|_h  \leq  CC_V g_f  $.
Now we write
∫e^-S_f   d μ_G_f   = 1  +   ∫(  e^-S_f -1  )  d μ_G_f
Estimate the last  integral as in lemma <ref> and get for $g_f$ sufficiently small

| ∫(  e^-S_f -1  )  d μ_G_f  |
≤       e^-S_f -1 _ h(G_f )
≤e^   S_f  _ h(G_f )     -1
≤e^  CC_V g_f      -1 < 1/2

This suffices and completes the proof.

§ FOURIER SERIES CONVENTIONS

We consider functions on the $d$-dimensional  torus $\bbT_j \equiv  \bbR^d/L^j \bbZ^d$.      If  $p \in \bbT_j^* \equiv  2\pi  L^{-j}    \bbZ^d$
then $e^{ipx} $ is invariant under $x_{\mu} \to x_{\mu} + L^j$ and so defines a function on the torus.    We have
$\int  e^{-ipx}  e^{iqx} dx = L^{dj} \de_{p,q} $ so $e_p(x)  = L^{-\frac12 dj} e^{ipx} $ form an orthonormal basis for $L^2(\bbT_j) $
and there is  the expansion
f(x)  = ∑_p ∈_j^*  (e_p,f )  e_p(x)  = ∑_p ∈_j^*  L^-dj    [   ∫e^-ipx   f(x) dx  ]   e^ipx
We write this as
f(x)  =  ∑_p ∈_j^*  L^-dj   f̃( p)   e^ipx  ≡∑'_p ∈_j^*    f̃(p)   e^ipx
f̃(p)  =    ∫e^-ipx   f(x) dx
We also have $\int \overline f(x) g(x)  dx = \sum_p (f,e_p)(e_p, g) $
which we write as
∫ f̅(x)   g(x) dx = ∑_p ∈_j^*  L^-dj  f̃(p)  g̃(p)  ≡∑'_p ∈_j^*   f̃(p)  g̃(p)

§ CHARACTERISTIC FUNCTIONS OF  GRASSMANN  INTEGRALS

Let $\cG_h$ be the Grassman algebra on some $\bbT_j$  as defined in section <ref>.   We consider integrals   $ \int [ \cdots ] d\mu$       on $\cG_h$ defined
to be continuous linear functionals on $\cG_h$.    For $F$ of the form (<ref>) we have for some constant $c$
|∫F d μ| ≤cF _h