\font\tenmib=cmmib10 \font\sevenmib=cmmib10 scaled 800 \font\titolo=cmbx12 \font\titolone=cmbx10 scaled\magstep 2 \font\cs=cmcsc10 \font\ss=cmss10 \font\sss=cmss8 \font\crs=cmbx8 \font\ninerm=cmr9 \font\ottorm=cmr8 \textfont5=\tenmib \scriptfont5=\sevenmib \scriptscriptfont5=\fivei \font\msxtw=msbm10 scaled\magstep1 \font\euftw=eufm10 scaled\magstep1 % \font\msytw=msym10 scaled\magstep1 % \font\msytww=msym8 scaled\magstep1 % \font\msytwww=msym7 scaled\magstep1 \font\indbf=cmbx10 scaled\magstep2 \newcount\mgnf\newcount\driver \driver=1 %dvips \mgnf=0 %ingrandimento \ifnum\mgnf=0 \magnification=\magstep0 \hsize=14truecm\vsize=24.truecm \parindent=0.3cm\baselineskip=0.45cm\fi \ifnum\mgnf=1 \magnification=\magstep1\hoffset=0.truecm \hsize=14truecm\vsize=24.truecm \baselineskip=18truept plus0.1pt minus0.1pt \parindent=0.9truecm \lineskip=0.5truecm\lineskiplimit=0.1pt \parskip=0.1pt plus1pt\fi \let\a=\alpha \let\b=\beta \let\g=\gamma \let\d=\delta \let\e=\varepsilon \let\z=\zeta \let\h=\eta 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\def\tttt{\hbox{\msytww T}} \def\ttt{\hbox{\msytwww T}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%% FIGURE fig1.ps %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \let\figini=\initfig \let\figfin=\endfig \figini{fig1} \write13<%!> \write13<%%BoundingBox 0 0 200 40> \write13 \write13<%uso x1 y1 x2 y2 origineP_1assexperP_2pilasu|P_2-P_1|> \write13<4 2 roll 2 copy translate exch 4 1 roll sub 3 1 roll > \write13 \write13<2 copy atan rotate 2 copy exch 4 1 roll mul > \write13<3 1 roll mul add sqrt } def> \write13<> \write13 \write13 \write13 \write13<3 1 roll exch 0 3 1 roll 0 0 moveto > \write13<{dup 2 index mul sin 4 mul lineto} > \write13 \write13<> \write13 \write13<0 0 moveto dup dup 0 exch 2 div lineto 0 > \write13 \write13<0 0 lineto fill stroke } def> \write13<> \write13 \write13 \write13 \write13<> \write13 \write13<0 0 moveto 0 2 copy lineto stroke > \write13 \write13 \write13<> \write13 \write13 \write13 \write13<> 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behaviour in the} \centerline{\titolone uncommensurate Hubbard-Holstein model} %\centerline{\titolone with an uncommensurate potential} \vskip1.truecm \centerline{{\titolo V. Mastropietro}\footnote{${}^\ast$}{\ottorm Supported by MURST, Italy.}} \centerline{Dipartimento di Matematica, Universit\`a di Roma ``Tor Vergata''} \centerline{Via della Ricerca Scientifica, I-00133, Roma} \vskip.2truecm \line{\vtop{ \line{\hskip1.5truecm\vbox{\advance \hsize by -3.1 truecm \0{\cs Abstract.} {\it A small denominator problem arising in the perturbative expansion for the Schwinger functions of a quantum model of interacting fermions in one dimension is solved by renormalization group techniques. We show that the Schwinger functions asymptotic behaviour is anomalous and described in terms of critical indices. }} \hfill} }} \vskip1.2truecm \centerline{\titolo 1. Introduction.} \*\numsec=1\numfor=1 \0{\bf 1.1} Peierls [P] and Fr\"ohlich [F] noted in the 50s that one dimensional metals are unstable at low temperature, in the sense that they can lower their energy by a periodic distorsions of the lattice with period ${2\pi\over 2p_F}$, if $p_F$ is the Fermi momentum. Such a distorsion is called {\it Charge Density Wave} (CDW), as both the lattice and the electrons charge density form a new periodic structure with period possibly bigger than the original lattice period $a$. Fr\"ohlich suggested that, if the period of the CDW is {\it incommensurate} with the original period of the lattice, the CDW can carry an electric current. The important role of uncommensurate CDW in conductivity was confirmed in [LRA]. Starting from the early 70s many experimental properties of anysotropic compounds were explained qualitively in terms of CDW. The CDW were observed in TTF-TCNQ by X-rays and the anomalous behaviour of conductivity in many compounds was explained in terms of them (see for instance [L] for a rewiew). The most simple model to understand the formation of a CDW is the {\it Holstein model} [H], whose hamiltonian is given by a thigth binding hamiltonian, describing fermions on a lattice, and an interaction with a classical phonon field. The interaction between electrons is completely neglected. Numerical simulations show, see for instance [AAR], that for small phonon fields the energy in such a model is minimized if the phonon field is a periodic function with period ${2\pi\over 2 p_F}$, either if the Fermi momentum is commensurate or incommensurate with the lattice periodicity \ie if ${p_F\over\pi a}$ is a rational or irrational number. A mathematical proof of the formation of a CDW in the Holstein model is up to now available only in the commensurate case, [KL], [BGM2]. Assuming the existence of CDW and their modelization by a periodic function with period ${2\pi\over 2p_F}$, a quantitative comparation with the experiments requires in any case the computation of the Holstein Schwinger functions, in terms of which the physical properties of the model can be obtained. The Schwinger functions properties could be obtained studying the solutions of the Schroedinger equation on a lattice with commensurate or uncommensurate periodic potential. This second case is much more difficult %While in the case of commensurate CDW the computation %of the Schwinger functions properties is an immediate consequence %of the Bloch wave theory (see for instance [BM1]), the situation is less %clear in the case of uncommensurate CDW. In this second case in fact %(which, as we saw, is the most interesting physically), one has to study %the spectrum of the finite difference Schroedinger equation with %an uncommensurate potential, a problem which as it is technically equivalent to the study of the Schroedinger equation in the continuum with a quasi-periodic potential, which is characterized by a {\it small divisor problem} and it is generally faced by KAM iterative techniques. Many properties are known about the solutions of the Schroedinger equation in a quasi periodic potential (see for instance [DS],[BLT],[JM] and the review [PF]), but the Schwinger functions properties of the Holstein model do not seem an immediate consequence of them. A different approach was developed then in [BGM1] for studying the Schwinger functions of the Holstein model using functional integration and renormalization group techniques. It was found that the Schwinger function decays for large distances faster than any power, and that there is a spectral gap at the Fermi surface. Of course in a realistic model for anysotropic compounds with a CDW one has to take into account also the interaction between electrons, as it is well known that in one dimension the interaction changes deeply the properties of the system. So one can consider the {\it Holstein-Hubbard model}, which is equal to the Holstein model but a term describing the short range interaction between fermions is added. Such a model is considered relevant for the understanding of the high-$T_c$ superconductivity, see [A]. We study the Holstein-Hubbard model in the incommensurate case, but for the (much easier) commensurate case similar results follow from the analysis (for a sligthly different model) in [BM2], [BM3]. %The small divisor problem associated to this model is treated %not by the usual KAM iterative techniques but by direct methods \ie by %proving directly the convergence of the series expressing the %Schwinger functions. Such methods are essentially the same used %for the proof of the converhence of the Lindstedt series for the invariant %tori un a mechanical system, see [G],[GM]. We find a well defined %perturbative expansion for the Schwinger functions and we computed in a %rigorous way their large distances asymptotic behaviour (no results %even heiristic are known, as far as we known, about the Schwinger %functions of this model). \vskip.5cm \0{\bf 1.2} The Schwinger functions of the Holstein-Hubbard model are studied writing them as a {\it functional Grassmanian integral} which is expressed by a erturbative series. The small denominator problem affecting this series is controlled, like in [BGM1], by using a sort of {\it Bryuno Lemma} [B] (see sec. 3.6 below) and using suitable {\it cancellations} to face the problem of the {\it resonances}. Such cancellations are implemented by Renormalization group techniques. The approach we follow is then very closely related to the "direct" methods developed in recent years for proving the KAM theorem by showing the convergence of the {\it Lindstedt series} expressing the invariant tori, see [E1],[G],[CF],[GM]. The main technical novelty of the present paper is that in the above quoted papers (including [BGM1]) the perturbative expansion can be expressed in terms of {\it Feynmann graphs} which are only {\it tree graphs} \ie with no loops: this is not surprising as the KAM theorem is a classical problem and the Holstein model is a non interacting model \ie bilinear in the fields. On the contrary the perturbative series for the Schwinger functions of the Holstein-Hubbard model are expressed in terms of Feynmann graphs with loops. So as far as we known this is the first case in which a small denominator problem in a fully interacting quantum theory is solved, and it is not clear to us if one can solve it also by the traditional iterative KAM techniques. The main idea is to combine such direct methods to study KAM problems with the Renormalization Group techniques developed to study interacting fermions starting from [BG] (similar techniques were introduced also in [FMRT]). \vskip.5cm \0{\bf 1.3} The hamiltonian of the {\it Holstein-Hubbard model} is % $$ H=H_0+u P+\l V+\nu N \; , \Eq(1.1)$$ $$H_0=\sum_{x,y\in\L} t_{xy}\,\psi^+_{x}\psi^-_{y} - \mu\sum_{x\in\L} \psi^+_{x}\psi^-_{x}$$ $$P=\sum_{x\in\L} \f_x\psi^+_{x}\psi^-_{x}\quad N=\sum_{x\in\L}\psi^+_{x}\psi^-_{x}$$ $$V=\sum_{x,y\in\L} v(x-y_)\psi^+_{x}\psi^-_{x}\psi^-_{y}\psi^+_{y}$$ % where $x,y$ are points on the one-dimensional lattice $\L$ with unit spacing, length $L$ and periodic boundary conditions; we shall identify $\L$ with $\{x\in Z:\ -[L/2]\le x \le [(L-1)/2]\}$. Moreover the matrix $t_{xy}$ is defined as $t_{xy}=\d_{x,y}-(1/2)[\d_{x,y+1}+\d_{x,y-1}]$, where $\d_{x,y}$ is the Kronecker delta, and $\m$ is the chemical potential. The fields $\psi_x^{\pm}$ are creation ($+$) and annihilation ($-$) fermionic fields, satisfying periodic boundary conditions: $\psi_x^{\pm}=\psi_{x+L}^{\pm}$. We define also $\psi_{\xx}^{\pm}=e^{tH}\psi_x^{\pm}e^{-Ht}$, with $\xx=(x,t)$, $-\b/2\le t \le \b/2$ for some $\b>0$; on $t$ antiperiodic boundary conditions are imposed. The term $P$ rapresents the interaction of the fermions with a classical phonon field. We are interested in studying potentials which, in the limit $L\to\io$, are of the form $\f_x=\bar\f(2px)$, where $\bar\f$ is a real function on the real line $2\pi$-periodic and $p/\p$ is an irrational number, so that the phonon field has a period which is incommensurate with the period of the lattice. We also impose that $\bar\f(u)$ is of mean zero (its mean value can be absorbed in the chemical potential), even and analytic in $u$, so that % $$\bar\f(u) = \sum_{0\neq n \in Z} \hat \f_n\,e^{inu} \; , \qquad |\hat \f_n| \le F_0 \, e^{-\x|n|}\; , \qquad \hat \f_n=\hat \f_{-n}=\hat \f_n^* \; . \Eq(1.2) $$ % At finite volume we need a potential satisfying periodic boundary conditions; hence, at finite $L$, we approximate $\f_x$ by % $$ \f^{(L)}_x = \sum_{n=-[L/2]}^{[(L-1)/2]} \hat \f_n\,e^{2inp_Lx}\; , \Eq(1.13) $$ % where $p_L$ tends to $p$ as $L\to\io$ and is of the form $p_L=n_L\pi/L$, with $n_L$ an integer, relatively prime with respect to $L$. The definition of $p_L$ implies that $2np_L$ is an allowed momentum (modulo $2\p$), for any $n$, and that the sum in \equ(1.13) is indeed a sum over all allowed values of $k$, except $k=0$. For technical reasons we need $p_L$ verifying the {\it Diophantine property} \equ(1.19) uniformly in $L$. In [BGM1], App. 1, a sequence of numbers verifying \equ(1.19) is constructed. The term $V$ in the hamiltonian rapresents the interaction of the fermions by a short range two body potential; in particular we assume $|v(x-y)|\le C e^{-\k |x-y|}$ for suitable values of the costants $C,\k$, and $\hat v(0)-\hat v(2p_F)\geq 0$. Finally $\nu$ is a {\it counterterm} which will be fixed to a $\l,u$-dependent value in order to fix the Fermi momentum \ie we fix $\nu$ so that the Fermi momentum in the interacting $\l\not=0$, with chemical potential $\mu+\nu$, is equal to the Fermi momentum of the free $\l=0$ system with chemical potential $\mu$. We assume that $u$ is positive while $\l$ can be positive or negative. \vskip.5cm %\*\numsec=1\numfor=1 \0{\bf 1.3} %We shall refer to [BGM1] for the definition of the {\it Schwinger functions}, in terms %of which all the physical properties of the model can be expressed, and %of the {\it Grassmanian integration} which allows us to compute them. Denote by $\|\a-\b\|_{T^1}$ the distance on $T^1$ of $\a,\b\in {T}^1$, and, for $\xx=(x,x_0), \yy=(y,y_0)\in R^2$, by $|\xx-\yy|$ the distance $|\xx-\yy|=\sqrt{(x-y)^2+(x_0-y_0)^2}$. The definition of the two-point Schwinger function is standard and it will be recalled below, see \equ(1.10f). If $E_0^n$ is the {\it ground state energy} \ie the minimum value of $H$ over the states $|\psi_n>$ with $n$ particles $N|\psi_n>=n|\psi_n>$, the {\it spectral gap} is $\D= E_0^{n+1}+E_0^{n-1}-2 E_0^n$. With the above definitions we shall prove the following theorem. \vskip.5cm \0{\cs Theorem} {\it Let us consider a sequence $L_i$, $i\in Z^+$, such that % $$ \lim_{i\to\io} L_i=\io \; , \qquad \lim_{i\to\io}p_{L_i}=p \; , $$ % and let $S^{L_i,\b}(\xx;\yy)$ be the two-points Schwinger function at finite temperature $\b^{-1}$. Suppose also that the fermions are spinless and that there is a positive integer $m$ such that $p_F=mp_{L_i}\, (\mod 2\p)$, $\hat \f_m\not=0$ and $p_{L_i}$ satisfies the {\sl diophantine condition} % $$ \|2n p_{L_i} \|_{T^1} \ge C_0 |n|^{-\t} \; , \qquad 0\neq n \in Z \;\qquad |n|\le {L_i\over 2} , \Eq(1.19) $$ % for some positive constants $C_0$ and $\t$ independent of $i$. Then it is possible to choose an $\e_0>0$ and a function $\nu\equiv \nu(\l,u)$ such that if $|u|,|\l|\le\e_0$, the following sentences are true: %and three functions %$\nu\equiv \nu(\l,u),\h_1\equiv \h_1(\l,u),\h_2\equiv \h_2(\l,u)$ %continuous in their arguments for $|\l|\le\e_0$ and such that %$\h_1\equiv\h_1(\l,u)=\b_3\l^2+O(\l^3)$ %and $\h_2\equiv \h_2(\l,u)=\b_1\l+O(\l^2)$. %Then if $|\l|\le\e_0$, %the following sentences are true: \vskip.2truecm (i) There exists the limit $\lim_{\b\to\io \atop i\to\io} S^{L_i,\b}(\xx;\yy) = S(\xx;\yy)$ \vskip.2cm (ii) $S(\xx;\yy)$ is continuous as a function of $\l,u$ and there exist three constants $K_1$, $K_2$, $K_3$ and, for any $N>1$, a constant $C_N$, such that, if $|\xx-\yy|\ge K_3|\hat u(p_F)|^{-1}$ $$|S(\xx,\yy)|\le {1\over \hat Z(p_F)} {K_1 |\hat u(p_F)|} {C_N \over 1 +( |\hat u(p_F)| \,|\xx-\yy|)^N}\Eq(z.1)$$ while for $|\xx-\yy|\le K_3|\hat u(p_F)|^{-1}$ one has $$|S(\xx,\yy)|\le {K_2 \over 1+|\xx-\yy|^{1+\h_3}}\Eq(z.2)$$ with $$\hat u(p_F)=u^{1+\h_2}\qquad \hat Z(p_F)=u^{-\h_1}\Eq(z.3)$$ and $\h_1\equiv \h_1(\l,u),\h_2\equiv \h_2(\l,u),\h_3\equiv \h_1(\l,u)(1+\h_2(\l,u))^{-1}$ continuous in their arguments for $|\l|\le\e_0$ and such that $\h_1\equiv\h_1(\l,u)=\b_1\l^2+O(\l^3)$ and $\h_2\equiv \h_2(\l,u)=\b_2\l+O(\l^2)$, with $\b_1,\b_2$ positive non vanishing constants. %\vskip.2truecm %(ii)There are two positive constants such that the discontinuty %at the Fermi surface is different from zero and it verifies %$$c_1 \hat Z(p_F)^{-1}\le Z^{-1}\le c_2 \hat Z(p_F)^{-1}$$ \vskip.2truecm (iii) We can write $$S(\xx;\yy)=S_1(\xx;\yy)+\tilde\e S_2(\xx;\yy)\; ,\Eq(1.21)$$ where % $$ S_1(\xx;\yy) = g^{(1)}(\xx;\yy) + \int {d\kk\over(2\p)^2}\, [1-\hat f_1(\kk)]\, \phi(k,x,\sigma)\,\phi^*(k,y,\sigma) \, {e^{-ik_0(x_0-y_0)} \over -ik_0+\e(k,\sigma)}$$ % with % $$ \eqalignno{ g^{(1)}(\xx;\yy) & = \int {d\kk\over(2\p)^2}\, \hat {f_1(\kk)\over \hat Z(k)}\, {e^{-ik_0(x_0-y_0)} \over -ik_0+ 1-\cos k-\mu}\; ,\cr \e(k,\sigma) & = [1-\cos(|k|-p_F)] \cos p_F \cr & +\sign (|k|-p_F) \, \sqrt{ [\sin(|k|-p_F)\sin p_F]^2+\hat u(k)^2} \; , \cr \phi(k,x,\hat u(k)) & =e^{-ikx} U(k,x,\hat u(k)) \; , & \eq(1.22) \cr U(k,x,\hat u(k)) & = e^{i\sign(k)p_Fx} \left[ \cos(p_Fx) \sqrt{ 1- {\sign(|k|-p_F)\hat u(k) \over \sqrt{(\sin(|k|-p_F)\sin p_F)^2+\hat u(k)^2}}} \right. \cr & \left. - i\,\sign(k)\sin(p_Fx) \sqrt{ 1+ {\sign(|k|-p_F)\hat u(k)\over\sqrt{(\sin(|k|-p_F)\sin p_F)^2+\hat u(k)^2}}} \; \right] \; . \cr} $$ % Here $\hat f_1(\kk)$ denotes a cutoff function with support far enough from the two singular points $\kk=(\pm p_F,0)$, and $\hat u(k),\hat Z(k)$ are two regular functions such that $|\hat u(k)-u|=O(u\l)|$ , $|\hat Z(k)^{-1}-1|=O(\l)$ for $||k|-p_F|>{p_F\over 2}$, and $\hat u(p_F), \hat Z(p_F)$ are given by \equ(z.3); moreover if $\l=0$ then $\hat u(k)=u$, $Z(\kk)=1$. Finally $\tilde\e=Max(|\l|,u,\hat u(p_F))$ and $S_1(\xx,\yy), S_2(\xx,\yy)$ obeys to the same bound \equ(z.2). Moreover if $\lim_{\l\to 0,u\to 0}S(\xx,\yy)=g(\xx,\yy)$ then $$|S_1(\xx;\yy)-{g(\xx;\yy)\over |\xx-\yy|^{\h_3}}|\le K_2 {Max(u,\hat u(p_F))\over \hat Z(p_F)}\log\hat u(p_F)\Eq(luttt)$$ \vskip.2cm (iv)For any $i$ there is a spectral gap $\D $ verifying $$\D\geq {\hat u(p_F)\over 2}\Eq(z.4)$$ } \* \0{\bf 1.4} Let us comment the quite elaborated theorem above. The point (ii) says that, like in the non interacting $\l=0$ case, one can distinguish two regions in the large distance behavior of the Schwinger function, discriminated by an intrinsic length which is $O(u^{1+\h_2})$ instead $O(u)$. In the second region (larger distances) the decay is faster than any power but, contrary to the free case, the {\it decay rate} is renormalized by the interaction as it is not $O(u)$ but $O(u^{1+\h_2})$. Moreover there is a factor $\hat Z(p_F)$ in the bound which can be interpretated as a {\it wave function renormalization}, which can be very large if $u$ is small ($\hat Z(p_F)\to 0$ if $u\to 0$). In the non interacting case of course $\hat Z(p_F)=1$. In the first region (smaller distances) the slower bound \equ(z.2) is found for the Schwinger function decay. Note that combing \equ(z.2) with \equ(luttt) we can conclude that for $|\xx-\yy|$ less than $O(u^{-1\over 2})$ (say) the Schwinger function decays as the free ($\l=u=0$) one divided by ${1\over |\xx-\yy|^{\h_3}}$, in the sense that $|S(\xx,\yy)-{g(\xx;\yy)\over |\xx-\yy|^{\h_3}}|\le {C\over|\xx-\yy|^{1+\h_3}}$ and noting that $\lim\sup_{|\xx-\yy|\to\io}|(\xx-\yy) g(\xx;\yy)|>0$. The faster than any power decay and the modification of the decay rate suggest the presence of an {\it anomalous gap} \ie $\D\simeq u^{1+\h_2}$. This is confirmed by (iv) in which a lower bound for the gap is obtained; note that , if the interaction is attractive ($\l\le 0)$, \equ(z.4) says that the ratio between the bare and interacting gap is $<<1$ and diverging as $u\to 0$. In the limit $u\to 0$ it is known [BGL] that the Schwinger function decays as the Luttinger model one, and this is in fact what we find in the limit $u\to 0$. At the end the interaction produces an anomalous gap and an anomalous wave function renormalization, and the anomaly is related to the appearance of critical indices, explicitely computed by convergent series. \vskip.5cm \0{\bf 1.5} The point iii) in the theorem allows us to make clearer the relationship of our results with the literature. In the $\l=0$ case the hamiltonian is the sum of many single particle hamiltonians and the eigenfunctions of the many body problem are a product of the eigenfunctions of the {\it finite difference Schroedinger equation} $$-\psi(x+1)-\psi(x-1)+u \phi_x\psi_x=E\psi_x$$ where $\phi_x=\bar \phi(2px)$, $\bar \phi(u)$ $2\pi$-periodic and $p$ Diophantine. By a number of works on the spectrum it is known that for certain values of $E$ the eigenfunctions are {\it quasi Bloch waves} of the form $\phi(k,x,u)=e^{i k x} U(x)$, with $k=k(E)$ and $U(x)=\bar U(p x)$ and $\bar U$ $2 \pi$-periodic. In particular this is true for $k=mp$, for $\l$ small enough, and the spectrum has a gap $O(u)$ in corrispondence of this $E$ (this was proved by [JM],[MP],[E] for the case of the Schoedinger equation in the continuum with a quasi periodic potential; but one can extend these results to the finite difference Schroedinger equation, see [BLT]). >From the decomposition \equ(1.21) we see that $S_1$ for $\l=0$ is essentially the Schwinger function corresponding to a quasi Bloch wave $\phi(k,x,u)$ with a gap $O(u)$ in the spectrum and, as we can expect that $S_1$ is the dominant part for large distances \ie for $k\simeq p_F=mp$, this is in agreement with the results about Schroedinger equation. If $\l\not=0$ we find that the Schwinger function has the same structure with the main difference that $u$ is replaced by $\hat u(k)$ and there is a factor ${1\over \hat Z(\kk)}$ more. It is natural to interpretate this result saying that the fermions for $k\simeq mp$ are {\it quasi particles} which are {\it interacting quasi Bloch waves} ${1\over \sqrt{\hat Z(\kk)}}\phi(k,x,\hat u(k))$. Of course to really prove this one has to prove that $S_1(\xx;\yy)$ is really the dominant term. %This means that the Schwinger function can be written as the sum of two %terms with $\lim_{\l,u\to 0} S_A=g$, $\lim_{\l,u\to 0} S_B=0$. Noting that %$\phi(\kk,\xx,u)$ are Bloch waves, up to terms $O(u)$, one could say, if %$S_A$ were the "dominant part" of the Schwinger functions, that the %interacting one-particle wavefunctions for $\kk$ near $p_F$ are %approximately ${\phi(\kk,\xx,u(k))\over Z(k)}$ \ie interacting %Bloch waves (in the same way if $\l=0$ the results in [BGM] says %that the one-particle wavefunction are approximately Bloch waves}. %Finally we note that there are no difference in the Schwinger functions %asynptotic behaiour if the CDW is commensurate or not. The case of %commensurate CDW is (essentially) trated in [BM] and an identica\L&=\{x\in\ZZZ:\ -[L/2]\le x \le [(L-1)/2]\}\cr}$$ %l %asymtotic behaviour is obtained. \vskip.5cm \0{\bf 1.6} The Schwinger function behaviour we find is not peculiar of the Holstein-Hubbard model in the uncommensurate case; one can prove similar statements in the commensurate case \ie large distance behaviour of the system is independent from the periodic or quasi periodic nature of the phonon field $\phi_x$. The commensurate case was studied in [BM2],[BM3] (there was studied the case of interacting fermions in the continuum space, but an adaptation to this case is trivial). The large distances Schwinger function behaviour we find is typical to a wide class of models (in statystical physics or particle physics), like the massive Thirring model or the Yukawa2 model (see [BM3],[BM4] and references therein). A renormalization group analysis shows in fact that such models are equivalent in the sense that the relevant part of their Beta function is equal. Finally some comments about the spin; the discussion is essentially identical to the one in [BM2] and we do not repeat it. In the spinning case the number of running coupling constants is larger than in the spinless case and the renormalization group flow is more complex. For repulsive interaction ($\l>0$) things do not change while in the opposite case the flow is umbounded and no conclusions can be drawn (except if $u>e^{-1\over k\l^2}$, the trivial case); so our results are valid for spinning fermions only if $\l>0$. The paper is self-consistent but we have omitted the proofs of some technical lemmas which can be found in the literature (especially in [BGPS]) in order to save space. \vskip2.truecm \centerline{\titolo 2. Multiscale decomposition and anomalous integration} \*\numsec=2\numfor=1 \0{\bf 2.1} As it is well known, the Schwinger functions can be written as power series in $\l$, convergent for $|\l| \le \e_\b$, for some constant $\e_\b$ (the only trivial bound of $\e_\b$ goes to zero, as $\b\to\io$). This power expansion is constructed in the usual way in terms of Feynman graphs, by using as {\sl free propagator} the function % $$\eqalign{ g^{L,\b}(\xx;\yy) &\= g^{L,\b}(\xx-\yy)= {{\rm Tr} \left[e^{-\b H_0} {\bf T} (\psi^-_\xx \psi^+_\yy)\right] \over {\rm Tr} [e^{-\b H_0}]} = \cr &={1\over L} \sum_{k\in {\cal D}_L} e^{-ik(x-y)} \left\{ {e^{-\t e(k)} \over 1+e^{-\b e(k)}} \indic(\t>0) - {e^{-(\b+\t) e(k)} \over 1+e^{-\b e(k)}} \indic(\t\le 0) \right\}\; , \cr}\Eq(1.3f)$$ % where $\t=x_0-y_0$, $\indic(E)$ denotes the indicator function ($\indic(E)=1$, if $E$ is true, $\indic(E)=0$ otherwise), $e(k)=1-\cos k -\mu$ and ${\cal D}_L\=\{k={2\pi n/L}, n\in Z, -[L/2]\le n \le [(L-1)/2]\}$. It is easy to prove that, if $x_0\not= y_0$, % $$g^{L,\b}(\xx-\yy)= \lim_{M\to\io} {1\over L\b} \sum_{\kk\in {\cal D}_{L,\b}} {e^{-i\kk\cdot(\xx-\yy)}\over -ik_0+\cos p_F-\cos k}\; , \Eq(1.4f)$$ % where $\kk=(k,k_0)$, $\kk\cdot\xx=k_0x_0+kx$, ${\cal D}_{L,\b}\={\cal D}_L \times {\cal D}_\b$, ${\cal D}_\b\=\{k_0=2(n+1/2)\pi/\b, n\in Z, -M\le n \le M-1\}$ and $p_F$ is the {\sl Fermi momentum}, defined so that $\cos p_F =1-\mu$ and $0\le p_F \le \p$. Hence, if we introduce a finite set of Grassmanian variables $\{\psi^\pm_\kk\}$, one for each of the allowed $\kk$ values, and a linear functional $P(d\psi)$ on the generated Grassmanian algebra, such that % $$\int P(d\psi) \psi^-_{\kk_1}\psi^+_{\kk_2} = L\b \d_{\kk_1,\kk_2} \hat g_{\kk_1}\;,\quad \hat g_\kk= {1\over -ik_0+\cos p_F-\cos k} \; ,\Eq(1.5f)$$ % we have % $${1\over L\b} \sum_{\kk\in {\cal D}_{L,\b}} \, e^{-i\kk\cdot(\xx-\yy)} \, \hat g_\kk = \int P(d\psi)\psi^-_\xx \psi^+_\yy \= g^{L,\b}(\xx;\yy) \; ,\Eq(1.6f)$$ % where the {\sl Grassmanian field} $\psi_\xx$ is defined by % $$\psi_\xx^{\pm}= {1\over L\b} \sum_{\kk\in {\cal D}_{L,\b}} \psi_\kk^{\pm} e^{\pm i\kk\cdot\xx}\; .\Eq(1.7f)$$ The ``Gaussian measure'' $P(d\psi)$ has a simple representation in terms of the ``Lebesgue Grassmanian measure'' $d\psi d\bar\psi$, defined as the linear functional on the Grassmanian algebra, such that, given a monomial $Q(\psi,\bar\psi)$ in the variables $\psi_\kk,\bar\psi_\kk$, % $$\int d\psi d\bar\psi Q(\psi,\bar\psi) =\cases{1& if $Q(\psi,\bar\psi)= \prod_\kk \psi_\kk\bar\psi_\kk \; ,$\cr 0& otherwise $\; .$\cr} \Eq(1.8f)$$ % We have % $$P(d\psi) = \Big\{ \prod_\kk (L\b\hat g_\kk) \Big\} \exp \Big\{-\sum_\kk (L\b \hat g_\kk)^{-1} \bar\psi_\kk \psi_\kk \Big\} d\psi d\bar\psi \; .\Eq(1.9f)$$ % Note that, since $\psi_\kk^2=\bar\psi_\kk^2=0$, $e^{-z \bar\psi_\kk \psi_\kk} =1-z \bar\psi_\kk \psi_\kk$, for any complex $z$. By using standard arguments (see, for example, [NO], where a different regularization of the propagator is used), one can show that the Schwinger functions can be calculated as expectations of suitable functions of the Grassmanian field with respect to the ``Gaussian measure'' $P(d\psi)$. In particular, the two-point Schwinger function can be written, if $x_0\not= y_0$, as % $$ S^{L,\b}(\xx;\yy) = \lim_{M\to\io} {\int P(d\psi)\, e^{-\VV(\psi)}\,\psi^-_{\xx}\psi^+_{\yy} \over\int P(d\psi)\,e^{-\VV (\psi)}} \; , \Eq(1.10f) $$ % where $\VV(\psi)=u P(\psi)+\l V(\psi)+\nu N(\psi)$ with % $$V(\psi)=\sum_{x,y\in\L}\int_{-\b/2}^{\b/2} dx_0\int_{-\b/2}^{\b/2} dy_0 v(x-y)\d(x_0-y_0)\psi_\xx^+\psi_\xx^-\psi_\yy^-\psi_\yy^+$$ $$ P(\psi)=\sum_{x\in\L} \int_{-\b/2}^{\b/2} dx_0 \Big[\f_x\psi_\xx^+ \psi^-_\xx \Big]\qquad N(\psi)=\sum_{x\in\L} \int_{-\b/2}^{\b/2} dx_0 \psi_\xx^+ \psi^-_\xx \; .\Eq(1.11f) $$ %and $\d(x_0-y_0)=\d_{x_0,y_0}$ (and in an analogous way %$\d(x-y)=L\d_{x,y}$. % If $x_0=y_0$, $S^{L,\b}(\xx;\yy)$ must be defined as the limit of \equ(1.10f) as $x_0-y_0\to 0^-$, as we shall understand always in the following. \vskip.5cm \0{\bf 2.2} We assume from now on $p_F=p_L$ \ie $m=1$ (this assumption is of course not restrective) and start by evalutating the partition function\ie the denominator of \equ(1.10f) $$\int P(d\psi) e^{-\VV(\psi)}\; , \Eq(2.1)$$ It is convenient to decompose the Grassmanian integration $P(d\psi)$ into a finite product of independent integrations: % $$ P(d\psi)=\prod_{h=h_\b }^1 P(d\psi^{(h)}) \; ,\Eq(2.1) $$ where $h_\b >-\io$ will be defined below (before \equ(2.9)) This can be done by setting % $$ \psi_\kk^{\pm}=\bigoplus_{h=h_\b }^1\psi_\kk^{(h)\pm} \; , \qquad \hat g_\kk=\sum_{h=h_\b }^1 \hat g^{(h)}_\kk \; , \Eq(2.2) $$ % where $\psi^{(h)\pm}_\kk$ are families of Grassmanian fields with propagators $\hat g^{(h)}_\kk$ which are defined in the following way. %We call moreover %$\psi_\kk^{(\le k)\pm}=\bigoplus_{h=h_\b }^k\psi_\kk^{(h)\pm}$ %and $g_\kk^{\le k}=\sum_{h=h_\b }^k \hat g^{(h)}_\kk$ % We introduce a {\sl scaling parameter} $\g>1$ and a function $\c(\kk') \in C^{\io}(T^1\times R)$, $\kk'=(k',k_0)$, such that, if $|\kk'|\=\sqrt{k_0^2+||k'||_{T^1}^2}$: % $$ \c(\kk') = \c(-\kk') = \cases{ 1 & if $|\kk'| a_0\; ,$\cr}\Eq(2.3)$$ % where $a_0=\min \{p_F/2, (\p-p_F)/2 \}$. This definition is such that the supports of $\c(k-p_F,k_0)$ and $\c(k+p_F,k_0)$ are disjoint and the $C^\io$ function on $T^1\times R$ % $$\hat f_1(\kk) \= 1- \c(k-p_F,k_0) - \c(k+p_F,k_0) \Eq(2.4)$$ % is equal to $0$, if $||k|-p_F||_{T^1}^2 +k_0^2t_0 \g^{h+1}$, and $f_h(\kk')= 1$, if $|\kk'| =t_0\g^h$. We finally define, for any $h\le 0$: % $$ \hat f_h(\kk) = f_h(k-p_F,k_0) + f_h(k+p_F,k_0)\; ,\Eq(2.7) $$ % $$ \hat g^{(h)}_\kk \= { \hat f_h(\kk) \over -ik_0+\cos p_F -\cos k} \; . \Eq(2.8) $$ Note that, if $\kk\in {\cal D}_{L,\b}$, then $|k_0|\ge \p/\b$, implying that $\hat f_h(\kk)=0$ for any $h< h_\b = \min \{h:t_0\g^{h+1} > \p/\b \}$. Hence, if $\kk\in {\cal D}_{L,\b}$, the definitions \equ(2.4) and \equ(2.7), together with the identity \equ(2.6), imply that % $$1=\sum_{h=h_\b }^1 \hat f_h(\kk) \; .\Eq(2.9)$$ The definition \equ(2.7) implies also that, if $h\le 0$, the support of $\hat f_h(\kk)$ is the union of two disjoint sets, $A_h^+$ and $A_h^-$. In $A_h^+$, $k$ is strictly positive and $||k-p_F||_{T^1}\le a_0\g^h \le a_0$, while, in $A_h^-$, $k$ is strictly negative and $||k+p_F||_{T^1}\le a_0\g^h$. Therefore, if $h\le 0$, we can write $\psi^{(h)\pm}_{\kk}$ as the sum of two independent Grassmanian variables $\psi_{\kk,\o}^{(h)\pm}$ with propagator % $$ \int P(d\psi^{(h)})\,\psi^{(h)-}_{\kk_1,\o_1} \psi^{(h)+}_{\kk_2,\o_2} = L\b \d_{\kk_1,\kk_2}\,\d_{\o_1,\o_2}\, \hat g^{(h)}_{\o_1}(\kk_1) \; , \Eq(2.10)$$ % so that % $$ \psi^{(h)\pm}_{\kk}=\bigoplus_{\o=\pm 1}\psi^{(h)\pm}_{\kk,\o} \; , \qquad \hat g^{(h)}_\kk=\sum_{\o=\pm 1} \hat g^{(h)}_\o(\kk) \; , \Eq(2.11) $$ % $$ \hat g^{(h)}_\o(\kk)={\theta(\o k) \, \hat f_h(\kk) \over -ik_0 + \cos p_F - \cos k }\; , \Eq(2.12) $$ % where $\theta(k)$ is the (periodic) step function. If $\o k> 0$, we will write in the following $k=k'+\o p_F$, where $k'$ is the {\sl momentum measured from the Fermi surface} and we shall define, if $h\le 0$, % $$ \tilde g^{(h)}_{\o}(\kk') \= \hat g^{(h)}_\o(\kk)= {f_h(\kk') \over -i k_0+v_0 \o\sin k'+(1-\cos k')\cos p_F } \; , \Eq(2.13) $$ % where $v_0=\sin p_F$. We call moreover $\psi_{\kk,\omega}^{(\le k)\pm}=\bigoplus_{h=h_\b }^k\psi_{\kk,\omega}^{(h)\pm}$ and $g_{\kk,\omega}^{\le k}=\sum_{h=h_\b }^k \hat g^{(h)}_{\kk,\omega}$ %In order to simplify the notation, it will be useful in the following to %denote $\hat g^{(1)}_\kk$ also as $\tilde g^{(1)}_+(\kk')$, with %$k=k'+p_F$. This kind of decomposition is completely standard in the theory of the $d=1$ Fermi system; we repeat it here only for clarity. We define $$e^{-\VV^{(0)}(\psi^{(\le 0)})}=\int P(d\psi^{(1)}) e^{-\VV^{(0)}(\psi^{(1)}+\psi^{(\le 0)})}$$ It is possible to prove , see [BGL], [BGPS] that $$\VV^{(0)}(\psi^{(\le 0)})=\l {1\over (L\b)^4}\sum_{\kk_1,...,\kk_4\in {\cal D}_{L,\b}} \hat v(\kk_1-\kk_2) \psi^{(\le 0)+}_{\kk_1} \psi^{(\le 0)-}_{\kk_2}\psi^{(\le 0)+}_{\kk_3}\psi^{(\le 0)-}_{\kk_4} \d(\kk_1+\kk_3-\kk_2-\kk_4)$$ $$+ {1\over L\b}\sum_{\kk\in {\cal D}_{L,\b}}(\nu+F(\kk)) \psi^{(\le 0)+}_{\kk}\psi^{(\le 0)-}_{\kk} +u\sum_{m=1}^\io \hat\phi_m {1\over L\b}\sum_{\kk\in {\cal D}_{L,\b}} \psi^{(\le 0)+}_{\kk} \psi^{(\le 0)-}_{\kk+2m \pp_L}+\psi^{(\le 0)+}_{\kk} \psi^{(\le 0)-}_{\kk-2m \pp_L}$$ $$+\sum_{m=1}^\io\sum_{n=0}^\io {1\over (L\b)^m}\sum_{\kk_1,..., \kk_m\in {\cal D}_{L,\b}} \psi^{(\le 0)\s_1}_{\kk_1}... \psi^{(\le 0)\s_m}_{\kk_m} W_{m,n}^{\le 0}(\kk_1,...,\kk_m;z)\d(\sum_{i=1}^n\s_i\kk_i+2n\pp_L)$$ where $\sigma_i=\pm$, $|F(\kk)|\le C|\l|$ and the kernels $W_{m,n}^{\le 0}(\kk_1,...,\kk_m;z)$ are $C^\io$ bounded functions such that $W_{m,n}=W_{m,-n}$ and $|W_{m,n}|\le C^m z^{\max(2,m/2-1)}$ if $z=Max(|\l|,u,|\nu|)$; moreover $\d(\kk)=L\b\d_{k_0}\d_{k}$. %(and in an analogous way %$\d(x-y)=L\d_{x,y}$. %The proof of the above statement is quite standard, %as one can repeat word by word, up to trivial modication, %the corresponding analysis for a similar model in [BGPS]. \vskip.5cm \0{\bf 2.2} We perform now the infrared integration $$\int P(d\psi^{(\le 0)})e^{-\VV^{(0)}(\psi^{(\le 0)})}\Eq(stel)$$ It is convenient, for reasons which will be clear below, to split $\VV$ in a {\it relevant} and {\it irrelevant} part $\VV=\LL\VV+\RR\VV$ where $\LL$, the {\it localization operator}, a linear operator defined in the following way:. \vskip.5cm 1) If $m>4$ then $$\LL\{{1\over (L\b)^m}\sum_{\kk'_1,...,\kk'_m\in {\cal D}_{L,\b}} W_{m,n}^{\le 0}(\kk'_1+\o_1 p_F,...)[\prod_{i=1}^m\psi^{(\le 0)\s_i}_{\kk'_i+\o_i p_F,\o_i}] \d(\sum_{i=1}^m \s_i(\kk'_i+\o_i p_F)+2np_L)\}=0$$ \vskip.5cm 2) If $m=4$ then $$\LL \{ {1\over (L\b)^4}\sum_{\kk'_1,...,\kk'_4\in {\cal D}_{L,\b}} W_{4,n}^{\le 0}(\kk'_1+\o_1 p_F,..., \kk'_4+\o_4 p_F)$$ $$[\prod_{i=1}^4 \psi^{(\le 0)\s_i}_{\kk'_i+\o_i p_F,\o_i}] \d(\sum_{i=1}^4 \s_i(\kk'_i+\o_i p_F)+2np_L)\}$$ $$=\d_{\sum_{i=1}^4\sigma_i\o_i p_F+2n p_L} {1\over (L\b)^4}\sum_{\kk'_1,...,\kk'_4\in {\cal D}_{L,\b}} W_{m,n}^{\le 0}(\o_1 p_F,..., \o_4 p_F)[\prod_{i=1}^4 \psi^{(\le 0)\s_i}_{\kk'_i+\o_i p_F,\o_i}]\d(\sum_{i=1}^4 \s_i \kk'_i)\Eq(loc1)$$ %$$\psi^{(\le 0)+}_{\kk'_1+\o_1 p_F,\o_1} %\psi^{(\le 0)-}_{\kk'_2+\o_2 p_F,\o_2} %\psi^{(\le 0)+}_{\kk'_3+\o_3 p_F,\o_3} %\psi^{(\le 0)-}_{\kk'_4+\o_4 p_F,\o_4} %\d(\sum_{i=1}^4 \s_i \kk'_i)$$ \vskip.5cm 3)If $m=2$ then $$\LL\{ {1\over (L\b)^2}\sum_{\kk'_1,\kk'_2\in {\cal D}_{L,\b}} W_{2,n}^{\le 0}(\kk'_1+\o_1 p_F, \kk'_2+\o_2 p_F)[\prod_{i=1}^2\psi^{(\le 0)\s_i}_{\kk'_i+\o_i p_F,\o_i}] \d(\sum_{i=1}^2 \s_i(\kk'_i+\o_i p_F)+2np_L)\}$$ $$=\d_{(\o_1-\o_2)p_F+2n p_L}{1\over (L\b)}\sum_{\kk'\in {\cal D}_{L,\b}} [W_{2,n}^{\le 0}(\o_1 p_F,\o_2 p_F)+\Eq(loc2)$$ $$+\o_1E(k'+\o_1 p_F)\partial_{k} W_{2,n}^{\le 0}(\o_1 p_F,\o_2 p_F)+ k^0 \partial_{k_0}W_{2,n}^{\le 0}(\o_1 p_F,\o_2 p_F)] [\prod_{i=1}^2\psi^{(\le 0)\s_i}_{\kk'_i+\o_i p_F,\o_i}]$$ \vskip.5cm where $E(k'+\o p_F)=v_0\o\sin k'+(1-\cos k')\cos p_F$ (the symbol $\partial_k,\partial_{k_0}$ means discrete derivativs). \vskip1cm We can write then the relevant part of the effective potential in the following way: $$\LL\VV^{(0)}=n_0 F_\nu^{(\le 0)}+s_0 F_\s^{(\le 0)}+z_0 F_\z^{(\le 0)}+a_0 F_\a^{(\le 0)}+i_0 F_\iota^{(\le 0)}+t_0 F_\t^{(\le 0)}+l_0 F_\l^{(\le 0)}\Eq(bin)$$ where $$F_\nu^{(\le 0)}=\sum_\o {1\over (L\b)}\sum_{\kk'\in {\cal D}_{L,\b}} \psi^{(\le 0)+}_{\kk'+\o p_F,\o} \psi^{(\le 0)-}_{\kk'+\o p_F,\o}$$ $$F_\s^{(\le 0)}=\sum_\o {1\over (L\b)}\sum_{\kk'\in {\cal D}_{L,\b}} \psi^{(\le 0)+}_{\kk'+\o p_F,\o} \psi^{(\le 0)-}_{\kk'-\o p_F,-\o}$$ $$F_\a^{(\le 0)}=\sum_\o {1\over (L\b)}\sum_{\kk'\in {\cal D}_{L,\b}} E(k'+\o p_F) \psi^{(\le 0)+}_{\kk'+\o p_F,\o} \psi^{(\le 0)-}_{\kk'+\o p_F,\o}$$ $$F_\z^{(\le 0)}=\sum_\o {1\over (L\b)}\sum_{\kk'\in {\cal D}_{L,\b}} (-i k_0) \psi^{(\le 0)+}_{\kk'+\o p_F,\o} \psi^{(\le 0)-}_{\kk'+\o p_F,\o}$$ $$F_\iota^{(\le 0)}=\sum_\o {1\over (L\b)}\sum_{\kk'\in {\cal D}_{L,\b}} E(k'+\o p_F) \psi^{(\le 0)+}_{\kk'+\o p_F,\o} \psi^{(\le 0)-}_{\kk'-\o p_F,-\o}$$ $$F_\t^{(\le 0)}=\sum_\o {1\over (L\b)}\sum_{\kk'\in {\cal D}_{L,\b}} (-i k_0) \psi^{(\le 0)+}_{\kk'+\o p_F,\o} \psi^{(\le 0)-}_{\kk'-\o p_F,-\o}$$ $$F_\l^{(\le 0)}={1\over (L\b)^4}\sum_{\kk'_1,...,\kk'_4\in {\cal D}_{L,\b}} \psi^{(\le 0)+}_{\kk'_1+p_F,1} \psi^{(\le 0)+}_{\kk'_1-p_F,-1} \psi^{(\le 0)-}_{\kk'_3+p_F,1} \psi^{(\le 0)-}_{\kk'_4-p_F,-1}\d(\sum_{i=1}^4\s_i\kk_i)$$ and $\l_0=\l(\hat v(0)-\hat v(2 p_F))+O(\l^2)$, $s_0=u+O(u\l)$, $t_0,i_0=O(u\l)$, $a_0.z_0=O(\l)$, $n_0=\nu+O(\l)$. We write \equ(stel) as % $$\int P_{Z_0}(d\psi^{\le 0}) \, e^{-\VV^{(0)}(\sqrt{Z_0}\psi^{\le 0})} \; , \Eq(3.6) $$ % where $Z_0=1$ and $P_{Z_0}(d\psi^{(\le 0)})$ is the Grassmanian integration with propagator % $$ \eqalign{ & g^{(\le 0)}(\xx;\yy) = \sum_{\o,\o'=\pm1} {1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \, {1\over Z_0}e^{-i\kk'\cdot(\xx-\yy)}\,e^{-i(\o x - \o' y)p_F} \, \tilde g^{(\le 0)}_{\o,\o'}(\kk') \; , \cr & \tilde g^{(\le 0)}_{\o,\o'}(\kk') = \d_{\o,\o'} \, \tilde g^{(\le 0)}_{\o}(\kk') \; , \cr} \Eq(3.7) $$ with % $$ \tilde g^{(\le 0)}_\o(\kk') = {C_0^{-1}(\kk') \over -ik_0 + (1-\cos k') \cos p_F + v_0\o \sin k' } \; , \qquad C_h^{-1}(\kk')=\sum_{j=h_\b}^h f_{j}(\kk') \; , \Eq(3.8) $$ % see \equ(2.12), \equ(2.13). We write % $$\int P_{Z_0}(d\psi^{(\le 0)}) \, e^{-\VV^0(\sqrt{Z_0}\psi^{(\le 0)})} = {1 \over \NN_0}\int \tilde P_{Z_{-1}}(d\psi^{(\le 0)}) \, e^{-\tilde \VV^{(0)}(\sqrt{Z_0}\psi^{\le 0})} \; , \Eq(3.9) $$ % where $\NN_0$ is a suitable constant and, again up to a constant, % $$ \eqalign{ &\tilde P_{Z_{-1}}(d\psi^{(\le 0)}) = \prod_{\kk}\prod_{\o=\pm1} d\psi^{(\le 0)+}_{\kk'+\o\pp_F,\o} d\psi^{(\le 0)-}_{\kk'+\o\pp_F,\o} \cr &\exp \Big\{ -\sum_{\o=\pm1} {1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \, C_0(\kk') Z_{-1}(\kk') \Big[\Big( -ik_0-(\cos k'-1)\cos p_F +\o v_0\sin k' \Big) \cr & \psi^{(\le0)+}_{\kk'+\o\pp_F,\o} \psi^{(\le0)-}_{\kk'+\o\pp_F,\o} + \sigma_{-1}(\kk') \, \psi^{(\le0)+}_{\kk'+\o\pp_F,\o} \psi^{(\le0)-}_{\kk'-\o\pp_F,-\o} \Big] \Big\} \; , \cr} \Eq(3.10) $$ % with $Z_{-1}(\kk')\sigma_{-1}(\kk')=C_0^{-1}(\kk')\,s_0$, $Z_{-1}(\kk')=Z_0(1+C_0^{-1}(\kk')z_0)$ and $\tilde \VV^{(0)}=\LL \tilde \VV^{(0)}+(1-\LL) \VV^{(0)}$, if % $$\LL\tilde \VV^{(0)}(\sqrt{Z_0}\psi)= n_0 F_\nu^{(\le 0)}+(a_0-z_0) F_\a^{(\le 0)}+i_0 F_\iota^{(\le 0)}+t_0 F_\t^{(\le 0)}+l_0 F_\l^{(\le 0)} \; . \Eq(3.11) $$ The r.h.s of \equ(3.9) can be written as % $$ {1 \over \NN_0}\int P_{Z_{-1}}(d\psi^{(\le -1)}) \int \tilde P_{Z_{-1}}(d\psi^{(0)}) \, e^{-\tilde \VV^{(0)}(\sqrt{Z_0}\psi^{(\le 0)})} \; , \Eq(3.12) $$ % where $ P_{Z_{-1}}(d\psi^{(\le -1)})$ and $\tilde P_{Z_{-1}}(d\psi^{(0)})$ are given by \equ(3.10) with $Z_{-1}(\kk')$ replaced by $Z_{-1}(0)\equiv Z_{-1}$ and $C_0(\kk')$ replaced with $C_{-1}(\kk')$ and $\tilde f_0^{-1}(\kk')$ respectively, if $$\tilde f_0(\kk')=Z_{-1}[{C_0^{-1}(\kk')\over Z_{-1}(\kk')}- {C_{-1}^{-1}(\kk')\over Z_{-1}}]$$ and $\psi^{(\le 0)}$ replaced with $\psi^{(\le -1)}$ and $\psi^{(0)}$ respectively. The Grassmanian integration $\tilde P_{Z_{-1}}(d\psi^{(0)})$ has propagator % $$ g^{(0)}(\xx;\yy) = \sum_{\o,\o'=\pm1} e^{-i(\o x - \o' y)p_F}\, g^{(0)}_{\o,\o'}(\xx;\yy) \; , \Eq(3.13) $$ % if % $$ g^{(0)}_{\o,\o'}(\xx;\yy)\=\int \tilde P_{Z_{-1}}(d\psi^{(0)})\, \psi^{(0)-}_{\xx,\o}\psi^{(0)+}_{\yy,\o'} \Eq(3.14) $$ % is given by % $$ g^{(0)}_{\o,\o'}(\xx;\yy)={1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \, e^{-i\kk'\cdot(\xx-\yy)}{\tilde f_0(\kk')\over Z_{-1}} [T_{0}^{-1}(\kk')]_{\o,\o'} \; , \Eq(3.15) $$ % where the $2\times2$ matrix $T_{0}(\kk')$ has elements % $$ \cases{ [T_{0}(\kk')]_{1,1} = \left(-ik_0-(\cos k'-1)\cos p_F+ v_0\sin k' \right) \; , & \cr [T_{0}(\kk')]_{1,2} = [T_{0}(\kk')]_{2,1} = \sigma_{-1}(\kk') \; , & \cr [T_{0}(\kk')]_{2,2} = \left(-ik_0-(\cos k'-1)\cos p_F-v_0\sin k'\right) \; , \cr} \Eq(3.16) $$ % which is well defined on the support of $f_0(\kk')$, so that, if we set % $$ A_{0}(\kk') = \det T_0(\kk') = [ -ik_0-(\cos k'-1)\cos p_F ]^2 - (v_0\sin k')^2 - [\sigma_{-1}(\kk')]^2 \; ,\Eq(3.17) $$ % then % $$ T_{0}^{-1}(\kk')= {1\over A_{0}(\kk') } \left( \matrix{ [\t_{0}(\kk')]_{1,1} & [\t_{0}(\kk')]_{1,2} \cr [\t_{0}(\kk')]_{2,1} & [\t_{0}(\kk')]_{2,2} \cr} \right) \; , \Eq(3.18) $$ % with % $$ \cases{ [\t_{0}(\kk')]_{1,1} = \left[-ik_0-(\cos k'-1) \cos p_F-v_0\sin k'\right] \; , & \cr [\t_{0}(\kk')]_{1,2} = [\t_{0}(\kk')]_{2,1} = -\sigma_{-1}(\kk') \; , & \cr [\t_{0}(\kk')]_{2,2} = \left[-ik_0-(\cos k'-1)\cos p_F + v_0 \sin k' \right] \; . & \cr} \Eq(3.19) $$ % We {\it rescale} the fields so that % $${1 \over \NN_0}\int P_{Z_{-1}}(d\psi^{(\le -1)}) \int \tilde P_{Z_{-1}}(d\psi^{(0)}) \, e^{-\hat \VV^{(0)} (\sqrt{Z_{-1}}\psi^{(\le 0)})}$$ so that, if the operator $\LL$ is defined as in \equ(loc1),\equ(loc2) $$ \LL\hat\VV^{(0)}(\psi)= \nu_0 F_\nu^{(\le 0)}+\d_0 F_\a^{(\le 0)}+\iota_0 F_\iota^{(\le 0)}+\t_0 F_\t^{(\le 0)}+\l_0 F_\l^{(\le 0)} \; . \Eq(3.11a) $$ where by definition $$\nu_o={Z_0\over Z_{-1}}n_0;\quad \d_0={Z_0\over Z_{-1}}(a_0-z_0);\quad \t_0={Z_0\over Z_{-1}} t_0\quad\iota_0={Z_0\over Z_{-1}}i_0;\quad \l_0=({Z_0\over Z_{-1}})^2 l_0$$ Then we perform the integration $$ \int \tilde P_{Z_{-1}}(d\psi^{(0)}) \, e^{-\hat\VV^{(0)} (\sqrt{Z_{-1}}\psi^{(\le 0)})} = e^{-\VV^{-1}(\sqrt{Z_{-1}}\psi^{(\le -1)}) + \tilde E_0} \; , \Eq(3.20) $$ % where $\tilde E_0$ is a suitable constant and % $$ \LL \VV^{(-1)}(\psi)= \g^{-1}\n_{-1} F_\nu^{(-1)}+s_{-1} F_\sigma^{(-1)} +a_{-1} F_\a^{(\le -1)}+ z_{-1} F_\z^{(\le -1)}+$$ $$i_{-1} F_\iota^{(\le -1)}+\t_{-1} F_\t^{(\le -1)}+ l_{-1} F_\l^{(\le -1)}\; \Eq(3.21) $$ % \0{\bf 2.3} The procedure can be iterated, and at each step one has to perform the integration $$\int P_{Z_h}(d\psi^{(\le h)}) \, e^{-\VV^{(h)}(\sqrt{Z_h}\psi^{(\le h)})}\Eq(3.22)$$ with $$ \eqalign{ P_{Z_{h}}(d\psi^{(\le h)}) = & \prod_{\kk}\prod_{\o=\pm1} d\psi^{(\le h)+}_{\kk'+\o\pp_F,\o} d\psi^{(\le h)-}_{\kk'+\o\pp_F,\o} \cr \exp \Big\{ &-\sum_{\o=\pm1} {1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \, C_0(\kk') Z_{h} \Big[\Big( -ik_0-(\cos k'-1)\cos p_F +\o v_0\sin k' \Big) \cr & \psi^{(\le0)+}_{\kk'+\o\pp_F,\o} \psi^{(\le0)-}_{\kk'+\o\pp_F,\o} + \sigma_{h}(\kk') \, \psi^{(\le0)+}_{\kk'+\o\pp_F,\o} \psi^{(\le0)-}_{\kk'-\o\pp_F,-\o} \Big] \Big\} \; , \cr} \Eq(3.10a) $$ Moreover, with notations analogues to \equ(loc1),\equ(loc2) \eq(bin) we can write $$\LL\VV^{(h)}(\psi)=\g^h n_h F_\nu^{(\le h)}+s_h F_\s^{(\le h)}+z_h F_\z^{(\le h)}+a_h F_\a^{(\le h)}+ i_h F_\iota^{(\le h)}+t_h F_\t^{(\le h)}+l_h F_\l^{(\le h)} \; . \Eq(3.11a) $$ We write $$\int P_{Z_h}(d\psi^{(\le h)}) \, e^{-\VV^{(h)}(\sqrt{Z_h}\psi^{(\le h)})}= \int \tilde P_{Z_{h-1}}(d\psi^{(\le h)}) \, e^{-\tilde\VV^{(h)}(\sqrt{Z_h}\psi^{(\le h)})} \Eq(3.22a)$$ where $$ \eqalign{ &\tilde P_{Z_{h-1}}(d\psi^{(\le h)}) = \prod_{\kk}\prod_{\o=\pm1} d\psi^{(\le h)+}_{\kk'+\o\pp_F,\o} d\psi^{(\le h)-}_{\kk'+\o\pp_F,\o} \cr &\exp \Big\{ -\sum_{\o=\pm1} {1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \, C_0(\kk') Z_{h-1}(\kk') \Big[\Big( -ik_0-(\cos k'-1)\cos p_F +\o v_0\sin k' \Big) \cr & \psi^{(\le0)+}_{\kk'+\o\pp_F,\o} \psi^{(\le0)-}_{\kk'+\o\pp_F,\o} + \sigma_{h-1}(\kk') \, \psi^{(\le0)+}_{\kk'+\o\pp_F,\o} \psi^{(\le0)-}_{\kk'-\o\pp_F,-\o} \Big] \Big\} \; , \cr} \Eq(3.10b) $$ with $Z_{h-1}(\kk')=Z_h(1+C_h^{-1}(\kk')z_h)$, $Z_{h-1}(\kk') \s_{h-1}(\kk')=Z_h( \s_h(\kk')+C_h^{-1}(\kk') s_h)$ and $\tilde \VV=\LL\tilde\VV+(1-\LL)\VV$ with $$\LL\tilde\VV^{(h)}(\psi)=\g^h n_h F_\nu^{(\le h)}+(a_h-z_h) F_\a^{(\le h)}+i_h F_\iota^{(\le h)}+t_h F_\t^{(\le h)}+l_h F_\l^{(\le h)} \; . \Eq(3.11b)$$ The r.h.s of \equ(3.22a) can be written as % $$ {1 \over \NN_h}\int P_{Z_{h-1}}(d\psi^{(\le h-1)}) \int \tilde P_{Z_{h-1}}(d\psi^{(h)}) \, e^{-\tilde \VV^{(h)}(\sqrt{Z_h}\psi^{(\le h)})} \; , \Eq(3.12a) $$ % where $ P_{Z_{h-1}}(d\psi^{(\le h-1)})$ and $\tilde P_{Z_{h-1}}(d\psi^{(h)})$ are given by \equ(3.10b) with $Z_{h-1}(\kk')$ replaced by $Z_{h-1}(0)\equiv Z_{h-1}$ and $C_h(\kk')$ replaced with $C_{h-1}(\kk')$ and $\tilde f_h^{-1}(\kk')$ respectively, if $$\tilde f_h(\kk')=Z_{h-1}[{C_h^{-1}(\kk')\over Z_{h-1}(\kk')}- {C_{h-1}^{-1}(\kk')\over Z_{h-1}}]$$ and $\psi^{(\le h)}$ replaced with $\psi^{(\le h-1)}$ and $\psi^{(h)}$ respectively. Note that $\tilde f_h(\kk')$ is a compact support function, with support $O(\g^h)$. The Grassmanian integration $\tilde P_{Z_{h-1}}(d\psi^{(h)})$ has propagator % $$ {g^{(h)}(\xx;\yy)\over Z_{h-1}} = \sum_{\o,\o'=\pm1} e^{-i(\o x - \o' y)p_F}\, {g^{(h)}_{\o,\o'}(\xx;\yy) \over Z_{h-1}}\; , \Eq(3.13a) $$ % if % $$ g^{(h)}_{\o,\o'}(\xx;\yy)\=\int \tilde P_{Z_{h-1}}(d\psi^{(h)})\, \psi^{(h)-}_{\xx,\o}\psi^{(h)+}_{\yy,\o'} \Eq(3.14) $$ % is given by % $$ g^{(h)}_{\o,\o'}(\xx;\yy)={1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \, e^{-i\kk'\cdot(\xx-\yy)} \tilde f_h(\kk') [T_{h}^{-1}(\kk')]_{\o,\o'} \; , \Eq(3.15a) $$ % where the $2\times2$ matrix $T_{h}(\kk')$ has elements % $$ \cases{ [T_{h}(\kk')]_{1,1} = \left(-ik_0-(\cos k'-1)\cos p_F+ v_0\sin k' \right) \; , & \cr [T_{h}(\kk')]_{1,2} = [T_{h}(\kk')]_{2,1} = \sigma_{h-1}(\kk') \; , & \cr [T_{h}(\kk')]_{2,2} = \left(-ik_0-(\cos k'-1)\cos p_F-v_0\sin k'\right) \; , \cr} \Eq(3.16a) $$ % which is well defined on the support of $f_0(\kk')$, so that, if we set % $$ A_{h}(\kk') = \det T_h(\kk') = [ -ik_0-(\cos k'-1)\cos p_F ]^2 - (v_0\sin k')^2 - [\sigma_{h-1}(\kk')]^2 \; ,\Eq(3.17) $$ % then % $$ T_{h}^{-1}(\kk')= {1\over A_{h}(\kk') } \left( \matrix{ [\t_{h}(\kk')]_{1,1} & [\t_{h}(\kk')]_{1,2} \cr [\t_{h}(\kk')]_{2,1} & [\t_{h}(\kk')]_{2,2} \cr} \right) \; , \Eq(3.18a) $$ % with % $$ \cases{ [\t_{h}(\kk')]_{1,1} = \left[-ik_0-(\cos k'-1) \cos p_F-v_0\sin k'\right] \; , & \cr [\t_{h}(\kk')]_{1,2} = [\t_{h}(\kk')]_{2,1} = -\sigma_{h-1}(\kk') \; , & \cr [\t_{h}(\kk')]_{2,2} = \left[-ik_0-(\cos k'-1)\cos p_F + v_0 \sin k' \right] \; . & \cr} \Eq(3.19a) $$ % Note that $\sigma_h(\kk')$ is a smooth function on $T^1\times R$ and, if $\kk'$ varies in the support of $C_h^{-1}(\kk')$ then there exists two positive constants $c_1,c_2$ such that: $$c_1 \sigma_h\le \sigma_h(\kk')\le c_2\sigma_h\Eq(ass)$$ if $\sigma_h\equiv \sigma_h(0)$, as $C_k^{-1}(\kk')=1$ for $k\ge h+1$. The large distance behaviour of the propagator \equ(3.13a) is given by the following lemma, see [BM2]: \vskip.5cm \0{\bf 2.4}{\rm LEMMA} {\it The propagator $g^h_{\o,\o'}(\xx-\yy)$ can be written as: $$g^h_{\o,\o}(\xx-\yy)=g^h_{L;\o}(\xx-\yy)+C_1(\xx-\yy)+C_2(\xx-\yy) \Eq(bb)$$ with $$g^h_{L;\o}(\xx-\yy)=\int d\kk {e^{i\kk\xx}\over -i k_0+\o v_0\kk'} \tilde f_h(\kk')$$ For any integer $N>1$ and for $|x-y|\le {L\over 2}$, $|x_0-y_0|\le {\b\over 2}$ it holds $|C_1(\xx-\yy)|\le {\g^{2h} C_N\over 1+(\g^h(\xx-\yy))^N}$ and $|C_2(\xx-\yy)|\le |{\sigma^h\over \g^h}|^2{\g^{h} C_N\over 1+(\g^h(\xx-\yy))^N}$. Moreover $$|g^h_{\o,-\o}(\xx-\yy)|\le |{\sigma^h\over \g^h}| {\g^{h} C_N\over 1+(\g^h(\xx-\yy))^N}$$} \vskip.5cm Note that $g^h_{L;\o}(\xx-\yy)$ coincides with the propagator "at scale $\g^h$" of the Luttinger model, see [BeGM]. This remark will be crucial for studying the Renormalization group flow, see sec.4. %\vskip1cm %{\rm Definition} Let be %$$\tilde C_h=\max_{i=0,1,2\atop k\ge h} |{\sigma_k\over\g^k}|^i$$ %The function $\tilde h\equiv \tilde h(C)$ \vskip.5cm \0{\bf 2.5} We {\it rescale} the fields so that % $${1 \over \NN_h}\int P_{Z_{h-1}}(d\psi^{(\le h-1)}) \int \tilde P_{Z_{h-1}}(d\psi^{(h)}) \, e^{-\hat\VV^{(h)} (\sqrt{Z_{h-1}}\psi^{(\le h)})}$$ so that $$\LL\hat\VV^{(h)}(\psi)= \g^h\nu_h F_\nu^{(\le h)}+\d_h F_\a^{(\le h)}+\iota_h F_\iota^{(\le h)}+\t_h F_\t^{(\le h)}+\l_h F_\l^{(\le h)} \; . \Eq(3.11ah) $$ where by definition $$\nu_h={Z_h\over Z_{h-1}}n_0;\quad \d_h={Z_h\over Z_{h-1}}(a_h-z_h);\quad \t_h={Z_h\over Z_{h-1}} t_h;\quad \iota_h= {Z_h\over Z_{h-1}}i_0;\quad \l_h=({Z_h\over Z_{h-1}})^2 l_h$$ We call $\vec v_h=(\nu_h,\d_h,\t_h,\iota_h,\l_h)$. We perform the integration $$ \int \tilde P_{Z_{h-1}}(d\psi^{(h)}) \, e^{-\hat\VV^{(h)} (\sqrt{Z_{h-1}}\psi^{(\le h)})} = e^{-\VV^{h-1}(\sqrt{Z_{h-1}}\psi^{(\le h-1)}) + \tilde E_h} \; , \Eq(3.20h) $$ % where $\tilde E_h$ is a suitable constant and % $$\LL \VV^{(h-1)}(\psi)= \g^{h-1} n_{h-1} F_\nu^{(h-1)}+s_{h-1} F_\sigma^{(h-1)} +a_{h-1} F_\a^{(\le h-1)}+$$ $$z_{h-1} F_\z^{(\le h-1)}+i_{h-1} F_\iota^{(\le h-1)}+\t_{h-1} F_\t^{(\le h-1)}+ l_{h-1} F_\l^{(\le -1)}\; , \Eq(3.21h) $$ % \vskip.5cm Note that the above procedure allows us to write the running coupling constants $\vec v_h$ in terms of $\vec v_k$, $k\ge h-1$ $$\vec v_h=\vec\b(\vec v_{h+1},...,\vec v_{0})\Eq(beta)$$ The function $\vec\b(\vec v_{h+1},...,\vec v_{0})$ is called {\it Beta function}. The {\it effective potential} $V^h(\psi)$ is a sum of terms of the form $${1\over (L\b)^m} \sum_{\kk'_1,...,\kk'_m \in {\cal D}_{L,\b}} \d(\sum_{i=1}^n \sigma_i(\kk'_i+\o_i p_F)+2mp) \hat W_{n,m}^{\le h}(\kk'_1,..,\kk'_n;\{\o\})\prod_{i=1}^n \psi^{\sigma_i (\le h)}_{\kk'_1+\o_i p_F,\o_i}\Eq(bbb)$$ where $\hat W_{n,m}^h(\kk'_1,..,\kk'_n;\{\o\})= W_{n,m}^h(\kk'_1+\o_1 p_F,..)$. %By the compact support properties of the $\psi^{\e_i \le h}_{\kk'_1+\o_i p_F,\o_i}$ The following lemmas will be useful in the future. \vskip.5cm \0{\bf 2.6} {\rm LEMMA} {\it Assume that $\sum_{i=1}^n \sigma_i \o_i p_F+2mp_L\not=0$ and $m\not=0$. Then \equ(bbb) is identically vanishing unless} $$|m|\geq C_1[\g^{-h\over\t}-n]\Eq(3.21hh)$$ \vskip.5cm {\rm PROOF} Remembering the compact support of the Grassmanian operator $\psi^{\sigma_i (\le h)}_{\kk'_i+\o_i p_F,\o_i}$ we can write, using \equ(1.19) $$a_0 n\g^h\geq |\sum_{i=1}^n \sigma_i\kk'_i|\geq |2mp_L+\sum_{i=1}^n\sigma_i \o_i p_F|_{T^1}\ge C_0( 2|m| p_L+n p_F)^{-\t}$$ from which \equ(3.21hh) holds. \qed \vskip.5cm \0{\bf 2.7} {\rm LEMMA} {\it Assume that $\sum_{i=1}^n \sigma_i \o_i p_F+2mp_L \not=0$ and $m=0$. Then \equ(bbb) is identically vanishing unless $h\geq \bar h$, if $\bar h$ is a suitable constant.} \vskip.5cm {\rm PROOF} Using again compact support of the Grassmanian operator we can write $$a_0 n\g^h\geq |\sum_{i=1}^n \sigma_i \o_i p_F|_{T^1}\geq 4p_F\qquad\qed$$ %\qed \vskip.5cm Let be $$h^*={\rm inf}\{h\ge h_\b:a_0\g^{h+1}\ge 4|\sigma_{h}|\}\Eq(h*)$$ >From Lemma 2.4 it follows trivially: \vskip1cm \0{\bf 2.8} {\rm LEMMA} {\it For $h> h^*$ for any integer $N>1$ it is possible to find a $C_N$ such that, for $|x-y|\le{L\over 2}$, $|x_0-y_0|\le{\b\over 2}$ $$|g^h_{\o,\o'}(\xx-\yy)|\le {C_N \g^h\over 1+(\g^h|\xx-\yy|)^N}$$} \vskip.5cm \0{\bf 2.9} In the next section we will see that, using the above lemmas and assuming that the running coupling constants are bounded, the integrations \equ(3.12a) are well defined for $0\ge h\ge h^*$. The integration of the scale from $h^*$ to $h_\b$ can be performed "in a single step" $$\int P_{Z_{h^*}}(d\psi^{(\le h^*)}) e^{-\VV^{h^*}(\sqrt{Z_{h^*}}\psi^{(\le h^*)}}= \int \tilde P_{Z_{h^*-1}}(d\psi^{(\le h^*)}) e^{-\tilde\VV^{h^*}(\sqrt{Z_{h^*}}\psi^{(\le h^*)}}\Eq(uu)$$ Calling $\int \tilde P_{Z_{h^*-1}}(d\psi^{(\le h^*)})\psi_\xx^{+(\le h^*)}\psi_\yy^{-(\le h^*)}= {g^{(\le h^*)}(\xx-\yy)\over Z_{h^*-1}}$, we prove in the next section that the integration in the r.h.s. in \equ(uu) is well defined. This will be proved by using the following lemma. \vskip.5cm \0{\bf 2.10}{\rm LEMMA} {\it Assume that $h^*$ is finite uniformly in $L,\b$ so that $|{\s_{h^*-1}\over\g^{h^*}}|\ge \k$, if $\k$ is a constant. Then it is possible to find a constant $C_N$ such that, for $|x-y|\le{L\over 2}$, $|x_0-y_0|\le{\b\over 2}$ $$|g^{\le h^*}_{\o,\o'}(\xx-\yy)|\le {C_N \g^{h^*}\over 1+(\g^{h^*}|\xx-\yy|)^N}$$} \vskip.5cm Comparing Lemma 2.8 and Lemma 2.10 we see that the propagator of the integration of all the scale between $h^*$ and $h_\b$ has the same bound as the propagator of the integration of a single scale greater than $h^*$. \vskip2.truecm \centerline{\titolo 3. Analiticity of the effective potential} \*\numsec=3\numfor=1 \0{\bf 3.1} We found convenient in order to discuss the analiticity of the effective potential $\VV^h$ to pass to the coordinate rapresentation. We use then the fields $$\psi^{(\le h)\sigma}_{\xx,\o}={1\over L\b}\sum_{k`\in {\cal D}_{L,\b}} e^{-i\sigma\kk'\xx} \psi^{(\le h)\sigma}_{\kk'+\o p_F,\o}$$ %Moreover we define: %$$W_{n,m}^{\le h}(\xx_1,...,\xx_n;\{\o\})\equiv \bar W_{n,m}^{\le %h}=$$ %$$\int \prod_{i=1}^n d\kk'_i %\prod_{i=1}^n e^{i\e_i\kk'_i\xx_i} %\d(\sum_{i=1}^n(\kk'_i+\o_i p_F)+2m p) \hat W_{n,m}^{\le h}(\kk'_1,...,\kk'_n;\{\o\})$$ %Note that if $\sum_{i=1}^n\o_i p_F+2m p=0$ then $\bar W_{n,m}^{\le h}$ is a %translation invariant function. and we can write the analogue of \equ(loc1) \equ(loc2) in coordinate space (we write the action of $\RR=1-\LL$): \vskip.5cm 1)if $n>4$ $$\RR \{\sum_{\xx_1,...,\xx_m\in\L} \prod_{i=1}^n\psi^{(\le h)\sigma_i}_{\xx_i,\o_i} W^{\le h}_{n,m}\}= \sum_{\xx_1,...,\xx_m\in\L} \prod_{i=1}^n\psi^{(\le h)\sigma_i}_{\xx_i,\o_i} W^{\le h}_{n,m}$$ %$$\equiv W^{\le h}_{n,m}(\xx_1-\xx_4,\xx_2-\xx_4,\xx_3-\xx_4)$$ \vskip.5cm 2) if $n=4$ and $\sum_{i=1}^4\s_i\omega_i p_F+2np_L\not=0$ one has $\RR=I$ otherwise %$W^{\le %h}_{4,m}(\xx_1,...,\xx_4,\{\o\})=W_{4,m}(\xx_1-\xx_4,...,\xx_3-\xx_4;\{\o\})$ %and %The $\RR$ operator, when its action is not trivial, %in the coordinate space can be written (simply %performing the Fourier transform). %In the case of eq(??), with $(\o_1+\o_2-\o_3-\o_4)=0$, $n=0$, and calling %$W_{4,0}(\xx_1-\xx_2,\xx_2-\xx_4,\xx_3-\xx_4;\{\o\})$ the Fourier transform of %$W_{4,0}(\kk'_1+\o_1 p_F,...,\kk'_4+\o_4 p_F)\d(\kk'_1+\kk'_2-\kk'_3-\kk'_4)$ %one obtains $$\RR \sum_{\xx_1,...,\xx_4\in\L}\prod_{i=1}^4\psi^{(\le h)\sigma_i}_{\xx_i,\o_i} W_{4,m}^{\le h}(\xx_1-\xx_4,\xx_2-\xx_4,\xx_3-\xx_4;\{\o\})= \; \Eq(za.1) $$ $$ \sum_{\xx_1,...,\xx_4\in\L}\prod_{i=1}^4\psi^{(\le h)\sigma_i}_{\xx_i,\o_i} W_{4,m}^{\le h}(\xx_1-\xx_4,\xx_2-\xx_4,\xx_3-\xx_4;\{\o\})-$$ $$-\d(x_1-x_2)\d(x_2-x_3)\d(x_3-x_4) \sum_{{\bf t}_1,{\bf t}_2,{\bf t}_3\in\L} W_{4,m}^{\le h}({\bf t}_1,{\bf t}_2,{\bf t}_3;\{\o\})\}$$ where we have used that if $\sum_{i=1}^4\s_i\omega_i p_F+2np_L=0$ the kernels $W^{\le h}_{4,m}$ are {\it traslation invariant}. %(\xx_1,...,\xx_4,\{\o\})=W_{4,m}(\xx_1-\xx_4,...,\xx_3-\xx_4;\{\o\})$ %and \vskip.5cm 3)in $n=2$ and $\sum_{i=1}^2\s_i\omega_i p_F+2np_L\not=0$ then $\RR=I$ otherwise $$\RR \{\sum_{\xx_1,\xx_2\in\L}\prod_{i=1}^2\psi^{(\le h)\sigma_i}_{\xx_i,\o_i} W_{2,m}^{\le h}(\xx_1-\xx_2;\{\o\})\}=\sum_{\xx_1,\xx_2\in\L} \prod_{i=1}^2\psi^{(\le h)\sigma_i}_{\xx_i,\o_i} [W^{\le h}_{2,0}(\xx_1-\xx_2;\{\o\})-$$ $$\d(\xx_1-\xx_2) \sum_{{\bf t}\in\L} W_{2,m}^{\le h}({\bf t};\{\o\})+\; \Eq(za.2)$$ $$\partial_{x_2^0} \d(\xx_1-\xx_2) \sum_{{\bf t}\in\L} t_0 W_{2,m}^{\le h}({\bf t};\{\o\})+ \partial_{x_2}\d(\xx_1-\xx_2) \sum_{{\bf t}\in\L} \o t W_{2,m}^{\le h}({\bf t};\{\o\})]+$$ $$\sum_{\xx_1\in\L}\psi^{(\le h)+}_{\xx,\o_1}\D\psi^{(\le h)-}_{\xx,\o_2} \sum_{{\bf t}\in\L} \o t W_{2,m}^{\le h}({\bf t};\{\o\})]$$ and again we have used that when the $\RR$ operation acts the kernels are traslation invariant; moreover if $\o E(k'+\o p_F)=2 p_F k'+\bar f(k')$ we have that $\D\psi^{(\le h)\sigma}_{\xx,\o}= {1\over L\b}\sum_{k`\in {\cal D}_{L,\b}} e^{-i\sigma\kk'\xx} \bar f(k') \psi^{(\le h)\sigma}_{\kk'+\o p_F,\o}$. %moreover $\bar \partial_{x_2}\d(\xx_1-\xx_2)$ is the distribution %$-\sum_{\kk'}\o E(k'+\o p_F)e^{i \kk'(\xx_1-\xx_2)}$. It is convenient to write %$\o E(k'+\o p_F)=2 p_F k+\bar f(k)$ with $|\bar f(k)|\le C k^2$. Then %$\bar \partial_{x_2}\d(\xx_1-\xx_2)=\partial_{x_2}\d(\xx_1-\xx_2)+\d^2\d(\xx_1-\xx_2)$. %and, in the distribution sense, $(x_1-x_2)^a \d^2\d(x_1-x_2)=0$ if $a>2$ \vskip.5cm \0{\bf 3.2} The following identities will be useful in the following \vskip.5cm $$(\xx_i-\xx_j)^a\RR\{\sum_{\xx_1,...,\xx_4\in\L} \prod_{i=1}^4\psi ^{(\le h)\sigma_i}_{\xx_i,\o_i} (W_{4,0}^{\le h}(\xx_1-\xx_4,\xx_2-\xx_4,\xx_3-\xx_4;\{\o\})\}=\Eq(za.3)$$ $$(\xx_i-\xx_j)^a\int \sum_{\xx_1,...,\xx_4\in\L} \prod_{i=1}^4\psi^{(\le h)\sigma_i}_{\xx_i,\o_i} (W_{4,0}^{\le h}(\xx_1-\xx_4,\xx_2-\xx_4,\xx_3-\xx_4;\{\o\})$$ %The following trivial identity are valid in the distribution sense: %$$(x_i-x_j)^a \RR(W_{4,0}(\xx_1-\xx_2,\xx_2-\xx_4,\xx_3-\xx_4;\{\o\})= %(x_i-x_j)^a W_{4,0}(\xx_1-\xx_2,\xx_2-\xx_4,\xx_3-\xx_4;\{\o\}) \; , %\Eq(za.3)$$ if $a$ is any integer $\geq 1$ and $i,j$ can take any value between $1$ and $4$. \vskip.5cm $$(\xx_1-\xx_2)^a \RR\{ \int \sum_{\xx_1,\xx_2\in\L} \prod_{i=1}^2\psi^{(\le h)\sigma_i}_{\xx_i,\o_i} W_{2,0}^{\le h}(\xx_1-\xx_2;\{\o\})\}=\; , \Eq(za.4)$$ $$\sum_{\xx_1,\xx_2\in\L}\prod_{i=1}^2\psi^{(\le h)\sigma_i}_{\xx_i,\o_i}[(\xx_1-\xx_2)^a W_{2,0}^h(\xx_1-\xx_2;\{\o\})]+$$ $$(\xx_1-\xx_2)^a \psi^{(\le h)+}_{\xx,\o_1}\D\psi^{(\le h)-}_{\xx,\o_2} \sum_{{\bf t}\in\L}\o t W_{2,0}^{\le h}({\bf t};\{\o\})$$ if $a$ is any integer $\geq 2$. Finally \vskip.5cm $$(x_1-x_2) \RR \{\sum_{\xx_1,\xx_2\in\L} \prod_{i=1}^2\psi^{(\le h)\sigma_i}_{\xx_i,\o_i} W_{2,0}^{\le h}(\xx_1-\xx_2;\{\o\})\}=\; \Eq(za.5)$$ $$\sum_{\xx_1,\xx_2\in\L}\prod_{i=1}^2\psi^{(\le h)\sigma_i}_{\xx_i,\o_i}[(x_1-x_2) W_{2,0}^h(\xx_1-\xx_2;\{\o\})-\d(\xx_1-\xx_2)\sum_{{\bf t}\in\L} t W^{\le h} ({\bf t};\{\o\})]$$ $$-(x_1-x_2)\psi^{(\le h)+}_{\xx,\o_1}\D\psi^{(\le h)-}_{\xx,\o_2} \sum_{{\bf t}\in\L} \o t W_{2,0}^h({\bf t};\{\o\})$$ and a similar equation if $(x_1-x_2)$ is replaced by $(x_1^0-x_2^0)$. \vskip.5cm %W_{2,0}(\xx_1-\xx_2;\{\o\})=(\xx_1-\xx_2)^a W_{2,0}(\xx_1-\xx_2;\{\o\}) %\; , \Eq(z.4)$$ %if $a$ is any integer value $\geq 2$; and %$$(x_1-x_2)\RR W_{2,0}(\xx_1-\xx_2;\{\o\})=(x_1-x_2) %W_{2,0}(\xx_1-\xx_2;\{\o\})-\d(\xx_1-\xx_2) %\int d\vec t t W_{2,0}(\vec t;\{\o\}) \; , \Eq(z.5)$$ %$$(x_1^0-x_2^0)\RR W_{2,0}(\xx_1-\xx_2;\{\o\})=(x_1-x_2) %W_{2,0}(\xx_1-\xx_2;\{\o\})-\d(\xx_1-\xx_2)\int d\vec t t^0 %W_{2,0}(\vec t;\{\o\}) \; , \Eq(z.6)$$ Of course one can integrate the $\d$ functions in \equ(za.1) \equ(za.2) so obtaining $$\sum_{\xx_1,...,\xx_4\in\L} \prod_{i=1}^4\psi^{(\le h)\sigma_i}_{\xx_1,\o_i} \RR(W_{4,m}^{\le h}(\xx_1-\xx_4,\xx_2-\xx_4,\xx_3-\xx_4;\{\o\})=\; , \Eq(z.7)$$ $$\sum_{\xx_1,...,\xx_4\in\L}W_{4,m}^{\le h}(\xx_1-\xx_2,\xx_2-\xx_4,\xx_3-\xx_4;\{\o\})$$ $$[\psi^{(\le h)\sigma_1}_{\xx_1,\o_1}\psi^{(\le h)\sigma_2}_{\xx_2,\o_2}\psi^{(\le h)\sigma_3}_{\xx_3,\o_3}\psi^{(\le h)\sigma_4}_{\xx_4,\o_4}-\psi^{(\le h)\sigma_1}_{\xx_1,\o_1}\psi^{(\le h)\sigma_2}_{\xx_1,\o_2}\psi^{(\le h)\sigma_3}_{\xx_1,\o_3}\psi^{(\le h)\sigma_4}_{\xx_1,\o_4}]$$ The square brakets in the above equation can be written as $$\psi^{(\le h)\sigma_1}_{\xx_1,\o_1} D^{(\le h)\sigma_2}_{\xx_{2,1},\o_2}\psi^{(\le h)\sigma_2}_{\xx_3,\o_3}\psi^{(\le h)\sigma_4}_{\xx_4,\o_4}$$ $$+\psi^{(\le h)\sigma_1}_{\xx_1,\o_1}\psi^{(\le h)\sigma_2}_{\xx_1,\o_2} D^{(\le h)\sigma_3}_{\xx_{3,1},\o_3}\psi^{(\le h)\sigma_4}_{\xx_4,\o_4}+ \psi^{(\le h)\sigma_1}_{\xx_1,\o_1}\psi^{(\le h)\sigma_2}_{\xx_1,\o_2}\psi^{(\le h)\sigma_3}_{\xx_1,\o_3} D^{(\le h)\sigma_4}_{\xx_{1,4},\o_4}$$ so that the effect of $\RR$ is essentially to change the coordinate of the $\psi^{(\le h)\sigma_i}_{\xx_i,\o}$ fields and to replace a field $\psi^{(\le h)\sigma_i}_{\xx_i,\o}$ with $D^{(\le h)\sigma_i}_{\xx_{ij}}$ where $$D_{\xx_{ij}}^{(\le h)\sigma_i}=\psi^{(\le h)\sigma_i}_{\xx_i,\o_i}-\psi^{(\le h)\sigma_i}_{\xx_j,\o_i}= (\xx_i-\xx_j)\int_0^1 du\vec\partial \psi^{(\le h)\sigma_i}_{\xx_{ji}(u),\o_i}\; , \Eq(za.8)$$ with $\xx_{ji}(u)=u \xx_j+(1-u) \xx_i$ and $\vec\partial=(\partial_x,\partial_{x_0})$. %We will see that in the estimates the presence of a field %$D_{x_{ij}}^\e$ instead of a field $\psi^e$ produces a gain. In the same time integrating the $\d$'s in \equ(za.2) one can see that the effect of $\RR$ is to change the coordinate of the $\psi_{x_i,\o}^{(\le h)\sigma}$ fields and to replace a field $\psi{(\le h)\sigma}$ with a field $A^{(\le h)\sigma}$ wich can be written as: $$(\xx_2-\xx_1)^2\int_0^1 dt_1 \int_{0}^{t_1} dt_2\partial^2\psi{(\le h)\sigma}_{x_{21}(t_2)}+(x_2-x_1)\D\psi_{x_2}^{(\le h)\sigma}\; , \Eq(za.9)$$ %with $\D\psi_{x_2}{(\le %h)\sigma}=\o\int d\kk \bar f(k) e^{-i \kk' x} \psi_{\kk'+\o %p_F}{(\le %h)\sigma}$. Note the two different ways to write the $\RR$ operation: in \equ(za.1),\equ(za.2) the $\RR$ acts on the kernels of the effective potential \ie it consists in replacing $W^h_{n,m}$ with $\RR W^h_{n,m}$ (the definition of $\RR W^h_{n,m}$ is evident from \equ(z.1),\equ(z.2)); on the other hand writing as in \equ(z.7) the $\RR$ operation act on the $\psi$ fields, leaving the kernel untouched. \vskip.5cm \0{\bf 3.3} From \equ(3.20h) we have that, if $E^T_{h+1}$ denotes the {\it truncated expectation} with propagator \equ(3.13a): $$\VV^h(\sqrt{Z_h}\psi^{\le h})=\sum_{n=1}^{\io}{1\over n!} (-1)^{n+1}E^T_{h+1}(\hat \VV^{h+1} (\sqrt{Z_h}\psi^{\le h+1},...)$$ where $\hat \VV^{h+1}$ is obtained from $\VV^{h+1}$ following the operations described in sec.2. %$$\hat V^h(\sqrt{Z_{h-1}} %\psi^{\le h})=\LL \hat V^h+\RR \hat V^h=\LL^* V^h(\sqrt{Z_{h-1}}\sqrt{Z_h\over Z_{h-1}} \psi^{\le h}) %+\RR V^h(\sqrt{Z_{h-1}} \sqrt{Z_h\over Z_{h-1}} \psi^{\le h})\Eq(z.99)$$ %where $\LL^* V^h$ differs from $\LL V^h$ only becouse it does not contain anymore %the terms $F_\s$ and $F_\d$. Iterating the above equation we obtain that the effective potential can be written in term of a {\it tree expansion}. $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ A tree $\t$ consists of a family of lines arranged to connect a partially ordered set of points ({\it vertices}). Every line has two vertices at the extreme except the first one which has only one vertex, the first vertex $v_0$. The other extreme $r$ is the {\it root} and it is not a vertex. From each vertex $v$ start $s_v$ lines; if $s_v=0$ the vertices is called {\it end-point}, if $s_v=1$ is called {\it trivial vertex} and if $s_v>1$ is called {\it non trivial vertex}. To each vertex $v$ we associate a {\it scale} $h_v$. The vertices of $\t$ are naturally (partially) ordered. Giving to the tree an orientation from left to right we write $v_1k}|\vec v_h|\le\bar\e_k$ and $\sup_{h>k}|{Z_h\over Z_{h-1}}|\le e^{c_1\bar\e_k^2}$, chosen $\g^{1\over\t}/2\ge 1$, then $$|\int d\xx^{P_{v_0}}\sum_{\t\in\t_{n,k}}|W^k(\t,P_{v_0},\xx^{P_{v_0}})| (1+\g^k d(P_{v_0})^N\le |\L| \g^{-kD(P_{v_0})}(C_N\bar\e_k)^N| e^{-\g^{-k/\t} c_2 s_{P_v}\over 2^{-k+3}}$$ where $s_{P_v}=1$ if $N_v\not=0$, $|P_v|=2$ and $0$ otherwise, $N$ is a positive integer, $x^{P_{v^0}}$ is the set of points associated to $P_{v_0}$, $c_1,c_2,C_N$ are constants (not depending on $L,\b$), $d(P_{v_0})$ is the length of the shortest tree connecting the set of points $\xx^{P_{v_0}}$, $|\L|=L\b$ and $$D(P_{v_0})=-2+\sum_{f\in P_{v_0}}(1/2+m_f)$$ where $m_f$ is the order of the derivative applied to the fields of label $f$, see \equ(3.13a)} \vskip.5cm \0{\bf 3.5} {\cs Proof} In order to write in an explicit form \equ(z11) it is convenient to start studying $\RR \hat V^h$: $$\RR \hat V^k(\t,\sqrt{Z_{k-1}}\psi^{(\le k)})={1\over s_{v_0}!}\sum_{P_{v_0}}(\sqrt{Z_{k-1}})^{|P_{v_0}|} \RR\{\tilde\psi^{(\le k)}(P_{v_0}) \sum_{ P_{v_0^1},..,P_{v_{0^s_{v_0}}} } \sqrt{Z_{k}\over Z_{k-1}}^{|P_{v_0}|}$$ $$\sum_{ Q_{v_0^1},..,Q_{v_0^{s_{v_0}}} } \E_{k+1}^T[\int d\xx_{v_0^1} \tilde\psi^{(k+1)}(P_{v_0^1}/Q_{v_0^1})\bar W^{k+1}(\t,P_{v_0^1},\xx_{v_0^1}),...,$$ $$\sum_{x_{v_0^{s_{v_0}}}\in\L} \tilde\psi^{(k+1)}(P_{v_0^{s_{v_0}}}/Q_{v_0^{s_{v_0}}})\bar W^{k+1}(\t,P_{v_0^{s_{v_0}}},\xx_{v_0^{s_{v_0}}})]\Eq(bab)$$ >From \equ(z.7)-\equ(za.9) we know that the effect of $\RR$ is simply to replace $\tilde\psi^{(\le k)}(P_{v_0})$ with $$\sum^*_{i,j,b}(\xx_i-\xx_j)_b^{a_{P_{v_0}}} \hat\psi^{\le k}(P_{v_0})\Eq(bab1)$$ where $b=0,1$ is an index distinguishing the space ot time component of $\xx$ (in the following this index and $\sum^*_{i,j,b}$ will be omitted), $\xx_i,\xx_j$ are two coordinates in $\xx^{P_{v_0}}$ and \vskip.5cm 1)$a_{P_{v_0}}=0$ and $\hat\psi^k(P_{v_0})=\tilde\psi^k(P_{v_0})$ if $|P_{v_0}|>4$ or the field index do not verify verify the Kronecker $\d$ in \equ(loc1), \equ(loc2) \vskip.5cm 2)if $|P_{v_0}|=4$ and the field index verifies the Kronecker $\d$ in \equ(loc1) then $a_{P_{v_0}}=1$ and $\hat\psi^k(P_{v_0})$ differs from $\tilde\psi^k(P_{v_0})$ becouse a field $\int_0^1 dt \partial\psi^\e_{x_{ij}(t)}$ replaces a $\psi$ field, and the remaning $\psi$ fields are applied on different coordinates among $\xx^{P_{v_0}}$, see \equ(z.7); \vskip.5cm 3)if $|P_{v_0}|=2$ and the field index verify the Kronecker $\d$ in \equ(loc2), $a=2$ and $\hat\psi^k(P_{v_0})$ differs from $\tilde\psi^k(P_{v_0})$ becouse a field $\int_0^1 dt_1 \int_0^{t_1} dt_2 \partial^2\psi_{x_{ij}(t_2)}$ or a ${\D\psi\over (x_1-x_2)}$ replaces a $\psi$ field, and the remaning $\psi$ fields are applied on different coordinates among $\xx^{P_{v_0}}$. %$\x_i,\xx_j$ are two coordinates among %$\cup_{f\in P_{v_0}}\xx(f)$ \vskip.5cm Let we call: $$\hat\psi(P)=\prod_{f\in P}\partial^{q(f)}_{\xx(f)}\psi_{\xx(f)}^{\e(f)}\Eq(bab7)$$ where $q=0,1,2,3$, $\partial^0=1$, $\partial^1=\partial_\xx$, $\partial^2_\xx=\partial_\xx \partial_\xx$, $\partial^3={1\over (x_1-x_2)}\D$. We remember the well known expansion of truncated expectation in term of interpolating parameters $s_t$, $t=1,..,k-1$: $$\E^T_{h}(\hat\psi(P_1),...,\hat\psi(P_k))=\sum_{T}\prod_{l\in T} \partial^{q_l}_{\xx_l} \partial^{q'_l}_{\yy_l} g^h(\xx_l-\yy_l)\int dP_{T}(s) det G^T(s)\Eq(bab2)$$ where $T$ is a set of lines forming an {\it anchored tree graph} between the cluster of vertices from which the fields labeled with $P_1,..,P_k$ emerge: this means that $T$ is a set of lines connecting two points in different clusters, which becomes a tree graph if one identifies all the points in the same cluster; if $l\in T$ $\xx_l,\yy_l$ are the end-points of the line and are such that $\xx_l=\xx_{ij}$ or $\yy_l=\xx_{i'j'}$, where $\xx_{ij}$ denotes the coordinate of the $i$-th field of the monomial $\tilde\psi(P_j)$. If $f$ is a field variable such that $\xx(f)\equiv \xx_l\equiv\xx_{ij} $, we write $q(f)\equiv q_l\equiv q_{ij}$. In the same way if $\bar f$ is such that $\xx(\bar f)\equiv \yy_l\equiv\xx_{i'j'} $ we say $q(\bar f)\equiv q'_l$. $G^T(s)$ is a $(n-k+1) \times (n-k+1)$ matrix (if $n$ is the total number of fields), whose elements are $G^T_{jij'i'}=S_{jj'} \partial^{q_{ij}}\partial^{q_{i'j'}} g^h(x_{ij}-x_{i'j'})$ with $x_{ij}-x_{i'j'}$ non belonging to $T$, $S_{jj'}=\prod_{i=j}^{j'-1} s_t$ and $dP_T(s)$ is a normalized measure which depends on $s_t$ and $T$ (for the well known explicit formula of $dP_T$ and for its derivation, one can look for istance [Le],[BGPS]). %Insterting this expression in \equ(bab) we obtain, with the above notations: We write in an explicit way the action of $\RR$ in \equ(bab) obtaining $$\RR V^k(\t,\sqrt{Z_{k-1}}\psi^{(\le k)})={1\over s_{v_0}!}\sum_{P_{v_0}} \sqrt{Z_{k-1}}^{|P_{v_0}|} \hat\psi^{(\le k)}(P_{v_0}) \sum_{ P_{v_0^1},..,P_{v_0^{s_{v_0}}}} \sqrt{Z_{k}\over Z_{k-1}}^{|P_{v_0}|}$$ $$\sum_{ Q_{v_0^1},..,Q_{v_0^{s_{v_0}}}} \sum_{\xx_{v_0}\in\L} [(\xx_i-\yy_j)^{a(P_{v_0})} \E^T_{k+1}(\tilde\psi^{k+1}(P_{v_0^1}/Q_{v_0^1}),..., \tilde\psi^{k+1}(P_{v_0^{s_{v_0}}}/Q_{v_0^{s_{v_0}}}) ) ]\Eq(bab3)$$ $$\bar W^{k+1}(\t_1,P_{v_0^1},\xx_{v_0^1})...\bar W^{k+1}(\t_{s_{v_0}},P_{v_0^{s_{v_0}}},\xx_{v_0^{s_{v_0}}})]$$ %if $x_i,y_j$ are two points among $x^{P_{v_0}}$ and $\sum_{\a,\b}^*$ %is present only if $a(P_{v_0})\not=0$. We can write, for any anchored tree graph $T_{v_0}$ connecting the clusters $L_{v_0^1}...L_{v_0^{s_{v_0}}}$: $$(\xx_i-\yy_j)=\sum_{l\in T}(\xx_{l}-\yy_{l})+\sum^*_{iji'j'}(\xx_{ij}-\yy_{i'j'})\Eq(bab7)$$ where $\sum^*_{iji'j'}$ is a sum such that there is no $l\in T_{v_0}$ such that $\xx_{ij}=\xx_l,\yy_{i'j'}=\yy_l$. % ore not the end-points of a %line $l\in T_{v_0}$. So we can write: $$\sum^*_{ij}(\xx_{ij}-\yy_{ij})^{a(P_{v_0})}\E^T_{k+1}(\tilde\psi^{k+1}(P_{v_0^1}/Q_{v_0^1}),... ) ]\bar W^{k+1}(\t_1,P_{v_0^1},\xx_{v_0^1})...\bar W^{k+1}(\t_{s_{v_0}},P_{v_0^{s_{v_0}}},\xx_{v_0^{s_{v_0}}})]=$$ $$[\sum_{T_{v_0}}\sum^*_{\{a\}}\{\prod_{l\in T_{v_0}} (\xx_l-\yy_l)^{a_l} g^{k+1}(\xx_l-\yy_l)] [\int dP_{T}(s) det G^T(s)]$$ $$|\xx^{P_{v_0^1}}|^{a_{v_0^1}}\bar W^{k+1}(\t_1,P_{v_0^1},\xx_{v_0^1})...|\xx_{v_0^{s_{v_0}}}|^{a_{v_0^{s_{v_0}}}}\bar W^{k+1}(\t_{s_{v_0}},P_{v_0^{s_{v_0}}},\xx_{v_0^{s_{v_0}}})]\Eq(zzz)$$ %Replacing this formula in eq.(!!) we obtain %$\RR V^k(\t,\sqrt{Z_k}\psi^{\le k})={1\over s_{v_0}!}\sum_{P_{v_0}}\sqrt{Z_{k}}^{|P_{v_0}|} %\hat\psi^k(P_{v_0}) \sum_{ P_{v_0^1},..,P_{v_0^s_{v_0}} } %\sqrt{Z_{k+1}\over Z_k}^{|P_{v_0}|}\sum_{ Q_{v_0^1},..,Q_{v_0^s_{v_0}} } \int d\xx_{v_0}$$ %$$\sum_{T}\prod_{l\in T} \sum^* %|x_l-y_l|*{a_l}g^{k+1}(x_l-y_l)\int dP_{T}(s) det G^T(s) |\xx^{P_{v_0^1}}|^{a_{v_0^1}}\bar %W(\t,P_{v_0^1},\xx^{P_{v_0^1}})...|\xx^{P_{v_0^{s_{v_0}}}|^{a_{v_0^{s_{v_0}}}}\bar %W(\t,P_{v_0^{s_{v_0}}},\xx^{P_{v_0^{s_{v_0}}}})]$$ where $|\xx^{P_{v_0^i}}|$ is the difference among two coordinates in $\xx^{P_{v_0^i}}$, and $\sum^*_{\{a\}}$ is the sum over the indices $a_l+a_{v_0^1}+..$ with the constraint that $a_l+a_{v_0^1}+...+ a_{v_0^{s_{v_0}}}=a(P_{v_0})$. We are now in position to iterate the above procedure studying each $|\xx^{P_{v_0^i}}|^{a_{v_0^i}}\bar W^{k+1}(\t_i,P_{v_0^i},\xx^{P_{v_0^i}})$. We can distinguish several cases \vskip.5cm 1)if $\t^i$ is a trivial tree, $a_{v_0^i}=0$ and $ |\xx^{P_{v}}|^{a_v}\bar W^{k+1}=\vec v_{k+1}$; %and the iterative procedure stops. \vskip.5cm 2)if $\t^i$ is not a trivial tree, there are the following possibilityes: \vskip.5cm 2a)$a_{v_0^i}=0$. In this case we repeat word by word the analysis for $W^k$ \equ(bab),\equ(bab3), with the trivial substitution $k\to k+1$, $v_0\to v_0^i$. Note that if $\RR\not=0$ the effect of the renormalization is of replacing in the truncated expectation in \equ(bab3) $\tilde\psi(P_{v_0^i}/Q_{v_0^i} )$ with $\hat\psi(P_{v_0^i}/Q_{v_0^i})$, defined in an analogous way as $\hat\psi(P_{v_0})$ \equ(bab1), and to produce a factor $(\xx_i-\xx_j)^{a_{P_{v_0^i}}}$, $ij\in P_{v_0^i}$ which is written as in \equ(bab7). %, differing from $\tilde\psi(P_{v_0^i})$ because, if $\RR$ %is given by eq.() one field $\psi^\e$ is replaced with $\int dt_1 %\partial\psi^\e_{x_{ij}(t_1)}$ and if $\RR$ %is given by eq.() one field $\psi^\e$ is replaced with $\int_0^1 dt_1\int_0^{t_1} dt_2 %\partial^2\psi^\e_{x_{ij}(t_1)}$. \vskip.5cm 2b)$a_{v_0^i}\not =0$. In this case we have to use \equ(za.3) \equ(za.4) \equ(za.5) and we do as in \equ(bab),\equ(bab3). Note that the propagators belonging $T_{v_0^i}$ are of the form $(\xx_l-\xx_l)^a g(\xx_l-\xx_l)$ with $a<2$. %write the analogue of eq.(). \vskip1cm Iterating this procedure we find at the end %$$\sum^* \sum_T\{\prod_{l\in T_{v_0^i}} (\xx_l-\yy_l)^{a_l} g^{k+1}(\xx_l-\yy_l) %[\int dP_{T}(s) det G^T(s)] %|\xx^{P_{v_0^1}}|^{a_{v_0^1}}\bar %W(\t,P_{v_0^1},\xx^{P_{v_0^1}})...|\xx^{P_{v_0^{s_{v_0}}}|^{a_{v_0^ %\vskip.5cm %2b)$a_{v_0^i}\not 0$. In this case we use eq %We can iterate this procedure, by noting that: %\vskip.5cm %1)in the case in which $\bar %W(\t,P_{v_0^1},\xx^{P_{v_0^1}})=\RR W(\t,P_{v_0^1},\xx^{P_{v_0^1}})$ %we can use eq(!!). %\vskip.5cm %2)the effect of the renormalization of the subtrees $\t_1,..,\t_{s_{v_0}}$ by %using eq.(!!) has the effect that $x_{ij},x_{i'j'}$ in the truncated extepctation %$\Epsilon^T_{k+1}$ can be possibly replaced by eq(!!). %\vskip.5cm %Iterating the above procedure and taking into account the above remarks we can write $$\hat V^k(\t,\sqrt{Z_{k-1}}\psi^{\le k})= \sum_{\xx_{v^0}\in\L} \sum_{P_{v^0}} \sqrt{Z_{k-1}}^{|P_{v_0}|}\tilde\psi^{\le k}(P_{v_0}) %\sum_{P_{v^0}} \sqrt{Z_{k-1}}^{|P_{v_0}|}\tilde\psi^{\le k}(P_{v_0}) \sum_{\{ P_{v} \}}\{\int dt\} \prod_{v not e.p.} |{Z_{h_v-1}\over Z_{h_v-2}}|^{|P_v|/2}{1\over s_v!}$$ $$\sum_{\{q\}}^* [\prod_{v not e.p.}{1\over s_v!} \tilde\E^T_{h_v}(\hat\psi(P_{v^1}/Q_{v^1}),... ,\hat\psi(P_{v^{s_v}}/Q_{v^{s_v}})] [\prod_{i\in V_1} (\vec v_{h_i})_a e^{2 i p_F x_i \g_a}\g^{h_i\d_a}][\prod_{i\in V_2} \hat\ph_{n_i} e^{in_i p\xx_i}]\Eq(pal)$$ where: \vskip.5cm 1)The symbol $\sum_{\{ P_{v} \}}$ denotes the sum over all the compatible choices of the sets $P_v$ on all the vertex of $\t$ except $v_0$; moreover $Q_v\subset P_v$, $P_v=\bigcup_i Q_{v^i}$. \vskip.5cm 2)$\hat\psi(P_{v}/Q_{v})=\prod_{f\in P_{v}/Q_{v}}\partial^{q_f}\psi^{\e(f)}_{\xx_{ij}(t)}$ where $q_f=0,1,2,3$ with the same conventions as in \equ(bab7). %if $|P_{v}|=2,4$ \ie if the renormalization act on the cluster $v$. Moreover $\xx_{ij}(t)=\sum_{i'}\e_{i'j}(t) \xx_{i'j}$ with $\sum_{i'}\e_{i'j(t)}=1$ and $\e_{i'j}(0)=\d_{i,i'}$. Finally $$\tilde\E^T_{h}(\hat\psi(P_1),...,\hat\psi(P_k))=\sum_{T_v}\sum^*_{\{a\}} [\prod_{l\in T_v} \partial^{q_l}_{\xx_l} \partial^{q'_l}_{\yy_l} (\xx_l-\yy_l)^{a_l}g^h(\xx_l-\yy_l)] \int dP_{T}(s) det {G}^T(s)$$ %where $T$ is a set of lines forming %an {\it anchored tree graph} between the cluster of vertices %from which the fileds labeled with $P_1,..,P_k$ %emerge: this means that $T$ is a set of lines connecting two points %in different clusters, which becomes a tree graph if one identifies %all the points in the same cluster; %if $l\in T$ $x_l,y_l$ are the end-points of the line and are %such that $x_l=x_{ij}$ or $y_l=x_{i'j'}$, where $x_{ij}$ denotes the coordinate of the $i$-th field %of the monomial $\tilde\psi(P_j)$. If $y_l=x_{i'j'}=x(\bar f)$ then %with $a_l\le 2$, %$a_l\not=0$ only if $l$ it is contained on a cluster in which the %renormalization act non trivially. Moreover $\sum_{l\in T}a_l\le 2$. where $\sum^*_{\{q \}}$ and $\sum^*_{\{a \}}$ have the following constraints. If $v$ is such that $|P_v|=4$ and the Kronecker $\d$ of \equ(loc1) is verified %\vskip.5cm %3)$\sum_{z}^*$ is a sum over the derivatives choices in the monomial $\hat\psi$ %and $\sum_{a}^*$ over the order of zero. The constraints on these sums are that %if $L_v$ is a cluster on which $\RR$ act as in eq(!!) then $\hat\psi(P_v)$ contains a field $\partial\psi_{x'_{ij}}$ and there is a line $\bar l\in T_{\bar v}$, $\bar v>v$, with $a_{\bar l}=1$ and for any $l\in T_{\hat v}$, $l\not=\bar l$, $\bar v\ge \hat v>v$ $a_l=0$. If $v$ is such that $|P_v|=2$ and the Kronecker $\d$ of \equ(loc2) is verified then $\hat\psi(P_v)$ contains a field $\partial^2\psi_{\xx_{ij}}$ or $\D\psi$ and there are two (possibly coinciding) lines $l_1\in T_{\bar v_1}$, $l_2\in T_{\bar v_2}$ with $\bar v_1>v$, $\bar v_2>v$ with $a_l=1$ and for any $l\in T_{\hat v}$, $l\not=l_1,l_2$ $\bar v_1\ge \hat v>v$, $\bar v_2\ge \hat v>v$ one has $a_l=0$. %in a %cluster $L_{\bar v}$ contained in $L_v$ such that $a_l=1$; morover for any $l\in L_{v_i}$, %with $L_{v_i}$ containing $_{\bar v}$ and contained in $L_v$ $a_l=0$. If %$L_v$ is a cluster on which $\RR$ act as in eq(!!) then $\hat\psi(P_v)$ %contains a field $\partial^2\psi_{x'_{ij}}$ and there are two lines $l$ %(possibly coingiding) %in a %cluster $L_{\bar v_1}$ and $L_{\bar v_1}$ %contained in $v$ such that $a_l=1$; morover for any $l\in L_{v_i}$, %with $L_{v_i}$ containing $L_{\bar v_1}$ or $L_{\bar v_2}$ %and contained in $L_{v}$ $a_l=0$. If $v$ is a cluster on which $\RR=0$ %then $\hat\psi(P_v)=\tilde\psi(P_v)$. %\vskip.5cm %3a)if $i\in P_v$ $z_i=0$ if the renormalization acts trivially on $v$; if it act as in %eq() then $z_i=1$ and if t act as in %eq() then $z_i=2$. %\vskip.5cm %3b) $\sum_{l\in T}a_l\le 2$ and $a_l\not=0$ only if $l$ %it is contained on a cluster in which the %renormalization act non trivially. If $l$ is contained in a cluster $\bar v$ %and $\hat v$ is the first cluster containing \vskip.5cm 4)$\{\int dt\}$ is a product of integral over the interpolation parameters. \vskip.5cm 5) $V_1$ is the set of the end points of the tree, and $V_2$ is the set of end points {\it not} associated to running coupling constants. Moreover $\g_a=\pm 1$ in correspondence of end-points of kind $\t$ or $\iota$ and zero otherwise, and $\d_a=1$ in correspondence of end-points of kind $\nu$ and zero otherwise. %associated to $\t$ to which is associated a term of the relevant %part of the effective potential, while $V_2\cup V_1$ is the set of all the end points. %$[\prod_i \sum_{n_i}\hat\ph_{n_1} %e^{inp\xx}$ is a product over the end points associated with scale %$0$ to which is associated $\RR V^0$. \vskip1cm Summing \equ(pal) over the index $n_i$ \ie summing over all the trees differing only for the choices of the addend $\RR V^0$ associated to a given end-point(\ie summing over trees with the same labels and differing only for the choices of $n_i$), we find %Remembering the definition of $\hat \t$ we have that $$ \int d\xx^{v_0} \sum_{P_{v^0}} \sqrt{Z_{k-1}}^{|P_{v_0}|}\tilde\psi^{\le k}(P_{v_0}) \sum_{\{ P_{v} \}}\{\int dt\}[\prod_{i\in V_2}\sum_{n_i}] %[\prod_{i\in V_1} v_{h_i}][\prod_{i\in V_2}\sum_{n_i} %\hat\ph_{n_i} %e^{in_ip\xx_i}] \prod_{v not e.p.} |{Z_{h_v-1}\over Z_{h_v-2}}|^{|P_v|/2}{1\over s_v!} \sum_{\{q \}}^*$$ $$\tilde\E^T_{h_v}(\hat\psi(P_{v^1}/Q_{v^1}),... ,\hat\psi(P_{v^{s_v}}/Q_{v^{s_v}}) [\prod_{i\in V_1} (\vec v_{h_i})_a e^{2 i p_F x_i \g_a}\g^{h_i\d_a}][\prod_{i\in V_2} \hat\ph_{n_i} e^{in_i p\xx_i}] \Eq(pal1)$$ %Given a cluster $L_v$ (with scale $h_v$), we define %$$N_v=\sum_{i\in L_v} n_i$$ %where $i$ are the end-points contained in $L_v$, %$n_i$ is the index of $\hat\ph_{n_i}$ if $i\in V_2$ %or $n_i=\pm 1$ if to $i$ %is associated a $F_\t$ or a $F_\iota$ term. By definition %$N_v=N_{v^1}+...+N_{v^{s_v}}$. It is convenient then to rewrite %the above expression as %$$ %\sum_{\{ P_{v} \}}[\sum_{\{N_v\}}\prod_{v {\rm not} e.p.}\d_{N_v,N_{v^1}+...+N_{v^{s_v}}}] %\{\int dt\}\int d\xx_{v^0} %[\prod_{i\in V_1} v_{h_i}] %[\sum_{\{N_v\}}\prod_{v {\rm not} e.p.}\d_{N_v,N_{v^1}+...+N_{v^{s_v}}} %\prod_{i\in V_2} \hat\ph_{n_i} %e^{in_ip\xx_i}] %\prod_{v not e.p.} |{Z_{h_v-1}\over Z_{h_v-2}}|^{|P_v|/2}$$ %$$\sum^*_{\{q\}} %[\prod_{v not e.p.}{1\over s_v!}\tilde\E^T_{h_v}(\hat\psi(P_{v^1}/Q_{v^1}),... %,\hat\psi(P_{v^{s_v}}/Q_{v^{s_v}}) %[\prod_{i\in V_1} v_{h_i}]\prod_{i\in V_2} \hat\ph_{n_i} %e^{in_ip\xx_i}] %\Eq(pal2)$$ %where $\sum_{\{N_v\}}$ is over the $N_v$ of all the clusters, including the end points. %By lemma 3.1 we know that $N_v\ge C[\g^{-h_v\over\t}-|P_v|]$ so that %$$ %\int d\xx^{v_0} %\sum_{P_{v^0}} \sqrt{Z_{k-1}}^{|P_{v_0}|}\tilde\psi^{\le k}(P_{v_0}) %\sum_{\{ P_{v} \}}[\sum_{\{N_v\geq C[\g^{-h_v\over\t}-|P_v|] %\}}\{\int dt\}$$ %$$\prod_{v {\rm not} e.p.}\d_{N_v,N_{v^1}+...+N_{v^{s_v}}}] %[\prod_{i\in V_1} v_{h_i}] %[\sum_{\{N_v\}}\prod_{v {\rm not} e.p.}\d_{N_v,N_{v^1}+...+N_{v^{s_v}}} %\prod_{i\in V_2} \hat\ph_{n_i} %e^{in_ip\xx_i}] %\prod_{v not e.p.} |{Z_{h_v-1}\over Z_{h_v-2}}|^{|P_v|/2}{1\over s_v!}$$ %$$[\sum_{\{q\}}^*\prod_{v not e.p.}{1\over s_v!} %\tilde\E^T_{h_v}(\hat\psi(P_{v^1}/Q_{v^1}),... %,\hat\psi(P_{v^{s_v}}/Q_{v^{s_v}})] %[\prod_{i\in V_1} v_{h_i}]\prod_{i\in V_2} \hat\ph_{n_i} %e^{in_ip\xx_i}] %\Eq(pal3)$$ We can bound $det G^T$ by the Grahm-Hadamard inequality ([Le],[BGPS]). %finding %$$|det G|\le \g^{{h\over 2}\sum_i \sum_{j=0}^2 (2j+1) |P_i^j|-h(k-1-s)} C^{\sum_i |P_i|-(k-1)}$$ %where $P^j$ is the subset of $P$ of the fields with a derivative of order $j$ %($\D$ is of order $2$). Once that we have bounded the determinant, we have to make the integration over the coordinates and over the interpolation variables. It is convenient to change variables from $\{\xx\}$ to $\{{\bf r}\}$, where $\{{\bf r}\}$ is the collection of the difference $\xx_l-\yy_l$ appearing in the factors $\prod_l \partial^{q_l}_{\xx_l} \partial^{q'_l}_{\yy_l} (\xx_l-\yy_l)^{a_l} g^{h}(\xx_l-\yy_l)$. Note that $\xx_l\equiv x_{ij}(t)$ so that the determinant of the Jacobian of this trasformation is a function of $t$, and one can worry about its integrability. However it is possible to show (see [BM1], App. 3) that such determinant is exactly $1$. %integrating over the difference of $T$ %and making the change of variable $\xx_l-\yy_l\to \g^h(\xx_l-\yy_l)$, if $h$ %is the scale of the cluster to which $l$ belongs, we find So we obtain a bound for $W^k$ \equ(zz10), estimating the propagators by lemma 2.8, using lemma 2.6 and \equ(n) and summing over $N_v$ istead over $n_i$: $$\L\g^{-k D(P_{v_0})} \sum_{\{ P_{v} \}}[\sum_{\{N_v\geq C[\g^{-h_v\over\t}-|P_v|] \}} \prod_{v {\rm not} e.p.}\d_{N_v,N_{v^1}+...+N_{v^{s_v}}}] [\prod_{v not e.p. } |{Z_{h_v-1}\over Z_{h_v-2}}|^{|P_v|\over 2}] $$ $$C^{|Q_v|-|P_v|} J(\t,P_{v_0},x_{v_0})\g^{-[D(P_v)+z_v(N_v,P_v)](h_v-h_{v'})} [\prod_{i\in V_1} (\vec v_{h_i})_a ][\prod_{i\in V_2} |\hat\ph_{n_i}|]$$ where $$\int d\{{\bf r}_l\} J(\t,P_{v_0},x_{v_0})=\int d\{{\bf r}_l\} \prod_{v not e.p.}{1\over s_v!}\sum_{T_v} \int dt \prod_{l\in T} \partial_{{\bf r_l}}^{q_l+q'_l}[ ({\bf r})_l^a \bar g^{h_v}(\vec r_l)]$$ and $D(P_v)=-2+\sum_{f\in P_v}(1/2+m_f)$, with $m_f$ being the order of derivatives applied to the field of label $f$, $v'$ is the vertex preceding $v$ on the tree (so that $h_{v'}=h_v-1$). The presence $z_v(N_v,P_v)$ is due to the renormalization procedure and it is defined as \vskip1cm 1)$z_v(N_v,P_v)=1$ if $|P_v|=4$, $\sum_{f\in P_v} m_f=0$ and $\sum_i\e_i\o_i p_F+2 N_v p=0$ \vskip.5cm 2)$z_v(N_v,P_v)=1$ if $|P_v|=2$, $\sum_{f\in P_v} m_f=1$ and $\sum_i\e_i\o_i p_F+2 N_v p=0$ \vskip.5cm 3)$z_v(N_v,P_v)=2$ if $|P_v|=2$, $\sum_{f\in P_v} m_f=0$ and $\sum_i\e_i\o_i p_F+2 N_v p=0$ \vskip1cm %Integrating we find %$$|\hat V^k(\hat\t,\sqrt{Z_{k-1}}\psi^{\le k})|\le \g^{-k D(p_{v_0}} %\sum_{\{ P_{v} \}}[\sum_{\{N_v\geq C(P_v)\g^{-h_v\over\t} \}} %\prod_{v {\rm not} e.p.}\d_{N_v,N_{v^1}+...+N_{v^{s_v}}}] %[\prod_{v not e.p. } |{Z_{h_v-1}\over Z_{h_v-2}}|^{|P_v|\over 2} $$ %$$\g^{-[D(P_v)+z_v(N_v,P_v)](h_v-h_{v'}} C^{\sum_i |P_{v^i}|-|P_v|} %\prod_{i\in V_1} v_{h_i}]\prod_{i\in V_2} |\hat\ph_{n_i}|]$$ %where $D(P_v)=-2+\sum_{f\in P_v}(1/2+m_f)$, with $m_f$ being the order of derivatives %applied to the field of label $f$, $v'$ is the vertex preceding $v$ on the tree (so that %$h_{v'}=h_v-1$). The presence $z_v(N_v,P_v)$ %is due to the renormalization procedure and it is defined as %\vskip1cm %1)$z_v(N_v,P_v)=1$ if $|P_v|=4$, $\sum_{f\in P_v} m_f=0$ and $\sum_i\e_i\o_i p_F+2 N_v p=0$ %\vskip.5cm %2)$z_v(N_v,P_v)=1$ if $|P_v|=2$, $\sum_{f\in P_v} m_f=1$ and $\sum_i\e_i\o_i p_F+2 N_v p=0$ %\vskip.5cm %3)$z_v(N_v,P_v)=2$ if $|P_v|=2$, $\sum_{f\in P_v} m_f=0$ and $\sum_i\e_i\o_i p_F+2 N_v p=0$ %\vskip1cm \vskip.5cm \0{\bf 3.6} We can write then $$\prod_{v not e.p.} \g^{-[D(P_v)+z_v(N_v,P_v)](h_v-h_{v'})}\le$$ $$\prod_{v not e.p.} \g^{-|P_v|\over 6}[\prod_{v\in T_4}\g^{h_v-h_{v'}}] [\prod_{v\in T_2}\g^{h_v-h_{v'}}][\prod_{v\in T_3}\g^{2(h_v-h_{v'})}]$$ where \vskip1cm 1)$T_4$ is the set of clusters with $|P_v|=4$, $\sum_{f\in P_v} m_f=0$ and $\sum_i\e_i\o_i p_F+2 N_v p\not =0$ \vskip.5cm 2)$T_2$ is the set of clusters with $|P_v|=2$, $\sum_{f\in P_v} m_f=1$ and $\sum_i\e_i\o_i p_F+2 N_v p\not =0$ \vskip.5cm 3)$T_3$ is the set of clusters with $|P_v|=2$, $\sum_{f\in P_v} m_f=0$ and $\sum_i\e_i\o_i p_F+2 N_v p\not =0$ \vskip.5cm Let we call {\it hard vertices} $\bar v$ the vertices such that $|P_{\bar v'}|\not=|P_{\bar v}|$; %there exists a $v$ following $\bar v$ such that for any %$\bar v\le \hat v\le v$ it holds $|P_{\hat v}|=|P_{\bar v}|=|P_{v}|$ and %$|P_{\bar v'}|\not=|P_{\bar v}|$, $|P_{\tilde v}|\not=|P_{v}|$ if $\tilde v'=v$. the set of such vertices is called $\TT$. Given $\t\in \t_{n,k}$ the number of hard vertices is bounded dy $C n$, if $C$ is a suitable constant. This can be easily checked remembering the graph rapresentation of the above series, see for instance [BGPS]; in each graph associated to the tree the clusters associated to hard vertices must differ at least for a line, and the number of lines is of course $O(n)$. To an hard vertex $\bar v$ we associate a {\it depth}, defined in the following way: if $\bar v$ is the first hard vertex preceding an end point on the tree then $D_{\bar v}=1$, otherwise $D_{\bar v}=1+max_{\bar v''}\{D_{\bar v''}\}$, where $\bar v''$ are hard vertices following $\bar v$ on the tree and such that there are no other hard vertices between $\bar v$ and anyone of the $\bar v''$. Note also that $D_{\bar v}\le -h_{\bar v}+2$. We can write $$\prod_{v not e.p.} \g^{-[D(P_v)+z_v(N_v,P_v)](h_v-h_{v'})}\le \prod_{v not e.p.} \g^{-|P_v|\over 6}[\prod_{\bar v\in \TT_4}\g^{-h_{\bar v'}}] [\prod_{\bar v \in \TT_2}\g^{-h_{\bar v'}}][\prod_{\bar v\in \TT_3} \g^{-2h_{\bar v'})}]$$ where $\bar v'$ is a the vertex preceding $\bar v$ and $\TT_i$ is the intersection between $\TT$ and $T_i$ . We write $\TT_i=\TT'_i\bigcup \TT''_i$ with $\TT'_i$ such that $N_v\not=0$ and $T''_i$ such that $N_v=0$. By Lemma 2.7 it follows that $$\prod_{\bar v\in \TT''_i}\g^{-h_{\bar v'}}\le C^n$$ On the other hand it is easy to show that $$\prod_i |\hat\ph_{n_i}|\le e^{\x n\over 2} \prod_i e^{-\x |n_i|/2}\prod_{\bar v\in \TT'_2\bigcup\TT'_3\bigcup\TT'_4} e^{-\x |N_{\bar v}|\over 2^{D_{\bar v}+1} }\Eq(fon)$$ In fact let us consider a hard vertex $\bar v_{1}$ not followed by any other hard vertex. Then we can write $$\prod_{i\in L_{\bar v_{1}}}e^{-\x |n_i|\over 2}\le e^{-\x |N_{\bar v_{1}} |\over 4} e^{-\x |N_{\bar v_{1}} |\over 4}$$ where the product is over the end-points contained in the cluster $\bar v_{1}$. So for the hard vertex $\bar v_1$ we have found the factor $e^{-\x |N_{\bar v_{1}} |\over 2^{D_{\bar v_1}+1}}$ and we have an extra factor $e^{-\x |N_{\bar v_{1}} |\over 4}$. Let us consider now a vertex $\bar v_{2}$ with depth $2$ \ie followed only by end points and hard vertex $\bar v_{i}$ with depth $1$; we can write $$\prod_{i\in L_{\bar v_2}}e^{-\x |n_i|\over 2} \prod_{\bar v_1:L_{\bar v_1} \i L_{\bar v_2}} e^{-\x |N_{\bar v_{1}} |\over 4}\le e^{-\x |N_{\bar v_{1}} |\over 8} e^{-\x |N_{\bar v_{1}} |\over 8}$$ so again the the factor $e^{-\x |N_{\bar v_{1}} |\over 2^{D_{\bar v_1}+1}}$ and an extra factor $e^{-\x |N_{\bar v_{1}}|\over 8}$. Proceding in this way we have \equ(fon), noting that to the end-points of kind $\iota,\t$ it is not associated any factor $e^{-\x}$ (this explains the factor $e^{\x n\over 2}$ in front of \equ(fon)). Using Lemma 2.6 and the fact that $D_{\bar v}\le -h_{\bar v'}+2$ we have $$\prod_i |\hat \ph_{n_i}|\le \prod_{\bar v\in \TT'_2\cup \TT'_3\cup \TT'_4} e^{-\x C_2 \g^{-h_{\bar v'}\over \t}\over 2^{-h_{\bar v'}+3}}\Eq(sp)$$ At the end, summing over $N_v$, we can write $$|\hat V^k(\hat\t,\sqrt{Z_{k-1}}\psi^{\le k})|\le C^n\e^n \g^{-k D(P_{v_0})}\sum_{\{P_v\}}\prod_{v not e. p.}\g^{-|P_v|\over 8}$$ $$\prod_{\bar v\in\TT'_1}\g^{-h_{\bar v'}} e^{-\x C_2 \g^{-h_{\bar v'}\over \t}\over 2^{-h_{\bar v'}+3}} \prod_{\bar v\in\TT'_2}\g^{-h_{\bar v'}}e^{-\x C_2 \g^{-h_{\bar v'} \over \t}\over 2^{-h_{\bar v'}+3}} \prod_{\bar v\in\TT'_3}\g^{-2h_{\bar v'}}e^{-\x C_2 \g^{-h_{\bar v'}\over \t} \over 2^{-h_{\bar v'}+3}}$$ Choosing $\g$ so that $\g^{1\over\t}/2>1$ and remembering that the hard vertices are $\le C_1 n$ we have that $$\prod_{\bar v\in\TT'_i}\g^{-2h_{\bar v'}} e^{-\x C_2 \g^{-h_{\bar v'}\over \t}\over 2^{-h_{\bar v'}+3}} \le C_2^n$$ %$$\sum_{r=0}^\io \g^{2r} e^{-\x C_2 \g^{r\over \t}\over 2^{3+3}}\le C$ By a standard calculation $$\sum_{\t\in\t_{k,n}}\sum_{\{ P_v\}}\prod_{v not e.p.} \g^{-|P_v|\over 8}\le C_3^n$$ and this completes the proof of Theorem 3.3 (the last factor in the bound comes from \equ(sp)). \vskip.5cm {\bf Remark:} Formula \equ(sp) is the analogue of {\it Brjuno lemma} for this problem; it ensures that the small denominator problem arising in the series for the uncommensurability of $\phi_x$, can be controlled taking into account the Diophantine condition. The same role is plaied by the original Brjuno lemma in the proof of the convergence of the Lindstedt series. \vskip1cm We have now to perform the last integration \equ(uu). \vskip.5cm \0{\bf 3.7}{\bf THEOREM} {\it If there exists constants $\k$, $C_2$, $\e$ such that ${\sigma_{h^*-1}\over\k}\ge \k$ and $|\vec v_{h^*}|\le\e$, $|{Z_{h^*}\over Z_{h^*-1}}|\le e^{C_2\e^2}$ than $$ |\int \tilde P_{Z_{h^*-1}}(d\psi^{(\le h^*)}) e^{-\tilde \VV^{h^*}(\sqrt{Z_{h^*}}\psi^{(\le h^*)})}|\le C_3 |\L| \g^{2 h^*}$$ where $C_3$ is a suitable constant} \vskip.5cm The proof of this theorem is rather straigthforward. We can write $$\int \tilde P_{Z_{h^*-1}}(d\psi^{\le h^*}) e^{-\tilde \VV^{h^*}(\sqrt{Z_{h^*-1}}\psi^{\le h^*})}=\sum_n{(-1)^n\over n!} E^T_{\le h^*}(\hat \VV^{h^*},...,\hat \VV^{h^*})=$$ $$=\sum_{n}\sum_{i_1,...,i_n}{(-1)^n\over n!} E^T_{\le h^*} (\hat\VV^{h^*}_{i_1},...,\hat\VV^{h^*}_{i_1})$$ and $\hat V^{h^*}_{i}$ are terms of the form $$\int \prod_{j=1}^i d\kk_i \prod_{j=1}^i\psi^{\e_i,\le h^*}_{\kk_i+\o_i p_F,\o_i}W^{h^*}_i(\kk_1,..,\kk_i) \d(\sum_{j=1}^i\e_j(\kk_j+\o_j p_F)+2np)\Eq(car)$$ It is easy to check that each addend in the above sum will be bounded by $$\g^{2h^*}\L C^n \prod_i \g^{h^*(i/2-2)m_i}\g^{h^* m_2^1} e^{-\x c_2 \g^{-h^*}|m_2^2|\over 2^{-h^*+3}}$$ %[\prod_{i\in V_2}\hat\ph_{n_1}][\prod_{i\in V_1} v_{h^*}^i$$ if $m_i$ is the number of terms of the form \equ(car), $m_2^1$ the number of such terms with $i=2$, $n=0$ and $m_2^2$ the number of terms with $i=2$, $n\not=0$. We have used that the number of loops is $\sum_i({i\over 2}-1)m_i+1$, the number of $g^{h^*}$ is $\sum_i {i m_i\over 2}$. %and we have taken into account the factor $\g^{h^*}$ associated, by Th. 3.4, to each %term \equ(car) with $i=2$ and $n=0$. %\equiv $$ %$$\sum_n\sum_{\t\in \hat\t_{n,k}}{1\over n!}W^{\le h^*}(\t)$$ %where $\hat\t_{n,k}$ are the trees such that from $k_{v_0^i}$ are n-end points corresponding %to one addend of $\tilde V^{h^*}$. %If $m_i^\t$ is the number of elements of the form $\int \prod_{j=1}^i d\kk_i %\prod_{j=1}^i\psi^{\e_i,\le h^*}_{\kk_i+\o_i p_F,\o_i}W^{h^*}(\kk_1,..,\kk_i) %\d(\sum_{j=1}^i\e_i(\kk_i+\o_i p_F)+2np)$ associated to $\t$, it is easy %to show by the Grahm-Hadamard inequality that %$$\g^{-2 h^*}|W^{\le h^*}(\t)|\le \L C^n %\g^{h^*(i/2-2)m_i}[\prod_{i\in V_2}\hat\ph_{n_1}][\prod_{i\in V_1} v_{h^*}^i$$ %So Th. 3 follows trivially if $m_2^\t=0$; if this is not the case we have to note that %we can call $m_2^1$ the set of verices of $v_h$ type and $m_2^2$ %the vertices of $\phi_n$ type. The factor $\g^{h^*}$ associated to the $\nu_h$ %factors allows us to rewrite the above equation as %$$\g^{-2 h^*}|W^{\le h^*}(\t)|\le \L C^n\e^n %\g^{-h^* m_2^2}[\prod_{i\in V_2}\hat\ph_{n_1}]$$ %By Lemma 2 we can write %$$\prod_{i\in V_2}\hat\ph_{n_1}\le \prod_{i\in V_2} e^{-\x |n_i|\over 2} e^{-\x C m_2^2 %\g^{-h^*/\t}}$$ %so that Th. 3 follows \vskip2.truecm \centerline{\titolo 4. The flow of the Renormalization group} \*\numsec=4\numfor=1 %Contrary to the preceding sections, in the following the assumption of %{\it spinless} fermions is crucial. We have seen at the end of sec.(4) %that, given a constant $\tilde C$, %{\it if} $\tilde C_h\leq %\tilde C$ the kernels of the effective potential $V^{h-1}$ and %the Beta function $\b^{h-1}$ are analytic as functions of their %arguments, if $\bar\e_h<\e<{1\over \tilde C C}$ and $\bar\e_h$ is %defined in eq.(\ref{cond})\cite{nota11}. %Of course there is no reason for which %$\tilde C_h\leq \tilde C$ for any $h$ so we call $h^*\equiv \tilde %h(\tilde C)$ the scale such that $\tilde C_{h^*}\leq \tilde C$ and %$\tilde C_{h^*-1}> \tilde C$; the scale $h^*$ can be computed once we %know the $h$-dependence of $\sigma_h$. Then, if for $h\geq h^*$ it %holds that $\bar\e_h<\e$ we have that the kernels of the effective %potential $V^{h-1}$ and the Beta function $\b^{h-1}$ are well defined %for $h\geq h^*$; to prove this, and compute $h^*$ we need informations %on the $h$-dependence of the running coupling constants, what is %provided by the study of the Beta function. \vskip.5cm \0{\bf 4.1} The convergence of the expansion for the partition function is proved by theorems 3.4,3.7 under suitable assumptions on the running coupling constants \ie that there exists a finite $h^*$ \equ(h*) such that, given a constant $\e$, then $\max_{k\ge h^*}|\vec v_{k}|le \e$ and $\max_{k\ge h^*}|{Z_k\over Z_{k-1}}|\le e^{c_2\e^2}$. In this section we prove that it is possible to choose $|\l|,u$ so small and a proper $\nu_0$ so that the above conditions are indeed verified. Let we say first that the property $\g>1$ can be used to show that there is a $\nu_0$ such that $|\nu_h|\le\e$ and $\nu_h=O(\g^h)$. %In this section we show that such assumptions %are indeed velid. %It is possible to choose %%$\nu_0$ (and so the %the counterterm $\nu$ so that $|\nu_h|<\e$ for any $0\geq h\geq h^*$; %in fact, given any sequence of running coupling constants verifying %$\max_{i,k\geq h} |\bar v_{k}|\leq \e$, $\max_{k\geq h}|{Z_k\over %Z_{k-1}}|\leq e^{\beta_1\e}$, In fact $\nu_h$ obeys to the equation $\nu_{h-1}=\g \nu_h+\b_\nu^h$, with $|\b_\nu^h|0$. The fact that $G^{1,h}_\l, G^{1,h}_\d$ are vanishing at the second order could generate a problem as one cannot exclude {\it a priori} there there is some non-vanishing term at some large order, and the evolution of the running coupling constants will depend critically on this unknown term. Fortunately it is possible to prove that: $$\lim_{h\to-\io} G^{1,h}_\l(g,\d;\ldots;g,\d)=0\Eq(van)$$ $$\lim_{h\to-\io} G^{1,h}_\d(g,\d;\ldots;g,\d)=0$$ % {\it i.e.} it is vanishing at {\it all} orders. In fact $G^{1,h}_\l$, $G^{1,h}_\d$ coincide, up to terms vanishing as $h\to-\io$ as $O(\g^h)$, with the correspondent quantities of the Luttinger model , and, by using the properties of the exact solution [ML],[M] one can prove that it is vanishing. This non perturbative result was proved in [BGPS] using some properties of the exact solution obtained in [BeGM],[BM1]. %(B.G.),(B.G.M.),(B.G.P.S.),(B.M.1). %s $|R^h_i|\leq K\e_h^2$, $i=\l,\d,\sigma,z$ and %$|G_\l^{2,h}|,|G_\d^{2,h}|,|G_z^{2,h}|,|G_\t^{2,h}|, %|G_\th^{2,h}|\leq K\e_h \tilde C_h$ %one finds, for $0\ge h\ge h^*$: As a straigthforward consequences of the above facts one can prove. \vskip.5cm \0{\bf 4.2}{\bf LEMMA}: There exist positive constants $c_1,c_2,c_3,c_4$ such that, if $\l,u$ are small enough: $$|g_{h-1}-g_{0}|< c_1\l^2$$ $$\l\b_1 c_3 h< \log\left({\sigma_{h-1}\over \sigma_0}\right)< \l\b_1 c_4 \Eq(4.2)$$ $$-\b_3 c_1\l^2h < \log(|Z_{h-1}|)< -\b_3 c_2\l^2h\qquad |\t_{h-1}-\t_{0}|< c_1|\l|$$ $$|\th_{h-1}-\th_{0}|0$ are suitable constants. Then, if $\l>0$ (for fixing ideas) there exists a $c_4<1$ such that $$\sigma_{h-1}\le u \g^{c_4\l\b_2(h-1)}{(1-\b_2 \l+c_b\e_h^2)\over \g^{-\l c_4\b_2}}\le u \g^{\l c_4\b_2(h-1)}$$ and a $c_3>1$ such that $$\sigma_{h-1}\ge u \g^{c_3\l\b_2(h-1)}{(1-\b_2 \l-c_a\e_h^2)\over \g^{-\l c_3\b_2}}\ge u \g^{c_4\b_2(h-1)}$$ We proceed in the same way for $Z_h$. Moreover : $$\l_h\le \l^2\sum_{h=h^*}^0\tilde {\s_h\over\g^h}\le \l^2\sigma_{h^*}\g^{-h^*} \sum_{h=h^*}^0\g^{-h+h^*}{\sigma_h\over\sigma_{h^*}}\le K$$ if $K$ is a constant. For the other constants we proceed in a similar way.\qed \vskip1.cm As a consequence of the above estimates on the running coupling constants fi find that $h^*$ is finite: $${\log_\g (u)\over 1-\l\b_2 c_\a}\leq h^*\leq {\log_\g (u)+1\over 1-\l\b_2 c_\b}\Eq(h**)$$ Noting that the above recursive relation of course implyes that $K_1\sigma_h\le\sigma_h(k)\le K_2\sigma_h$ the proof of the convergence for the exapnsion for the partition function is completed. %The above inequality proves the assumptions done to prove Th2, Th 3, if $\l$ %is choosen small enough.. %In fact it possible to choose $\l$ small %enough so that the series for $V^{h-1}$ and $\b^{h-1}$ , if $h\geq %h^*$, are convergent %; moreover from eq.(\ref{c1}) it is possible to %obtain an upper and lower bound for $h^*$: %All the above considerations hold for $\tilde C_h\leq tilde C$, %\hbox{\it i.e.\ } for $h\geq h^*$; from eq.(\ref{c1}) we see that %\begin{equation}\label{h*} %c_3 {\log_\g (\tilde C^{-1}u)\over %1-\l\b_1}\leq h^*\leq c_4{\log_\g (\tilde C^{-1} u)+1\over 1-\l\b_1} %\end{equation} %The above analysis is performed by approximating the anomalous %propagator eq.(\ref{proan}) with the Luttinger model propagator by %eq.(\ref{22}). This approximation is of course not reasonable for %large $|h|$ corresponding to momenta negligible with respect to the %$\sigma_h$ term due to the periodic potential. But for the scales $h< %h^*$ the bound\cite{(B.M.2)}: %\begin{equation}\label{x} %|g^{(< h^*)}(x-y)|\le A{\g^{h^*}\over Z_{h^*}}\left({\g^{h^*} %\over\sigma_{h^*}}\right) %{C_N\over 1+\sigma_{h^*}^{N}|x-y|^N} %\end{equation} %holds and, fro\TT'_3m eq.(43),(44), %$\left({\g^{h^*}\over\sigma_{h^*}}\right)\leq {K\over \tilde C}$, if %$K$ is a constant. In other words the propagator $g^{(\le %h^*)}_{\o,\o'}(x-y)$ obeys to the same bound of %$g^{(h)}_{\o,\o'}(x-y)$ for $h\ge h^*$; our choice of $h^*$ is made %just to obtain this. %The integration of the scale %between $-\i$ and $h^*$ is equivalent to the %integration of a single scale in a not filled band theory, so that for %the same considerations at the end of sec.(4) also the series expressing %\begin{eqnarray}\label{78} %\int\{{\cal D}\psi^{(< h^*)} %&e&^{-\int dk \psi^{+(h^*$ is bounded by %$(\bar\e_{h^*})^n \tilde C^n C_1^n$ while the $n$-th order term %of the series eq.(\ref{78}) is bounded %by $(\bar\e_{h^*-1})^n ({C_2\over \tilde C})^n$ so that there is a %non ambiguous way to fix %$\tilde C$ so that $\bar\e_{h^*}$ is the largest possible. \vskip.5cm \0{\bf 4.3} From the proof of the convergence of the partition function it is possible, repeating essentially word by word the analogous proof in [BGPS], to deduce the convergence of the Schwinger function. We do not repeat such computations here. %The Schwinger function eq. admits a perturbative expansion %similar to the one of the partition function, whose convergence %follows from the partition function expansion convergence %(B.G.P.S.); It holds that $$S(x,y)=S^{u.v.}(x,y)+ \sum_{h=h^*}^0\sum_{\o_1,\o_2}e^{i{\pi}(\o_1\xx-\o_2\yy)} \cdot({g^{(h)}_{\o_1,\o_2}(x-y)\over Z_h} +{\bar S^{(h)}_{\o_1,\o_2}(x,y)\over Z_h})\Eq(S)$$ where we call $g^{(\leq h^*)}(x-y)$ simply $g^{(h^*)}(x-y)$ and the first addend is obtained by the integration of $\psi^{(1)}$ ( and it is bounded by ${C_N\over 1+|\xx-\yy|^N}$; $\bar S^{(h)}_{\o_1,\o_2}(x,y)$ is given by a sum of Feynmann graphs similar to the ones contributing to the effective potential, see sec.3, with the difference that: \vskip.5cm 1) they have two external lines, to which the propagators $g^{(h_1)}_{\o_1,\o'}(x-x')$ or $g^{(h_2)}_{\o'',\o_2}(y-y')$ are associated, with $h_1,h_2\geq h$ and if $x',y'$ are respectively the coordinates of the vertex the external lines is entering in or coming out. %Note %that $\bar S^{(h)}_{\o_1,\o_2}(x-y)$ is translation invariant; the %reason is that, for the compact support properties (in momentum space) %of the propagators, if $\kk_1,\kk_2$ are the momenta associated to the %external lines, then $\kk_1-\kk_2=2n p_F$, with $n=0,\pm 1$. \vskip.5cm 2) $h$ is the smallest scale of the propagators contributing to $\bar S^{(h)}_{\o_1,\o_2}(x-y)$ (and not, as for the graphs for the effective potential, the scale of the external lines). No $\RR$ acts on clusters containing the external lines. \vskip.5cm %Note that $\sigma_{h^*}, Z_{h^*}$ depend on $\tilde C$, but it is easy %to check that they can be written as $\sigma_{h^*}=\bar %\sigma_{h^*}(1+\l f_1)$, $Z_{h^*}=\bar Z_{h^*}(1+\l f_2)$, with %$|f_1|, |f_2|$ bounded by some constant and $\bar \sigma_{h^*}$, $\bar %Z_{h^*}$ {\it independent} on $\tilde C$. One can check, by repeating the computations in [BGPS], sec.6, that $|\bar S^{(h)}_{\o,\o}(x,y)|\leq A \tilde\e \g^h{C_N\over 1+(\g^h|x-y|)^N}$ if $\tilde\e= \max(u,u^{1+\h_1},|\l|)$. %; the extra %factor ${\sigma_h\over\g^h}$ in the second bound follows from the fact %that in the graph contributing to $\bar S^{(h)}_{\o,-\o}(x,y)$ there %is at least a non diagonal propagator, which obeys, see the lines %to a bound similar to the diagonal propagator but %with a factor more ${\sigma_k\over\g^k}$. We are now ready to prove Theorem 1. First of all, we note that the r.h.s. of \equ(S) has a meaning also if we take the formal limit $i\to\io$, $\b \to\io$, (recall that $L=L_i$), by doing the substitution % $${1\over L_i\b} \sum_{\kk\in {\cal D}_{L_i,\b}} \rightarrow \int_{T^1\times R} {d\kk\over (2\p)^2}\; . \Eq(4.21)$$ % %In fact the integral over $\kk$ is well defined, since %$k\in T^1$ and $k_0$ belongs to a bounded set, except in the case %$h=1$, and, for $h=1$, each graph contributing to %$K_{\phi,\phi}^{(0)}(\xx;\yy)$ decays at least as $|k_0|^{-2}$ for large %values of $|k_0|$, (see comments between \equ(4.6) and \equ(4.6a)). The substitution \equ(4.21), applied to \equ(S), allows to define $g^{(h)}(\xx;\yy)$ in the limit $i\to\io$, $\b\to\io$, at least for $h\le 0$. For $h=1$, one has to be careful, since the integral over $k_0$ is not absolutely convergent. However it is easy to prove, by using standard well known arguments, that the limit as $i\to\io$ and $\b\to\io$ of $g^{(1)}(\xx;\yy)$ is well defined for $x_0\not= y_0$ and has the same discontinuity in $x_0=y_0$ of the same limit taken on the free propagator \equ(1.3f). %We shall denote this %limit, as usual, by doing again the substitution \equ(4.21) in the finite $L$ %and $\b$ expression (see again comments between \equ(4.6) and \equ(4.6a)). The previous considerations says that, for $\l$ real and small enough, there exists the limit % $$S(\xx;\yy) = \lim_{\b\to\io \atop i\to\io} S^{L_i,\b}(\xx;\yy) \Eq(4.22)$$ % and that this limit is obtained by doing the substitution \equ(4.21) in all quantities appearing in the r.h.s. of \equ(S). In fact let us consider one of the finite $L=L_i$ and $\b$ quantities appearing in the r.h.s. of \equ(S); if we interpret it as a Riemann sum of the corresponding $L_i=\b=\io$ quantity it is easy to prove that the difference between the two quantities can be bounded by a constant times $(1/L_i+1/\b)$. In fact one gets essentially the same bounds up to a factor $\g^{-h}(1/L_i+1/\b)$, coming from the comparison of the integral and the corresponding Riemann sum; we shall not give the details, which are completely straightforward. \vskip.5cm \0{\bf 4.3} Let us define %\begin{equation} $$\h_1=-{\log(\bar Z_{h^*})\over \log u}\qquad 1+\h_2= {\log(\sigma_{h^*})\over \log u}$$ %%\end{equation} which are respectively $\b_1\l^2+O(\l^3)$ and $\b_2\l+O(\l^2)$. %One can check [BGPS] %that $|\bar S^{(h)}_{\o,\o}(x,y)|\leq A \tilde\e \g^h{C_N\over 1+(\g^h|x-y|)^N}$ %and $|\bar %S^{(h)}_{\o,-\o}(x-y)|\leq A \tilde\e \sigma_h {C_N\over %1+(\g^h|x-y|)^N}$, if $\tilde\e= \max(u,u^{1+\h_1},|\l|)$ ; the extra %factor ${\sigma_h\over\g^h}$ in the second bound follows from the fact %that in the graph contributing to $\bar S^{(h)}_{\o,-\o}(x,y)$ there %is at least a non diagonal propagator, which obeys, see the lines %to a bound similar to the diagonal propagator but %with a factor more ${\sigma_k\over\g^k}$. The bounds \equ(z.1) can be obtained by \equ(S). In fact if $1\le |\xx-\yy|\le\hat u(p_F)^{-1}$ it holds, for $\g^{-h_x-1}<|x-y|\leq \g^{-h_x}$, $h_x>h^*$, %if $|g^{( %+\bar S^{(h)}_{\o_1,\o_2}(x,y)|\leq \bar g^{(h)}(x-y)$: $$ \sum_{h=0}^{h^*} |{ g^{(h)}_{\o_1,\o_2}(x-y)\over Z_h} +{\bar S^{(h)}_{\o_1,\o_2}(x,y)\over Z_h})|\le A_1\left[\sum_{h=h^*}^{h_x-1} {\g^h\over Z_h}+\sum_{h=h_x}^{0}{\g^h\over Z_h}{C_N\over \g^{Nh}|x-y|^N}\right]\le A_2\g^{h_x(1-\h_3)}$$ On the other hand if $|\xx-\yy|\ge \hat u(p_F)^{-1}$: $$ \sum_{h=0}^{h^*} |{ g^{(h)}_{\o_1,\o_2}(x-y)\over Z_h} +{\bar S^{(h)}_{\o_1,\o_2}(x,y)\over Z_h}) | \le {C_N\over |x-y|^N}\sum_{h=h^*+1}^1{\g^{-(N-1)h}\over Z_h} A_2{\g^{h^*}\over Z_{h^*}}\le$$ $${C_N\over (\g^{h^*}|x-y|)^N} $$ %The occupation number discontinuity can be obtained noting that, from the %theory of the Bloch waves, $\phi(p_F^+,\xx)=\cos(p_F\xx)+O(u)$ and %$\phi(p_F^-,\xx)=i\sin(p_F\xx)+O(u)$ so that the occupation number %discontinuity eq.(\ref{Z}) can be written as: %$$Z^{-1}={1\over L}\int %d\xx d\yy [\cos(p_F(\xx+\yy)+r(u)]S^{L,\b}(x,y)$$ %with $r(u)=O(u)$. %By inserting in the above expression eq.(\ref{sfb}) we obtain several %terms that we can bound in the following way, if $K_i$ are positive %constants: %$${r(u)\over L}\int d\xx d\yy \sum_{h=h^*}^0\sum_{\o_1,\o_2} %|g^{(h)}_{\o_1,\o_2}(x,y) %+\bar S^{(h)}_{\o_1,\o_2}(x,y))|\leq$$ %$$\qquad\leq K_1 u \sum_{h=h^*}^0 {1\over Z_h}\leq K_2 u\log u\leq K_3 u^{\h_1}$$ % %$${1\over L}\int d\xx d\yy \cos(p_F(\xx+\yy)) %\sum_{h=h^*}^0\sum_{\o_1,\o_2} %|\bar S^{(h)}_{\o_1,\o_2}(x,y))|\leq %K_4 \sum_{h=h^*}^0 {\sigma_h\over Z_h \g^h}\leq \tilde\e K_5 %u^{\h_1}$$ %in which we take into account that in the sum only the terms %$\o_1=-\o_2$ survive; %$${1\over L}\int d\xx d\yy %\cos(p_F(\xx+\yy))g^{(h)}_{\o_1,\o_2}(x-y) %=\int dk_0 %\sum_{h=h^*}^0\tilde f_h(0,k_0) {1\over Z_h} %(\GG_h(0,k_0)^{-1})_{\o,-\o}$$ %verifies the bound eq.(12); finally: % %$$ %{1\over L}\int d\xx d\yy [\cos(p_F(\xx+\yy)+r(u)] %S^{u.v.}(x,y) %\leq K_7 u %$$ >From the above bounds and lemma 2.6 it follows that $$\sum_{h=0}^{h^*}\sum_{\o_1,\o_2}e^{i{p_F}(\o_1\xx-\o_2\yy)} {g^{(h)}_{\o_1,\o_2}(x-y)\over Z_h}-\sum_{\o}e^{i{p_F}(\o(x-y)} {1\over (L\b)}\sum_{\kk'\in {\cal D}_{L,\b}} {f_h(\kk')\over -ik_0+\o k'}{1\over|\kk'|^{\h_3}}\le$$ $$C\sum_{h=0}^{h^*}[{\sigma_h\over Z_h}]\le C_1{\tilde\e\over \hat Z(p_F)}\log \hat u(P_F)$$ so proving \equ(luttt). \vskip.5cm \0{\bf 4.3} Let us prove the decomposition \equ(1.21). By diagonalizing the quadratic form $\hat g^{(h)}(x,y)=\sum_{\o_1,\o_2}e^{i{p_F}(\o_1\xx-\o_2\yy)} g^{(h)}_{\o_1,\o_2}(x-y)$ it is possible to see that: $$\hat g^{(h)}(x,y)= {1\over (L\b)}\sum_{\kk'_1\in {\cal D}_{L,\b}} e^{ik'(x-y)} {\tilde f_h(k')\over Z_h} [{F_{xy}(k',\sigma_h)\over A+B} +{F_{xy}(-k',-\sigma_h)\over A-B}]$$ where $$F_{xy}(k',\sigma_h)=\hat \phi(k',\xx,\sigma_h)\hat \phi(k',-\yy,\sigma_h)$$ $$\hat \phi(k',\xx,\sigma_h)={1\over\sqrt{2B}}[\sqrt{B-C}e^{i p_F\xx} - {e^{-i p_F\xx}\over \sqrt{B-C}}]$$ and $A=-i k_0+\kk^{'2}$, $B=\sqrt{(C^2+\s_h^2)}$ and $C=2\pi\kk'$. We can rewrite the above integral in terms of the $k$ variable. Recall that, if $\o={\rm sign}(\kk)$ and $f_h(k')\not =0$, then $k=\o p_F+\kk'$. Hence %\begin{eqnarray}\label{86} $$\hat g^h(x-y)=\sum_{\o_1,\o_2}e^{i{p_F}(\o_1\xx-\o_2\yy)} g^h_{\o_1,\o_2}(x-y)=$$ $$={1\over (L\b)}\sum_{\kk{\cal D}_{L,\b}} {\tilde f_h(k)\over Z_h(k)} {\hat\phi(\kk,\xx,\sigma_h))\hat\phi(\kk,-\yy,\sigma_h) e^{ik_0(x_0-y_0)} \over -ik_0-(\hat\e(k,\sigma_h)-p_F^2)}$$ where $\hat\e(k,\sigma_h),u(\kk,\xx,\sigma_h)$ are defined in the theorem 1. %%\begin{eqnarray}\label{bl1} %$$\hat\e(k,\sigma_h)=\left(|\kk|-{\pi}\right)^2+ %+2\pi{\rm sign}\left(|\kk|-{\pi}\right)\sqrt{ %\left(|\kk|-{\pi}\right)^2+\sigma_h^2}+\pi^2 %$$ %and %\begin{equation}\label{bl} %$$\hat\phi(\kk,\xx,\sigma_h))=e^{i\kk\xx} u(\kk,\xx,\sigma_h)$$ %$$u(\kk,\xx,\sigma_h)&=&e^{-i{\rm sign}(\kk)\pi\xx} %\left[\cos{(\pi\xx)}\sqrt{1+ %{{\rm sign}(|\kk|-{\pi})\sigma_h \over\sqrt{ %(|\kk|-{\pi})^2+\sigma_h^2}}}+\right.$$ %$$\left.+i{\rm sign}(\kk)\sin{(\pi\xx)}\sqrt{1- %{{\rm sign}(|\kk|-{\pi})\sigma_h \over\sqrt{ %(|\kk|-{\pi})^2+\sigma_h^2}}}\,\right] %$$ %In Ref.(B.M.2) %by a careful analysis of the Bloch waves it is %shown that %$$|\hat\phi(\kk,\xx,u)-\phi(\kk,\xx,u)|=O(u)\quad |\hat\e(\kk,u)-\e(\kk,u)|=O(u^2)$$ %where $\phi(\kk,\xx,u)$ are the Bloch waves \hbox{\it i.e.\ } %the solutions of eq.(4) %and $\e(\kk,u)$ the dispersion relation. The fourth statement of the %theorem then follows. \vskip.5cm \0{\bf 4.4} Finally the lower bound on the spectral gap can be obtained showing that we can prove the boundedness on $S(x,y;k_0)$ for $k_0$ complex, for $|\Im(k_0)|\le{\sigma_{h^*}\over 2}$. The expansion disussed before is not suitable for this, as the functions $\hat f_h(\kk)$ are not analytic in $k_0$. However one can repeat the above analysis using a decomposition $\hat f_h(k)$ \ie not depending on $k_0$ (see [BGM] app.4 in which are discussed the (obvious) modifications in the bounds changing the cut-off functions). So in the support of $\hat f_h(k)$, for $h>h^*$ we have that, if $|\h|\le {\s_{h^*}\over 2}$ $$|\Re[A_h(k',k_0+i\h]|\ge a_0\g^{2h}+\s_h^2-\h^2\ge {\g^{2h}\over 4}$$ as, by \equ(h*), $\g^h>\g^{h^*}\ge {4\s_{h^*}\over a_0}$. On the other hand if $h\le h^*$ $$|\Re[A_h(k`,k_0+i\h]|\ge {c_1\over\g^{h^*}}$$ for some constant $c_1$, so everothng is essentially unchanged. %by repeating %the computations in the preceding sections using an analytic partition %of the unity instead of a $C^\infty$ one, %see sec.3. In fact in this way one can %prove that the Fourier transform of the 2-point Schwinger function in bounded %as a function of $k_0$ in a strip of the imaginary plane of width %${\hat u(p_F)\over 2}$. %We preferred a $C^\infty$ decomposition, as it makes the %exposition %more readable and intuitive, but %there should be no problem to use an analytic decomposition. \centerline{\titolo References} \* \halign{\hbox to 1.2truecm {[#]\hss} & \vtop{\advance\hsize by -1.25 truecm \0#}\cr A& {S. Aubry in Polarons and Bipolarons in {\it Hich $T_c$ superconductors and related materials}, editors Salye E, Alexandrov A., Liang W. }\cr AAR& {S. Aubry, G. Abramovici, J. Raimbaut: Chaotic polaronic and bipolaronic states in the adiabatic Holstein model, {\it J. Stat. Phys.} {\bf 67}, 675--780 (1992). }\cr B& {A.D. Bryuno: The analytic form of differential equations I, %Transactions of the Moscow Mathematical Society {\it Trans. Moscow Math. Soc.} {\bf 25}, 131--288 (1971); II, {\it Trans. Moscow Math. Soc.} {\bf 26}, 199--239 (1972). }\cr BG& {G.Benfatto, G.Gallavotti: Perturbation theory of the Fermi surface in a quantum liquid. A general quasiparticle formalism and one dimensional systems, {\it J. Stat. Phys.} {\bf 59}, 541--664 (1990). }\cr BGL& {G.Benfatto, G.Gallavotti, J. Lebowitz: Disorder in the 1D Holstein model, {\it Hel.Phys. Acta} (1992). }\cr BGM1& {G.Benfatto,G.Gentile,V.Mastropietro: to appear on {\it J. Stat. Phys.} (1997). }\cr BGM2& {G.Benfatto,G.Gentile,V.Mastropietro: preprint (1997). }\cr BGPS& {G.Benfatto, G.Gallavotti, A.Procacci, B.Scoppola: Beta functions and Schwinger functions for a many fermions system in one dmension, {\it Comm.Math.Phys.} {\bf 160}, 93--171 (1994). }\cr BM1& {F.Bonetto,V.Mastropietro: Beta function and anomaly of the Fermi surface for a d=1 system of interacting fermions in a periodic potential, {\it Comm.Math.Phys.} {\bf 172}, 57-93 (1995).}\cr BM2& {F.Bonetto,V.Mastropietro: Filled band Fermi systems {\it Mat.Phys.Elect.Jour} {\bf 2}, 1--43 (1996 ).}\cr BM3& {F.Bonetto,V.Mastropietro: Critical indices in a d=1 filled band Fermi system, {\it Phys. Rev. B} {\bf 56},3,1296-1308 (1997).}\cr % BM4& {F.Bonetto,V.Mastropietro: Critical indices for the Yukawa2 quantum field theory, {\it Nucl.Phys.B} {\bf 497}, 541-554 (1997). }\cr BeGM& {G.Benfatto, G.Gallavotti,V.Mastropietro: Renormalization group and the Fermi surface in the Luttinger model, {\it Phys. Rev. B} {\bf 45}, 5468-5480, (1992). }\cr BLT& {J. Belissard, R. Lima, D. Testard: A metal-insulator transition for almost Mathieu model, {\it Comm. Math. Phys.} {\bf 88}, 207--234 (1983). }\cr % CF& {L. Chierchia, C. Falcolini: A direct proof of a theorem by Kolmogorov in hamiltonian systems, %Annali della Scuola Normale Superiore di Pisa {\it Ann. Scuola Norm. Super. Pisa Cl. Sci.} {\bf 21}, No. 4, 541--593 (1995). }\cr D& {H. Davenport: {\sl The Higher Arithmetic}, Dover, New York, 1983.}\cr % DS& {E.I. Dinaburg, Ya.G. Sinai: On the one dimensional Schroedinger equation with a quasiperiodic potential, {\it Funct. Anal. and its Appl.} {\bf 9}, 279--289 (1975). }\cr % E& {L.H. Eliasson: Floquet solutions for the one dimensional quasi periodic Schroedinger equation, {\it Comm. Math. Phys.} {\bf 146}, 447--482 (1992). }\cr E1& {L.H. Eliasson: Absolutely convergent series expansions for quasi-periodic motions, Report 2--88, 1--31, Department of Mathematics, University of Stockholm (1988) and {\it Mat.Phys.Elect.Jour} {\bf 2} .}\cr FMRS& {J. Feldman, J.Magnen, V.Rivasseau, E.Trubowitz: An infinite volume expansion for many fermions Freen functions, {\it Helv.Phys.Acta.} {\bf 65} 679-721,(1992)}\cr G1& {G. Gallavotti: Twistless KAM tori, %Communications in Mathematical Physics {\it Comm. Math. Phys.} {\bf 164}, 145--156 (1994). }\cr G2& {G. Gallavotti: Renormalization group and ultraviolet stability for scalar fields via renormalization group methods {\it Rev. Mod. Phys.} {\bf 57} 471-562 (1985) }\cr % GM& {G. Gentile, V. Mastropietro: Methods for the analysis of the Lindstedt series for KAM tori and renormalizability in classical mechanics. A review with some applications, {\it Rev. Math. Phys.} {\bf 8}, 393--444 (1996). }\cr % H& {T. Holstein: Studies of polaron motion, part 1. The molecular-crystal model. {\it Ann. Phys.} {\bf 8}, 325--342, (1959). }\cr % JM& {R.A. Johnson, J. Moser: The rotation number for almost periodic potentials, {\it Commun. Math. Phys.} {\bf 84}, 403--438 (1982).}\cr % KL& {T. Kennedy, E.H. Lieb: An itinerant electron model with crystalline or magnetic long range order, {\it Physica A} {\bf 138}, 320--358 (1986). }\cr % %L& {E.H. Lieb: %A model for crystallization: a variation %of the Hubbard model, %{\it Physica A} {\bf 140}, 240--250 (1986). }\cr % LAMR& {Lee P.A., Rice P.M., Anderson P.W., Solid State Comm. 14,703,1974. }\cr % L& { Lee P.A. Nature 291,11-12,1981. }\cr % %LM& {J. L. Lebowitz, N. Macris: %Peierls instability and low temperature phases %of the Static Holstein Model: rigorous results, %{\it J. Stat. Phys.} {\bf 76}, 91--123 (1994). }\cr % ML& {V.Mastropietro: Interacting soluble Fermi systems in one dimension, {\it Nuovo Cim.B} {\bf 1}, 304-312, (1994). }\cr ML& {D.Mattis, E.Lieb: Exact solution of a many fermion system and its associated boson field, {\it Journ.Math.Phys.} {\bf 6}, 304-312, (1965). }\cr % MP& {J. Moser, J. P\"oschel: An extension of a result by Dinaburg and Sinai on quasi periodic potentials, {\it Comment. Math. Helv.} {\bf 59}, 39--85 (1984). }\cr % NO& {J.W. Negele, H. Orland: {\sl Quantum many-particle systems}, Addison-Wesley, New York, 1988. }\cr % PF& {L. Pastur, A. Figotin: {\sl Spectra of random and almost periodic operators}, Springer, Berlin, 1991. }\cr % P& {R.E. Peierls: {\sl Quantum theory of solids}, Clarendon, Oxford, 1955. }\cr % %T& {E.C. Titchmarsh: %{\sl Eigenfunctions expansions associated with second %order differential equations}, %Clarendon, Oxford, 1955. }\cr % %To& {M. Toda: %{\sl Theory of nonlinear lattices}, Springer, Berlin, 1958. }\cr } % \* \ciao {\bf BIBLIOGRAPHY} \vskip.5cm [P] Peierls,R.E. "Quantum theory of solids", Oxford University press,London, 1955 \vskip.5cm [F] Frohlich H, Proc. R. Soc., A223,296,1954 \vskip.5cm [LRA] Lee P.A., Rice P.M., Anderson P.W., Solid State Comm. 14,703,1974 \vskip [L] Lee P.A. Nature 291,11-12,1981 \vskip [A] Aubry S. in {\it Polarons and Bipolarons in hich $T_c$ superconductors and related materials}, editors Salye E, Alexandrov A., Liang W \ciao \ciao \section{The flow of the renormalization group} Contrary to the preceding sections, in the following the assumption of {\it spinless} fermions is crucial. We have seen at the end of sec.(4) that, given a constant $\tilde C$, {\it if} $\tilde C_h\leq \tilde C$ the kernels of the effective potential $V^{h-1}$ and the Beta function $\b^{h-1}$ are analytic as functions of their arguments, if $\bar\e_h<\e<{1\over \tilde C C}$ and $\bar\e_h$ is defined in eq.(\ref{cond})\cite{nota11}. Of course there is no reason for which $\tilde C_h\leq \tilde C$ for any $h$ so we call $h^*\equiv \tilde h(\tilde C)$ the scale such that $\tilde C_{h^*}\leq \tilde C$ and $\tilde C_{h^*-1}> \tilde C$; the scale $h^*$ can be computed once we know the $h$-dependence of $\sigma_h$. Then, if for $h\geq h^*$ it holds that $\bar\e_h<\e$ we have that the kernels of the effective potential $V^{h-1}$ and the Beta function $\b^{h-1}$ are well defined for $h\geq h^*$; to prove this, and compute $h^*$ we need informations on the $h$-dependence of the running coupling constants, what is provided by the study of the Beta function. It is possible to choose %$\nu_0$ (and so the the counterterm $\nu$ so that $|\nu_h|<\e$ for any $0\geq h\geq h^*$; in fact, given any sequence of running coupling constants verifying $\max_{i,k\geq h} |\bar v_{k}|\leq \e$, $\max_{k\geq h}|{Z_k\over Z_{k-1}}|\leq e^{\beta_1\e}$, $\nu_h$ obeys to the equation $\nu_{h-1}=\g \nu_h+\b_\nu^h$, with $|\b_\nu^h|0$. The fact that $\{ G^{1,h}_\l\}_2=0$, $\{ G^{1,h}_\d\}_2=0$ could generate a problem as one cannot exclude {\it a priori} there there is some non-vanishing term at some large order, and the evolution of the running coupling constants will depend critically on this unknown term. Fortunately it is possible to prove that: \begin{eqnarray}\label{van} &&\lim_{h\to-\i} G^{1,h}_\l(g,\d;\ldots;g,\d)=0\nn\\ &&\lim_{h\to-\i} G^{1,h}_\d(g,\d;\ldots;g,\d)=0\nn \end{eqnarray} % {\it i.e.} it is vanishing at {\it all} orders. In fact $G^{1,h}_\l$, $G^{1,h}_\d$ coincide, up to terms vanishing as $h\to-\i$ as $O(\g^h)$, with the correspondent quantities of the Luttinger model , and, by using the properties of the exact solution \cite{(M.L.),(M.)} one can prove that it is vanishing\cite{(B.G.),(B.G.M.),(B.G.P.S.),(B.M.1),nota13} Finally, as $|R^h_i|\leq K\e_h^2$, $i=\l,\d,\sigma,z$ and $|G_\l^{2,h}|,|G_\d^{2,h}|,|G_z^{2,h}|,|G_\t^{2,h}|, |G_\th^{2,h}|\leq K\e_h \tilde C_h$ one finds\cite{(B.M.2)}: \begin{eqnarray}\label{c1} -c_1\l^2&<|g_{h-1}-g_{0}|<&c_2\l^2\nn\\ \l\b_1 c_4 h&\leq\left|\log\left(\frac{\sigma_{h-1}} {\sigma_0}\right)\right| \leq&\l\b_1 c_3 h\nn\\ -\b_3 c_3\l^2h&\leq \log(|Z_{h-1}|)\leq&-\b_3 c_4\l^2h\\ -c_1|\l|& <|\t_{h-1}-\t_{0}|<&c_2|\l|\nn\\ -c_1|\l|& <|\th_{h-1}-\th_{0}|<&c_2|\l|\nn\\ -c_1|\l|& <|\d_{h-1}-\d_{0}|<&c_2|\l|\nn \end{eqnarray} for suitable constants $c_1, c_2>0$ and $c_4>c_3>0$, \hbox{\it i.e.\ } the flow is essentially described by the second order truncation of the beta function. We use that $\sum_{h=h^*}^0\tilde C_h\leq K$, if $K$ is a constant. >From eq.(\ref{c1}) it follows that it possible to choose $\l$ small enough so that the series for $V^{h-1}$ and $\b^{h-1}$ , if $h\geq h^*$, are convergent; moreover from eq.(\ref{c1}) it is possible to obtain an upper and lower bound for $h^*$: %All the above considerations hold for $\tilde C_h\leq tilde C$, %\hbox{\it i.e.\ } for $h\geq h^*$; from eq.(\ref{c1}) we see that \begin{equation}\label{h*} c_3 {\log_\g (\tilde C^{-1}u)\over 1-\l\b_1}\leq h^*\leq c_4{\log_\g (\tilde C^{-1} u)+1\over 1-\l\b_1} \end{equation} The above analysis is performed by approximating the anomalous propagator eq.(\ref{proan}) with the Luttinger model propagator by eq.(\ref{22}). This approximation is of course not reasonable for large $|h|$ corresponding to momenta negligible with respect to the $\sigma_h$ term due to the periodic potential. But for the scales $h< h^*$ the bound\cite{(B.M.2)}: \begin{equation}\label{x} |g^{(< h^*)}(x-y)|\le A{\g^{h^*}\over Z_{h^*}}\left({\g^{h^*} \over\sigma_{h^*}}\right) {C_N\over 1+\sigma_{h^*}^{N}|x-y|^N} \end{equation} holds and, from eq.(43),(44), $\left({\g^{h^*}\over\sigma_{h^*}}\right)\leq {K\over \tilde C}$, if $K$ is a constant. In other words the propagator $g^{(\le h^*)}_{\o,\o'}(x-y)$ obeys to the same bound of $g^{(h)}_{\o,\o'}(x-y)$ for $h\ge h^*$; our choice of $h^*$ is made just to obtain this. The integration of the scale between $-\i$ and $h^*$ is equivalent to the integration of a single scale in a not filled band theory, so that for the same considerations at the end of sec.(4) also the series expressing \begin{eqnarray}\label{78} \int\{{\cal D}\psi^{(< h^*)} &e&^{-\int dk \psi^{+(h^*$ is bounded by $(\bar\e_{h^*})^n \tilde C^n C_1^n$ while the $n$-th order term of the series eq.(\ref{78}) is bounded by $(\bar\e_{h^*-1})^n ({C_2\over \tilde C})^n$ so that there is a non ambiguous way to fix $\tilde C$ so that $\bar\e_{h^*}$ is the largest possible. The Schwinger function eq.(\ref{sf}) admits a perturbative expansion similar to the one of the partition function, whose convergence follows from the partition function expansion convergence \cite{(B.G.P.S.)}; it holds that \begin{eqnarray}\label{sfb} S(x,y)&=&S^{u.v.}(x,y)+ \sum_{h=h^*}^0\sum_{\o_1,\o_2}e^{i{\pi}(\o_1\xx-\o_2\yy)}\cdot\nn\\ &&\cdot(g^{(h)}_{\o_1,\o_2}(x-y) +\bar S^{(h)}_{\o_1,\o_2}(x,y)) \end{eqnarray} where we call $g^{(\leq h^*)}(x-y)$ simply $g^{(h^*)}(x-y)$ and the first addend is bounded by ${C_N\over 1+|x-y|^N}$; $\bar S^{(h)}_{\o_1,\o_2}(x,y)$ is given by a sum of Feynmann graphs similar to the ones contributing to the effective potential, see sec.3, with the difference that: \begin{enumerate} \item they have two external lines, to which the propagators $g^{(h_1)}_{\o_1,\o'}(x-x')$ or $g^{(h_2)}_{\o'',\o_2}(y-y')$ are associated, with $h_1,h_2\geq h$ and if $x',y'$ are respectively the coordinates of the vertex the external lines is entering in or coming out. %Note %that $\bar S^{(h)}_{\o_1,\o_2}(x-y)$ is translation invariant; the %reason is that, for the compact support properties (in momentum space) %of the propagators, if $\kk_1,\kk_2$ are the momenta associated to the %external lines, then $\kk_1-\kk_2=2n p_F$, with $n=0,\pm 1$. \item $h$ is the smallest scale of the propagators contributing to $\bar S^{(h)}_{\o_1,\o_2}(x-y)$ (and not, as for the graphs for the effective potential, the scale of the external lines). \item No $\RR$ acts on clusters containing the external lines. \end{enumerate} Note that $\sigma_{h^*}, Z_{h^*}$ depend on $\tilde C$, but it is easy to check that they can be written as $\sigma_{h^*}=\bar \sigma_{h^*}(1+\l f_1)$, $Z_{h^*}=\bar Z_{h^*}(1+\l f_2)$, with $|f_1|, |f_2|$ bounded by some constant and $\bar \sigma_{h^*}$, $\bar Z_{h^*}$ {\it independent} on $\tilde C$. Let us define \begin{equation} \h_1=-{\log(\bar Z_{h^*})\over \log u}\qquad 1+\h_2= {\log(\bar\sigma_{h^*})\over \log u} \end{equation} which are respectively $O(\l^2)$ and $O(\l)$, with ${\rm sign}(\h_2)={\rm sign}(\l)$. One can check\cite{(B.G.P.S.)} that $|\bar S^{(h)}_{\o,\o}(x,y)|\leq A \tilde\e {\g^h\over Z_h}{C_N\over 1+(\g^h|x-y|)^N}$ and $|\bar S^{(h)}_{\o,-\o}(x-y)|\leq A \tilde\e {\sigma_h\over Z_h} {C_N\over 1+(\g^h|x-y|)^N}$, if $\tilde\e= \max(u,u^{1+\h_1},|\l|)$ ; the extra factor ${\sigma_h\over\g^h}$ in the second bound follows from the fact that in the graph contributing to $\bar S^{(h)}_{\o,-\o}(x,y)$ there is at least a non diagonal propagator, which obeys, see the lines after eq.(\ref{22}), to a bound similar to the diagonal propagator but with a factor more ${\sigma_k\over\g^k}\leq {\sigma_h\over\g^h}$, $k\geq h$. We now prove the statements of sec.2. The bounds eq.(9)(\ref{bo2}) can be obtained by eq.(\ref{sfb}), as for $\g^{-h_x-1}<|x-y|\leq \g^{-h_x}$, $h_x>h^*$, if $|g^{(h)}_{\o_1,\o_2}(x-y) +\bar S^{(h)}_{\o_1,\o_2}(x,y)|\leq \bar g^{(h)}(x-y)$: \begin{eqnarray} \sum_{h=0}^{h^*} \bar g^{(h)}(x-y)&\leq& A_1\left[\sum_{h=h^*}^{h_x-1} {\g^h\over Z_h}+\sum_{h=h_x}^{0}{\g^h\over Z_h}{C_N\over \g^{Nh}|x-y|^N}\right]\leq\nn\\ &\leq& A_2\g^{h_x(1-\h_3)} \end{eqnarray} while for $|x-y|\geq \sigma_{h^*}^{-1}$: \begin{eqnarray} \sum_{h=0}^{h^*} \bar g^{(h)}(x-y)&\leq& {C_N\over |x-y|^N}\sum_{h=h^*+1}^1{\g^{-(N-1)h}\over Z_h} \nn\\ \leq& A_2{\g^{h^*}\over Z_{h^*}}{C_N\over (\g^{h^*}|x-y|)^N} \end{eqnarray} %The occupation number discontinuity can be obtained noting that, from the %theory of the Bloch waves, $\phi(p_F^+,\xx)=\cos(p_F\xx)+O(u)$ and %$\phi(p_F^-,\xx)=i\sin(p_F\xx)+O(u)$ so that the occupation number %discontinuity eq.(\ref{Z}) can be written as: %$$Z^{-1}={1\over L}\int %d\xx d\yy [\cos(p_F(\xx+\yy)+r(u)]S^{L,\b}(x,y)$$ %with $r(u)=O(u)$. %By inserting in the above expression eq.(\ref{sfb}) we obtain several %terms that we can bound in the following way, if $K_i$ are positive %constants: %\begin{eqnarray} %&&{r(u)\over L}\int d\xx d\yy \sum_{h=h^*}^0\sum_{\o_1,\o_2} %|g^{(h)}_{\o_1,\o_2}(x,y) %+\bar S^{(h)}_{\o_1,\o_2}(x,y))|\leq\nn\\ %&&\qquad\leq K_1 u \sum_{h=h^*}^0 {1\over Z_h}\leq K_2 u\log u\leq K_3 u^{\h_1} %\end{eqnarray} %\begin{eqnarray} %&&{1\over L}\int d\xx d\yy \cos(p_F(\xx+\yy)) %\sum_{h=h^*}^0\sum_{\o_1,\o_2} %|\bar S^{(h)}_{\o_1,\o_2}(x,y))|\leq\nn\\ %&&\qquad \leq K_4 \sum_{h=h^*}^0 {\sigma_h\over Z_h \g^h}\leq \tilde\e K_5 %u^{\h_1} %\end{eqnarray} %in which we take into account that in the sum only the terms %$\o_1=-\o_2$ survive; %\begin{eqnarray} %&&{1\over L}\int d\xx d\yy %\cos(p_F(\xx+\yy))g^{(h)}_{\o_1,\o_2}(x-y)=\\ %&&\quad=\int dk_0 %\sum_{h=h^*}^0\tilde f_h(0,k_0) {1\over Z_h} %(\GG_h(0,k_0)^{-1})_{\o,-\o}\nn %\end{eqnarray} %verifies the bound eq.(12); finally: %\begin{equation} %{1\over L}\int d\xx d\yy [\cos(p_F(\xx+\yy)+r(u)] %S^{u.v.}(x,y) %\leq K_7 u %\end{equation} Let us prove the decomposition eq.(\ref{prep}). By diagonalizing the quadratic form $\hat g^{(h)}(x,y)=\sum_{\o_1,\o_2}e^{i{\pi}(\o_1\xx-\o_2\yy)} g^{(h)}_{\o_1,\o_2}(x-y)$ it is possible to see that: $$\hat g^{(h)}(x,y)=\int dk' e^{ik'(x-y)} {\tilde f_h(k')\over Z_h} [{F_{xy}(k',\sigma_h)\over A+B}+$$ $$+{F_{xy}(-k',-\sigma_h)\over A-B}]$$ where $$F_{xy}(k',\sigma_h)=\hat \phi(k',\xx,\sigma_h)\hat \phi(k',-\yy,\sigma_h)$$ $$\hat \phi(k',\xx,\sigma_h)={1\over\sqrt{2B}}[\sqrt{B-C}e^{i p_F\xx}$$ $$- {e^{-i p_F\xx}\over \sqrt{B-C}}]$$ and $A=-i k_0+\kk^{'2}$, $B=\sqrt{(C^2+\s_h^2)}$ and $C=2\pi\kk'$. We can rewrite the above integral in terms of the $k$ variable. Recall that, if $\o={\rm sign}(\kk)$ and $f_h(k')\not =0$, then $k=\o p_F+\kk'$. Hence \begin{eqnarray}\label{86} &&\hat g^h(x-y)=\sum_{\o_1,\o_2}e^{i{\pi}(\o_1\xx-\o_2\yy)} g^h_{\o_1,\o_2}(x-y)=\nn\\ &&=\int dk {\tilde f_h(k)\over Z_h(k)} {\hat\phi(\kk,\xx,\sigma_h))\hat\phi(\kk,-\yy,\sigma_h) e^{ik_0(x_0-y_0)} \over -ik_0-(\hat\e(k,\sigma_h)-\pi^2)} \end{eqnarray} where: \begin{eqnarray}\label{bl1} \hat\e(k,\sigma_h)&=&\left(|\kk|-{\pi}\right)^2+\\ &+&2\pi{\rm sign}\left(|\kk|-{\pi}\right)\sqrt{ \left(|\kk|-{\pi}\right)^2+\sigma_h^2}+\pi^2\nn \end{eqnarray} and \begin{equation}\label{bl} \hat\phi(\kk,\xx,\sigma_h))=e^{i\kk\xx} u(\kk,\xx,\sigma_h) \end{equation} \begin{eqnarray} u(\kk,\xx,\sigma_h)&=&e^{-i{\rm sign}(\kk)\pi\xx}\cdot\nn\\ &&\cdot\left[\cos{(\pi\xx)}\sqrt{1+ {{\rm sign}(|\kk|-{\pi})\sigma_h \over\sqrt{ (|\kk|-{\pi})^2+\sigma_h^2}}}+\right.\\ &&\left.+i{\rm sign}(\kk)\sin{(\pi\xx)}\sqrt{1- {{\rm sign}(|\kk|-{\pi})\sigma_h \over\sqrt{ (|\kk|-{\pi})^2+\sigma_h^2}}}\,\right]\nn \end{eqnarray} In Ref.\ \onlinecite{(B.M.2)} by a careful analysis of the Bloch waves it is shown that \begin{eqnarray} \label{am1} &&|\hat\phi(\kk,\xx,u)-\phi(\kk,\xx,u)|=O(u)\nn\\ &&|\hat\e(\kk,u)-\e(\kk,u)|=O(u^2) \end{eqnarray} where $\phi(\kk,\xx,u)$ are the Bloch waves \hbox{\it i.e.\ } the solutions of eq.(4) and $\e(\kk,u)$ the dispersion relation. The fourth statement of the theorem then follows. Finally the lower bound on the spectral gap can be obtained by repeating the computations in the preceding sections using an analytic partition of the unity instead of a $C^\infty$ one, see sec.3. In fact in this way one can prove that the Fourier transform of the 2-point Schwinger function in bounded as a function of $k_0$ in a strip of the imaginary plane of width ${\hat u(p_F)\over 2}$. We preferred a $C^\infty$ decomposition, as it makes the exposition more readable and intuitive, but there should be no problem to use an analytic decomposition. \vskip4.cm \pagina \centerline{\titolo References} \* \halign{\hbox to 1.2truecm {[#]\hss} & \vtop{\advance\hsize by -1.25 truecm \0#}\cr A& {S. Aubry in Polarons and Bipolarons in hich $T_c$ superconductors and related materials}, editors Salye E, Alexandrov A., Liang W. }\cr AAR& {S. Aubry, G. Abramovici, J. Raimbaut: Chaotic polaronic and bipolaronic states in the adiabatic Holstein model, {\it J. Stat. Phys.} {\bf 67}, 675--780 (1992). }\cr % BLT& {J. Belissard, R. Lima, D. Testard: A metal-insulator transition for almost Mathieu model, {\it Comm. Math. Phys.} {\bf 88}, 207--234 (1983). }\cr % D& {H. Davenport: {\sl The Higher Arithmetic}, Dover, New York, 1983.}\cr % DS& {E.I. Dinaburg, Ya.G. Sinai: On the one dimensional Schroedinger equation with a quasiperiodic potential, {\it Funct. Anal. and its Appl.} {\bf 9}, 279--289 (1975). }\cr % E& {L.H. Eliasson: Floquet solutions for the one dimensional quasi periodic Schroedinger equation, {\it Comm. Math. Phys.} {\bf 146}, 447--482 (1992). }\cr % G& {G. Gallavotti: Twistless KAM tori, {\it Comm. Math. Phys.} {\bf 164}, 145--156 (1994). }\cr % GM& {G. Gentile, V. Mastropietro: Methods for the analysis of the Lindstedt series for KAM tori and renormalizability in classical mechanics. A review with some applications, {\it Rev. Math. Phys.} {\bf 8}, 393--444 (1996). }\cr % H& {T. Holstein: Studies of polaron motion, part 1. The molecular-crystal model. {\it Ann. Phys.} {\bf 8}, 325--342, (1959). }\cr % JM& {R.A. Johnson, J. Moser: The rotation number for almost periodic potentials, {\it Commun. Math. Phys.} {\bf 84}, 403--438 (1982).}\cr % KL& {T. Kennedy, E.H. Lieb: An itinerant electron model with crystalline or magnetic long range order, {\it Physica A} {\bf 138}, 320--358 (1986). }\cr % %L& {E.H. Lieb: %A model for crystallization: a variation %of the Hubbard model, %{\it Physica A} {\bf 140}, 240--250 (1986). }\cr % LMR& {Lee P.A., Rice P.M., Anderson P.W., Solid State Comm. 14,703,1974. }\cr % L& Lee P.A. Nature 291,11-12,1981. }\cr % LM& {J. L. Lebowitz, N. Macris: Peierls instability and low temperature phases of the Static Holstein Model: rigorous results, {\it J. Stat. Phys.} {\bf 76}, 91--123 (1994). }\cr % MP& {J. Moser, J. P\"oschel: An extension of a result by Dinaburg and Sinai on quasi periodic potentials, {\it Comment. Math. Helv.} {\bf 59}, 39--85 (1984). }\cr % NO& {J.W. Negele, H. Orland: {\sl Quantum many-particle systems}, Addison-Wesley, New York, 1988. }\cr % PF& {L. Pastur, A. Figotin: {\sl Spectra of random and almost periodic operators}, Springer, Berlin, 1991. }\cr % P& {R.E. Peierls: {\sl Quantum theory of solids}, Clarendon, Oxford, 1955. }\cr % %T& {E.C. Titchmarsh: %{\sl Eigenfunctions expansions associated with second %order differential equations}, %Clarendon, Oxford, 1955. }\cr % %To& {M. Toda: %{\sl Theory of nonlinear lattices}, Springer, Berlin, 1958. }\cr } % \* \ciao \vskip4.cm {\bf BIBLIOGRAPHY} \vskip.5cm [P] Peierls,R.E. "Quantum theory of solids", Oxford University press,London, 1955 \vskip.5cm [F] Frohlich H, Proc. R. Soc., A223,296,1954 \vskip.5cm [LRA] Lee P.A., Rice P.M., Anderson P.W., Solid State Comm. 14,703,1974 \vskip [L] Lee P.A. Nature 291,11-12,1981 \vskip [A] Aubry S. in {\it Polarons and Bipolarons in hich $T_c$ superconductors and related materials}, editors Salye E, Alexandrov A., Liang W \ciao At the end, note that our results are independent on the constant $\tilde C$, which is arbitrary; in fact $\h_1,\h_2$ as well as the constants entering in the bounds of the theorem do not depend on $\tilde C$, which only affects the convergence radius of the series \hbox{\it i.e.\ } there is an {\it optimal} way to choose it; analogue considerations can be made for the parameter $\g$. \section{Conclusive remarks} A renormalization group treatment allows us to clarify this intuition of {\it equivalence} between different models in this context. Namely we can consider two models {\it equivalent} if their Beta functions coincide up to terms $O(\g^h)$. This means that, see eq.(48), the critical indices coincides at least at the first order. In this sense we think that the above analysis makes clear that the model with Hamiltonian eq.(1) in the spinless case and at $p_F={q\pi\over a}$ is equivalent to the massive Luttinger model eq.(\ref{ml}); one can repeat for this model the same renormalization group analysis almost without changes: the only (trivial) differences are that the dispersion relation is linear (so for instance the terms quadratic in $\kk$ in eq.(37) are not present) and that $\n_h=\t_h=\th_h\equiv 0$ for simmetry reasons. In other words the relation between the filled band model eq.(1) and the massive Luttinger model eq.(15) is the same of the one\cite{(B.G.M.)} between the $u=0$ model eq.(1) and the Luttinger model. Another equivalent model is a relativistic quantum field model, the Yukawa$_2$ model, describing a relativistic massive fermion in $d=1$. One can easily convince himself of this fact noting that the anomalous propagator eq.(\ref{proan}) can be written as \begin{equation}\label{19a} g^{(h)}(x-y)=g^{(h)}_{Y}(x-y)+ C_1^h(x-y) \end{equation} where \begin{equation}\label{you} g^{(h)}_Y(x-y)={1\over Z_h}\int dk \tilde f_h(k) e^{ik(x-y)} {\not k+\sigma_h I\over k^2+\sigma_h^2} \end{equation} with $\not k=i k_0\g_0+2\pi\kk \g^1$, if $\g_0,\g_1$ are the $\g$-matrices for a relativistic $d=1$ Fermi field with light velocity $2\pi$; $g_{Y}^{(h)}$ is dominant as $|C_1^{(h)}(x-y)|\leq A{\g^{2h}\over Z_h}{C_N\over 1+(\g^h|x-y|)^N}$ with $A$ a suitable constant matrix, \hbox{\it i.e.\ } it obeys to a similar bound with an extra factor $\g^h$; $g^h_{Y}$ is just the infrared propagator at scale $h$ of the Yukawa$_2$ model ($\sigma_h$ is just the fermion mass at scale $h$) and it is trivial to check that the two models are equivalent according to the above definition. As a conclusion let us discuss briefly the spinning case. The only difference with the spinless case treated here is that there are more running coupling constants corresponding to quartic monomials in the fields, and the second order beta function is in this case much more complex. One can verify that the dynamical system correspondent to the second order Beta function has a variety of behaviors; it seems that only for special choices of the potential everything might stay unchanged, with respect to the spinless case, while in general new ideas seem necessary. At the end, note that our results are independent on the constant $\tilde C$, which is arbitrary; in fact $\h_1,\h_2$ as well as the constants entering in the bounds of the theorem do not depend on $\tilde C$, which only affects the convergence radius of the series \hbox{\it i.e.\ } there is an {\it optimal} way to choose it; analogue considerations can be made for the parameter $\g$. \section{Conclusive remarks} A renormalization group treatment allows us to clarify this intuition of {\it equivalence} between different models in this context. Namely we can consider two models {\it equivalent} if their Beta functions coincide up to terms $O(\g^h)$. This means that, see eq.(48), the critical indices coincides at least at the first order. In this sense we think that the above analysis makes clear that the model with Hamiltonian eq.(1) in the spinless case and at $p_F={q\pi\over a}$ is equivalent to the massive Luttinger model eq.(\ref{ml}); one can repeat for this model the same renormalization group analysis almost without changes: the only (trivial) differences are that the dispersion relation is linear (so for instance the terms quadratic in $\kk$ in eq.(37) are not present) and that $\n_h=\t_h=\th_h\equiv 0$ for simmetry reasons. In other words the relation between the filled band model eq.(1) and the massive Luttinger model eq.(15) is the same of the one\cite{(B.G.M.)} between the $u=0$ model eq.(1) and the Luttinger model. Another equivalent model is a relativistic quantum field model, the Yukawa$_2$ model, describing a relativistic massive fermion in $d=1$. One can easily convince himself of this fact noting that the anomalous propagator eq.(\ref{proan}) can be written as \begin{equation}\label{19a} g^{(h)}(x-y)=g^{(h)}_{Y}(x-y)+ C_1^h(x-y) \end{equation} where \begin{equation}\label{you} g^{(h)}_Y(x-y)={1\over Z_h}\int dk \tilde f_h(k) e^{ik(x-y)} {\not k+\sigma_h I\over k^2+\sigma_h^2} \end{equation} with $\not k=i k_0\g_0+2\pi\kk \g^1$, if $\g_0,\g_1$ are the $\g$-matrices for a relativistic $d=1$ Fermi field with light velocity $2\pi$; $g_{Y}^{(h)}$ is dominant as $|C_1^{(h)}(x-y)|\leq A{\g^{2h}\over Z_h}{C_N\over 1+(\g^h|x-y|)^N}$ with $A$ a suitable constant matrix, \hbox{\it i.e.\ } it obeys to a similar bound with an extra factor $\g^h$; $g^h_{Y}$ is just the infrared propagator at scale $h$ of the Yukawa$_2$ model ($\sigma_h$ is just the fermion mass at scale $h$) and it is trivial to check that the two models are equivalent according to the above definition. As a conclusion let us discuss briefly the spinning case. The only difference with the spinless case treated here is that there are more running coupling constants corresponding to quartic monomials in the fields, and the second order beta function is in this case much more complex. One can verify that the dynamical system correspondent to the second order Beta function has a variety of behaviors; it seems that only for special choices of the potential everything might stay unchanged, with respect to the spinless case, while in general new ideas seem necessary. \vskip4.cm {\bf BIBLIOGRAPHY} \vskip.5cm [P] Peierls,R.E. "Quantum theory of solids", Oxford University press,London, 1955 \vskip.5cm [F] Frohlich H, Proc. R. Soc., A223,296,1954 \vskip.5cm [LRA] Lee P.A., Rice P.M., Anderson P.W., Solid State Comm. 14,703,1974 \vskip [L] Lee P.A. Nature 291,11-12,1981 \vskip [A] Aubry S. in {\it Polarons and Bipolarons in hich $T_c$ superconductors and related materials}, editors Salye E, Alexandrov A., Liang W \ciao \vskip2.truecm \centerline{\titolo 2. Multiscale decomposition and anomalous integration} \*\numsec=2\numfor=1 We start by evalutating the partition function $$\int P(d\psi) e^{\VV(\psi)}\; , \Eq(2.1)$$ It is convenient to decompose the Grassmanian integration $P(d\psi)$ into a finite product of independent integrations: % $$ P(d\psi)=\prod_{h=h_\b }^1 P(d\psi^{(h)}) \; ,\Eq(2.1) $$ where $h_\b >-\io$ will be defined below (before \equ(2.9)) This can be done by setting % $$ \psi_\kk^{\pm}=\bigoplus_{h=h_\b }^1\psi_\kk^{(h)\pm} \; , \qquad \hat g_\kk=\sum_{h=h_\b }^1 \hat g^{(h)}_\kk \; , \Eq(2.2) $$ % where $\psi^{(h)\pm}_\kk$ are families of Grassmanian fields with propagators $\hat g^{(h)}_\kk$ which are defined in the following way. We introduce a {\sl scaling parameter} $\g>1$ and a function $\c(\kk') \in C^{\io}(\TTT^1\times \RRR)$, $\kk'=(k',k_0)$, such that, if $|\kk'|\=\sqrt{k_0^2+||k'||_{\ttt^1}^2}$: % $$ \c(\kk') = \c(-\kk') = \cases{ 1 & if $|\kk'| a_0\; ,$\cr}\Eq(2.3)$$ % where $a_0=\min \{p_F/2, (\p-p_F)/2 \}$. This definition is such that the supports of $\c(k-p_F,k_0)$ and $\c(k+p_F,k_0)$ are disjoint and the $C^\io$ function on $\TTT^1\times \RRR$ % $$\hat f_1(\kk) \= 1- \c(k-p_F,k_0) - \c(k+p_F,k_0) \Eq(2.4)$$ % is equal to $0$, if $||k|-p_F||_{\ttt^1}^2 +k_0^2t_0 \g^{h+1}$, and $f_h(\kk')= 1$, if $|\kk'| =t_0\g^h$. We finally define, for any $h\le 0$: % $$ \hat f_h(\kk) = f_h(k-p_F,k_0) + f_h(k+p_F,k_0)\; ,\Eq(2.7) $$ % $$ \hat g^{(h)}_\kk \= { \hat f_h(\kk) \over -ik_0+\cos p_F -\cos k} \; . \Eq(2.8) $$ Note that, if $\kk\in {\cal D}_{L,\b}$, then $|k_0|\ge \p/\b$, implying that $\hat f_h(\kk)=0$ for any $h< h_\b = \min \{h:t_0\g^{h+1} > \p/\b \}$. Hence, if $\kk\in {\cal D}_{L,\b}$, the definitions \equ(2.4) and \equ(2.7), together with the identity \equ(2.6), imply that % $$1=\sum_{h=h_\b }^1 \hat f_h(\kk) \; .\Eq(2.9)$$ The definition \equ(2.7) implies also that, if $h\le 0$, the support of $\hat f_h(\kk)$ is the union of two disjoint sets, $A_h^+$ and $A_h^-$. In $A_h^+$, $k$ is strictly positive and $||k-p_F||_{\ttt^1}\le a_0\g^h \le a_0$, while, in $A_h^-$, $k$ is strictly negative and $||k+p_F||_{\ttt^1}\le a_0\g^h$. Therefore, if $h\le 0$, we can write $\psi^{(h)\pm}_{\kk}$ as the sum of two independent Grassmanian variables $\psi_{\kk,\o}^{(h)\pm}$ with propagator % $$ \int P(d\psi^{(h)})\,\psi^{(h)-}_{\kk_1,\o_1} \psi^{(h)+}_{\kk_2,\o_2} = L\b \d_{\kk_1,\kk_2}\,\d_{\o_1,\o_2}\, \hat g^{(h)}_{\o_1}(\kk_1) \; , \Eq(2.10)$$ % so that % $$ \psi^{(h)\pm}_{\kk}=\bigoplus_{\o=\pm 1}\psi^{(h)\pm}_{\kk,\o} \; , \qquad \hat g^{(h)}_\kk=\sum_{\o=\pm 1} \hat g^{(h)}_\o(\kk) \; , \Eq(2.11) $$ % $$ \hat g^{(h)}_\o(\kk)={\theta(\o k) \, \hat f_h(\kk) \over -ik_0 + \cos p_F - \cos k }\; , \Eq(2.12) $$ % where $\theta(k)$ is the (periodic) step function. If $\o k> 0$, we will write in the following $k=k'+\o p_F$, where $k'$ is the {\sl momentum measured from the Fermi surface} and we shall define, if $h\le 0$, % $$ \tilde g^{(h)}_{\o}(\kk') \= \hat g^{(h)}_\o(\kk)= {f_h(\kk') \over -i k_0+v_0 \o\sin k'+(1-\cos k')\cos p_F } \; , \Eq(2.13) $$ % where $v_0=\sin p_F$. In order to simplify the notation, it will be useful in the following to denote $\hat g^{(1)}_\kk$ also as $\tilde g^{(1)}_+(\kk')$, with $k=k'+p_F$. This kind of decomposition is completely standard in the theory of the $d=1$ Fermi system; we repeat it here only for clarity. We define $$e^{\VV^{(0)}(\psi^{(\le 0)})}=\int P(d\psi^{(1)}) e^{\VV^{(0)}(\psi^{(1)}+\psi^{(\le 0)})}$$ It is possible to prove that $$-\VV^{(0)}(\psi^{(\le 0)})=-\l\int d\kk_1 d\kk_2 d\kk_3 d\kk_4 \hat v(\kk_1-\kk_2) \psi^{(\le 0)+}_{\kk_1} \psi^{(\le 0)-}_{\kk_2}\psi^{(\le 0)+}_{\kk_3}\psi^{(\le 0)-}_{\kk_4}\d(\kk_1+\kk_3-\kk_2-\kk_4)+$$ $$+\int d\kk (\nu+F(\kk)) psi^{(\le 0)+}_{\kk}\psi^{(\le 0)-}_{\kk} +u\sum_{m=1}^\io \hat\phi_m \int d\kk [\psi^{(\le 0)+}_{\kk} \psi^{(\le 0)-}_{\kk+2m \pp_L}+\psi^{(\le 0)+}_{\kk} \psi^{(\le 0)-}_{\kk-2m \pp_L}+$$ $$\sum_{m=1}^\io\sum_{n=0}^\io \int d\kk_1 d\kk_{m}\psi^{(\le 0),\s_1}_{\kk_1}... \psi^{(\le 0),\s_m}_{\kk_m} W(\kk_1,...,\kk_m;z)\d(\sum_{i=1}^n\s_i\kk_i+2n\pp_L)$$ where $|F(\kk)|\le C|\l|$ and the kernels $W(\kk_1,...,\kk_m;z)$ are $C^\io$ bounded functions such that $W_{m,n}=W_{m,-n}$ and $|W_{m,n}|\le C^m z^{\max(2,m/2-1)}$ if $z=Max(|\l|,u,|\nu|)$. The proof of the above statement is quite standard, as one can repeat word by word, up to trivial modication, the corresponding analysis for a similar model in [BGPS]. We perform now the infrared integration $$\int P(d\psi^{(\le 0)})e^{\VV(\psi^{(\le 0)})}$$ It is convenient, for reasons which will be clear below, to split $\VV$ in a {\it relevant} and {\it irrelevant} part $\VV=\LL\VV+\RR\VV$ where $\LL$, the {\it localization operator}, is defined in the following way. $${1\over (L\b)^m}\sum_{\kk'_1,...,\kk'_m}W_{m,n}(\kk'_1+\o_1 p_F,..., \kk'_m+\o_m p_F)\prod_{i=1}^m\psi^{(\le 0)\s_i}_{\kk'_i+\o_i p_F,\o_i} \d(\sum_{i=1}^m \s_i(\kk'_i+\o_i p_F)+2np_L)=0\qquad {\rm if} m>4$$ \vskip.5cm $${1\over (L\b)^4}\sum_{\kk'_1,...,\kk'_4} W_{4,n}(\kk'_1+\o_1 p_F,..., \kk'_4+\o_4 p_F)$$ $$\psi^{(\le 0)+}_{\kk'_1+\o_1 p_F,\o_1} \psi^{(\le 0)-}_{\kk'_2+\o_2 p_F,\o_2} \psi^{(\le 0)+}_{\kk'_3+\o_3 p_F,\o_3}\psi^{(\le 0)-}_{\kk'_4+\o_4 p_F, \o_4} \d(\sum_{i=1}^4 \s_i(\kk'_i+\o_i p_F)+2np_L)=\Eq(loc1)$$ $$\d_{(\o_1-\o_2+\o_3-\o_4)p_F+2n p_L} {1\over (L\b)^4}\sum_{\kk'_1,...,\kk'_4} W_{m,n}(\o_1 p_F,..., \o_4 p_F)\psi^{(\le 0)+}_{\kk'_1+\o_1 p_F,\o_1}$$ $$\psi^{(\le 0)-}_{\kk'_2+\o_2 p_F,\o_2} \psi^{(\le 0)+}_{\kk'_3+\o_3 p_F,\o_3} \psi^{(\le 0)-}_{\kk'_4+\o_4 p_F,\o_4} \d(\sum_{i=1}^4 \s_i \kk'_i)$$ \vskip.5cm $${1\over (L\b)^2}\sum_{\kk'_1,\kk'_2}W_{2,n}(\kk'_1+\o_1 p_F, \kk'_2+\o_2 p_F)\psi^{(\le 0)+}_{\kk'_1+\o_1 p_F,\o_1} \psi^{(\le 0)-}_{\kk'_2+\o_2 p_F,\o_2} \d(\sum_{i=1}^2 \s_i(\kk'_i+\o_i p_F)+2np_L)=$$ $$\d_{(\o_1-\o_2)p_F+2n p_L}{1\over (L\b)}\sum_{\kk'} [W_{2,n}(\o_1 p_F,\o_2 p_F)+\Eq(loc2)$$ $$+E(k'+\o_1 p_F)\partial_{\kk} W_{2,n}(\o_1 p_F,\o_2 p_F)+k^0 \partial_{k_0}W_{2,n}(\o_1 p_F,\o_2 p_F)] \psi^{(\le 0)+}_{\kk'+\o_1 p_F,\o_1} \psi^{(\le 0)-}_{\kk'+\o_2 p_F,\o_2}$$ \vskip.5cm We can write then the relevant part of the effective potential in the following way: $$-\LL\VV^{(0)}=n_0 F_\nu^{(\le 0)}+s_0 F_\s^{(\le 0)}+z_0 F_\z^{(\le 0)}+a_0 F_\a^{(\le 0)}+i_0 F_\iota^{(\le 0)}+t_0 F_\t^{(\le 0)}+l_0 F_\l^{(\le 0)}$$ where. $$F_\nu^{(\le 0)}=\sum_\o {1\over (L\b)}\sum_{\kk'} \psi^{(\le 0)+}_{\kk'+\o p_F,\o} \psi^{(\le 0)-}_{\kk'+\o p_F,\o}$$ $$F_\s^{(\le 0)}=\sum_\o {1\over (L\b)}\sum_{\kk'} \psi^{(\le 0)+}_{\kk'+\o p_F,\o} \psi^{(\le 0)-}_{\kk'-\o p_F,-\o}$$ $$F_\a^{(\le 0)}=\sum_\o {1\over (L\b)}\sum_{\kk'} E(k'+\o p_F) \psi^{(\le 0)+}_{\kk'+\o p_F,\o} \psi^{(\le 0)-}_{\kk'+\o p_F,\o}$$ $$F_\z^{(\le 0)}=\sum_\o {1\over (L\b)}\sum_{\kk'} (-i k_0) \psi^{(\le 0)+}_{\kk'+\o p_F,\o} \psi^{(\le 0)-}_{\kk'+\o p_F,\o}$$ $$F_\iota^{(\le 0)}=\sum_\o {1\over (L\b)}\sum_{\kk'} E(k'+\o p_F) \psi^{(\le 0)+}_{\kk'+\o p_F,\o} \psi^{(\le 0)-}_{\kk'-\o p_F,-\o}$$ $$F_\t^{(\le 0)}=\sum_\o {1\over (L\b)}\sum_{\kk'} (-i k_0) \psi^{(\le 0)+}_{\kk'+\o p_F,\o} \psi^{(\le 0)-}_{\kk'-\o p_F,-\o}$$ $$F_\l^{(\le 0)}={1\over (L\b)^4}\sum_{\kk'_1,...,\kk'_4} \psi^{(\le 0)+}_{\kk'_1+p_F,1} \psi^{(\le 0)+}_{\kk'_1-p_F,-1} \psi^{(\le 0)-}_{\kk'_3+p_F,1} \psi^{(\le 0)-}_{\kk'_4-p_F,-1}$$ We have to perform the following integration % $$\int P_{Z_0}(d\psi^{\le 0}) \, e^{\VV^{(0)}(\sqrt{Z_0}\psi^{\le 0})} \; , \Eq(3.6) $$ % where $Z_0=1$ and $P_{Z_0}(d\psi^{(\le 0)})$ is the Grassmanian integration with propagator % $$ \eqalign{ & g^{(\le 0)}(\xx;\yy) = \sum_{\o,\o'=\pm1} {1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \, {1\over Z_0}e^{-i\kk'\cdot(\xx-\yy)}\,e^{-i(\o x - \o' y)p_F} \, \tilde g^{(\le 0)}_{\o,\o'}(\kk') \; , \cr & \tilde g^{(\le 0)}_{\o,\o'}(\kk') = \d_{\o,\o'} \, \tilde g^{(\le 0)}_{\o}(\kk') \; , \cr} \Eq(3.7) $$ with % $$ \tilde g^{(\le 0)}_\o(\kk') = {C_0^{-1}(\kk') \over -ik_0 + (1-\cos k') \cos p_F + v_0\o \sin k' } \; , \qquad C_h^{-1}(\kk')=\sum_{j=h_\b}^h f_{j}(\kk') \; , \Eq(3.8) $$ % see \equ(2.13), \equ(2.15). We write % $$\int P_{Z_0}(d\psi^{(\le 0)}) \, e^{\VV^0(\sqrt{Z_0}\psi^{(\le 0)})} = {1 \over \NN_0}\int \tilde P_{Z_{-1}}(d\psi^{(\le 0)}) \, e^{\tilde \VV^{(0)}(\sqrt{Z_0}\psi^{\le 0})} \; , \Eq(3.9) $$ % where $\NN_0$ is a suitable constant and, again up to a constant, % $$ \eqalign{ \tilde P_{Z_{-1}}(d\psi^{(\le 0)}) = & \prod_{\kk}\prod_{\o=\pm1} d\psi^{(\le 0)+}_{\kk'+\o\pp_F,\o} d\psi^{(\le 0)-}_{\kk'+\o\pp_F,\o} \cr \exp \Big\{ &-\sum_{\o=\pm1} {1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \, C_0(\kk') Z_{-1}(\kk') \Big[\Big( -ik_0-(\cos k'-1)\cos p_F +\o v_0\sin k' \Big) \cr & \psi^{(\le0)+}_{\kk'+\o\pp_F,\o} \psi^{(\le0)-}_{\kk'+\o\pp_F,\o} - \sigma_{-1}(\kk') \, \psi^{(\le0)+}_{\kk'+\o\pp_F,\o} \psi^{(\le0)-}_{\kk'-\o\pp_F,-\o} \Big] \Big\} \; , \cr} \Eq(3.10) $$ % with $Z_{-1}(k')\sigma_{-1}(\kk')=C_0^{-1}(\kk')\,s_0$, $Z_{-1}(\kk')=Z_0(1+C_0^{-1}(\kk')z_0$ and $\tilde \VV^{(0)}=\LL \tilde \VV^{(0)}+(1-\LL) \VV^{(0)}$, if % $$ -\LL\tilde \VV^{(0)}(\sqrt{Z_0}\psi)= n_0 F_\nu^{(\le 0)}+(a_0-z_0) F_\a^{(\le 0)}+i_0 F_\iota^{(\le 0)}+t_0 F_\t^{(\le 0)}+l_0 F_\l^{(\le 0)} \; . \Eq(3.11) $$ The r.h.s of \equ(3.9) can be written as % $$ {1 \over \NN_0}\int P_{Z_{-1}}(d\psi^{(\le -1)}) \int \tilde P_{Z_{-1}}(d\psi^{(0)}) \, e^{\tilde \VV^{(0)}(\sqrt{Z_0}\psi^{(\le 0)})} \; , \Eq(3.12) $$ % where $ P_{Z_{-1}}(d\psi^{(\le -1)})$ and $\tilde P_{Z_{-1}}(d\psi^{(0)})$ are given by \equ(3.10) with $Z^{-1}(\kk')$ replaced by $Z^{-1}(0)\equiv Z^{-1}$ and $C_0(\kk')$ replaced with $C_{-1}(\kk')$ and $\tilde f_0^{-1}(\kk')$ respectively, if $$\tilde f_0(\kk')=Z_{-1}[{C_0^{-1}(\kk')\over Z_{-1}(\kk')}- {C_{-1}^{-1}(\kk')\over Z_{-1}}]$$ and $\psi^{(\le 0)}$ replaced with $\psi^{(\le -1)}$ and $\psi^{(0)}$ respectively. The Grassmanian integration $\tilde P_{Z_{-1}}(d\psi^{(0)})$ has propagator % $$ g^{(0)}(\xx;\yy) = \sum_{\o,\o'=\pm1} e^{-i(\o x - \o' y)p_F}\, g^{(0)}_{\o,\o'}(\xx;\yy) \; , \Eq(3.13) $$ % if % $$ g^{(0)}_{\o,\o'}(\xx;\yy)\=\int \tilde P_{Z_{-1}}(d\psi^{(0)})\, \psi^{(0)-}_{\xx,\o}\psi^{(0)+}_{\yy,\o'} \Eq(3.14) $$ % is given by % $$ g^{(0)}_{\o,\o'}(\xx;\yy)={1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \, e^{-i\kk'\cdot(\xx-\yy)}{\tilde f_0(\kk')\over Z_{-1}} [T_{0}^{-1}(\kk')]_{\o,\o'} \; , \Eq(3.15) $$ % where the $2\times2$ matrix $T_{0}(\kk')$ has elements % $$ \cases{ [T_{0}(\kk')]_{1,1} = \left(-ik_0-(\cos k'-1)\cos p_F+ v_0\sin k' \right) \; , & \cr [T_{0}(\kk')]_{1,2} = [T_{0}(\kk')]_{2,1} = - \sigma_{-1}(\kk') \; , & \cr [T_{0}(\kk')]_{2,2} = \left(-ik_0-(\cos k'-1)\cos p_F-v_0\sin k'\right) \; , \cr} \Eq(3.16) $$ % which is well defined on the support of $f_0(\kk')$, so that, if we set % $$ A_{0}(\kk') = \det T_0(\kk') = [ -ik_0-(\cos k'-1)\cos p_F ]^2 - (v_0\sin k')^2 - [\sigma_{-1}(\kk')]^2 \; ,\Eq(3.17) $$ % then % $$ T_{0}^{-1}(\kk')= {1\over A_{0}(\kk') } \left( \matrix{ [\t_{0}(\kk')]_{1,1} & [\t_{0}(\kk')]_{1,2} \cr [\t_{0}(\kk')]_{2,1} & [\t_{0}(\kk')]_{2,2} \cr} \right) \; , \Eq(3.18) $$ % with % $$ \cases{ [\t_{0}(\kk')]_{1,1} = \left[-ik_0-(\cos k'-1) \cos p_F-v_0\sin k'\right] \; , & \cr [\t_{0}(\kk')]_{1,2} = [\t_{0}(\kk')]_{2,1} = \sigma_{-1}(\kk') \; , & \cr [\t_{0}(\kk')]_{2,2} = \left[-ik_0-(\cos k'-1)\cos p_F + v_0 \sin k' \right] \; . & \cr} \Eq(3.19) $$ % We {\it rescale} the fields so that % $${1 \over \NN_0}\int P_{Z_{-1}}(d\psi^{(\le -1)}) \int \tilde P_{Z_{-1}}(d\psi^{(0)}) \, e^{\hat \VV^{(0)} (\sqrt{Z_{-1}}\psi^{(\le 0)})}$$ so that $$ -\LL\hat\VV^{(0)}(\psi)= \nu_0 F_\nn^{(\le 0)}+\d_0 F_\a^{(\le 0)}+\iota_0 F_\iota^{(\le 0)}+\t_0 F_\t^{(\le 0)}+\l_0 F_\l^{(\le 0)} \; . \Eq(3.11a) $$ where by definition $$\nu_o={Z_0\over Z_{-1}}n_0;\qquad \d_0={Z_0\over Z_{-1}}(a_0-z_0);\qquad \t_0={Z_0\over Z_{-1}} t_0;\qquad \i_0={Z_0\over Z_{-1}}i_0;\qquad \l_0=({Z_0\over Z_{-1}})^2 l_0$$ Then we perform the integration $$ \int \tilde P(d\psi^{(0)}) \, e^{\hat\VV^{(0)} (\sqrt{Z_{-1}}\psi^{(\le 0)})} = e^{\VV^{-1}(\sqrt{Z_{-1}}\psi^{(\le -1)}) + \tilde E_0} \; , \Eq(3.20) $$ % where $\tilde E_0$ is a suitable constant and % $$ -\LL \VV^{(-1)}(\psi)= \g^{-1}\n_{-1} F_\nu^{(-1)}+s_{-1} F_\sigma^{(-1)} +a_{-1} F_\a^{(\le -1)}+ z_{-1} F_\z^{(\le -1)}+i_{-1} F_\i^{(\le -1)}+\t_{-1} F_\t^{(\le -1)}+ l_{-1} F_\l^{(\le -1)}\; , \Eq(3.21) $$ % The procedure can be iterated, and at each step one has to perform the integration $$\int P_{Z_h}(d\psi^{(\le h)}) \, e^{\VV^{(h)}(\sqrt{Z_h}\psi^{(\le h)})}\Eq(3.22)$$ with $$ \eqalign{ P_{Z_{h}}(d\psi^{(\le h)}) = & \prod_{\kk}\prod_{\o=\pm1} d\psi^{(\le h)+}_{\kk'+\o\pp_F,\o} d\psi^{(\le h)-}_{\kk'+\o\pp_F,\o} \cr \exp \Big\{ &-\sum_{\o=\pm1} {1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \, C_0(\kk') Z_{h} \Big[\Big( -ik_0-(\cos k'-1)\cos p_F +\o v_0\sin k' \Big) \cr & \psi^{(\le0)+}_{\kk'+\o\pp_F,\o} \psi^{(\le0)-}_{\kk'+\o\pp_F,\o} - \sigma_{h}(\kk') \, \psi^{(\le0)+}_{\kk'+\o\pp_F,\o} \psi^{(\le0)-}_{\kk'-\o\pp_F,-\o} \Big] \Big\} \; , \cr} \Eq(3.10a) $$ Moreover $$-\LL\VV^{(h)}(\psi)=\g^h n_h F_\nu^{(\le h)}+s_h F_\s^{(\le h)}+z_h F_\z^{(\le h)}+a_h F_\a^{(\le h)}+ i_h F_\iota^{(\le h)}+t_h F_\t^{(\le h)}+l_h F_\l^{(\le h)} \; . \Eq(3.11a) $$ We write $$\int P_{Z_h}(d\psi^{(\le h)}) \, e^{\VV^{(h)}(\sqrt{Z_h}\psi^{(\le h)})}= \int \tilde P_{Z_{h-1}}(d\psi^{(\le h)}) \, e^{\tilde\VV^{(h)}(\sqrt{Z_h}\psi^{(\le h)})} \Eq(3.22a)$$ where $$ \eqalign{ \tilde P_{Z_{h-1}}(d\psi^{(\le h)}) = & \prod_{\kk}\prod_{\o=\pm1} d\psi^{(\le h)+}_{\kk'+\o\pp_F,\o} d\psi^{(\le h)-}_{\kk'+\o\pp_F,\o} \cr \exp \Big\{ &-\sum_{\o=\pm1} {1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \, C_0(\kk') Z_{h-1}(\kk') \Big[\Big( -ik_0-(\cos k'-1)\cos p_F +\o v_0\sin k' \Big) \cr & \psi^{(\le0)+}_{\kk'+\o\pp_F,\o} \psi^{(\le0)-}_{\kk'+\o\pp_F,\o} - \sigma_{h-1}(\kk') \, \psi^{(\le0)+}_{\kk'+\o\pp_F,\o} \psi^{(\le0)-}_{\kk'-\o\pp_F,-\o} \Big] \Big\} \; , \cr} \Eq(3.10b) $$ with $Z_{h-1}(\kk')=Z_h(1+C_h^{-1}(\kk')z_h)$, $Z_{h-1}(\kk')\s_{h-1}(\kk')=Z_h( \s(\kk')_h+C_h^{-1}(\kk') s_h$ and $\tilde \VV=\LL\tilde\VV+(1-\LL)\VV$ with $$-\LL\tilde\VV^{(h)}(\psi)=\g^h n_h F_\nu^{(\le h)}+(a_h-z_h) F_\a^{(\le h)}+i_h F_\iota^{(\le h)}+t_h F_\t^{(\le h)}+l_h F_\l^{(\le h)} \; . \Eq(3.11b)$$ The r.h.s of \equ(3.22a) can be written as % $$ {1 \over \NN_h}\int P_{Z_{h-1}}(d\psi^{(\le h-1)}) \int \tilde P_{Z_{h-1}}(d\psi^{(h)}) \, e^{\tilde \VV^{(h)}(\sqrt{Z_h}\psi^{(\le h)})} \; , \Eq(3.12a) $$ % where $ P_{Z_{h-1}}(d\psi^{(\le h-1)})$ and $\tilde P_{Z_{h-1}}(d\psi^{(h)})$ are given by \equ(3.10b) with $Z^{h-1}(\kk')$ replaced by $Z^{h-1}(0)\equiv Z^{-1}$ and $C_h(\kk')$ replaced with $C_{h-1}(\kk')$ and $\tilde f_h^{-1}(\kk')$ respectively, if $$\tilde f_h(\kk')=Z_{h-1}[{C_h^{-1}(\kk')\over Z_{h-1}(\kk')}- {C_{h-1}^{-1}(\kk')\over Z_{h-1}}$$ and $\psi^{(\le h)}$ replaced with $\psi^{(\le h-1)}$ and $\psi^{(h)}$ respectively. Note that $\tilde f_h(\kk')$ is a compact support function, with support $O(\g^h)$. The Grassmanian integration $\tilde P_{Z_{h-1}}(d\psi^{(h)})$ has propagator % $$ {g^{(h)}(\xx;\yy)\over Z_{h-1}} = \sum_{\o,\o'=\pm1} e^{-i(\o x - \o' y)p_F}\, {g^{(h)}_{\o,\o'}(\xx;\yy) \over Z_{h-1}}\; , \Eq(3.13a) $$ % if % $$ g^{(h)}_{\o,\o'}(\xx;\yy)\=\int \tilde P_{Z_{h-1}}(d\psi^{(h)})\, \psi^{(h)-}_{\xx,\o}\psi^{(h)+}_{\yy,\o'} \Eq(3.14) $$ % is given by % $$ g^{(h)}_{\o,\o'}(\xx;\yy)={1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \, e^{-i\kk'\cdot(\xx-\yy)} \tilde f_h(\kk') [T_{h}^{-1}(\kk')]_{\o,\o'} \; , \Eq(3.15a) $$ % where the $2\times2$ matrix $T_{h}(\kk')$ has elements % $$ \cases{ [T_{h}(\kk')]_{1,1} = \left(-ik_0-(\cos k'-1)\cos p_F+ v_0\sin k' \right) \; , & \cr [T_{h}(\kk')]_{1,2} = [T_{h}(\kk')]_{2,1} = - \sigma_{h-1}(\kk') \; , & \cr [T_{h}(\kk')]_{2,2} = \left(-ik_0-(\cos k'-1)\cos p_F-v_0\sin k'\right) \; , \cr} \Eq(3.16a) $$ % which is well defined on the support of $f_0(\kk')$, so that, if we set % $$ A_{h}(\kk') = \det T_h(\kk') = [ -ik_0-(\cos k'-1)\cos p_F ]^2 - (v_0\sin k')^2 - [\sigma_{h-1}(\kk')]^2 \; ,\Eq(3.17) $$ % then % $$ T_{h}^{-1}(\kk')= {1\over A_{h}(\kk') } \left( \matrix{ [\t_{h}(\kk')]_{1,1} & [\t_{h}(\kk')]_{1,2} \cr [\t_{h}(\kk')]_{2,1} & [\t_{h}(\kk')]_{2,2} \cr} \right) \; , \Eq(3.18a) $$ % with % $$ \cases{ [\t_{h}(\kk')]_{1,1} = \left[-ik_0-(\cos k'-1) \cos p_F-v_0\sin k'\right] \; , & \cr [\t_{h}(\kk')]_{1,2} = [\t_{h}(\kk')]_{2,1} = \sigma_{h-1}(\kk') \; , & \cr [\t_{h}(\kk')]_{2,2} = \left[-ik_0-(\cos k'-1)\cos p_F + v_0 \sin k' \right] \; . & \cr} \Eq(3.19a) $$ % We {\it rescale} the fields so that % $${1 \over \NN_h}\int P_{Z_{h-1}}(d\psi^{(\le h-1)}) \int \tilde P_{Z_{h-1}}(d\psi^{(h)}) \, e^{\hat \VV^{(h)} (\sqrt{Z_{h-1}}\psi^{(\le h)})}$$ so that $$ -\LL\hat\VV^{(h)}(\psi)= \g^h\nu_h F_\nn^{(\le h)}+\d_h F_\a^{(\le h)}+\iota_h F_\iota^{(\le h)}+\t_h F_\t^{(\le h)}+\l_h F_\l^{(\le h)} \; . \Eq(3.11ah) $$ where by definition $$\nu_h={Z_h\over Z_{h-1}}n_0;\qquad \d_h={Z_h\over Z_{h-1}}(a_h-z_h);\qquad \t_h={Z_h\over Z_{h-1}} t_h;\qquad \iota_h= {Z_h\over Z_{h-1}}i_0;\qquad \l_h=({Z_h\over Z_{h-1}})^2 l_h$$ Then we perform the integration $$ \int \tilde P_{Z_{h-1}}(d\psi^{(h)}) \, e^{\hat\VV^{(h)} (\sqrt{Z_{h-1}}\psi^{(\le h)})} = e^{\VV^{h-1}(\sqrt{Z_{h-1}}\psi^{(\le h-1)}) + \tilde E_h} \; , \Eq(3.20h) $$ % where $\tilde E_h$ is a suitable constant and % $$ -\LL \VV^{(h-1)}(\psi)= \g^{h-1}\n_{-1} F_\nu^{(h-1)}+s_{h-1} F_\sigma^{(h-1)} +a_{h-1} F_\a^{(\le h-1)}+ z_{h-1} F_\z^{(\le h-1)}+i_{h-1} F_\iota^{(\le h-1)}+\t_{h-1} F_\t^{(\le h-1)}+ l_{h-1} F_\l^{(\le -1)}\; , \Eq(3.21h) $$ % \vskip.5cm The effective potential $V^h(\psi)$ is a sum of terms of the form $$\int \prod_{i=1}^n d\kk'_i \d(\sum_{i=1}^n \e_i(\kk'_i+\o_i p_F)+2mp) \bar W_{n,m}^h(\kk'_1,..,\kk'_n;\{\o\})\prod_{i=1}^n \psi^{\e_i \le h}_{\kk'_1+\o_i p_F,\o_i}\Eq(bbb)$$ where $\bar W_{n,m}^h(\kk'_1,..,\kk'_n;\{\o\})= W_{n,m}^h(\kk'_1+\o_1 p_F,..)$. %By the compact support properties of the $\psi^{\e_i \le h}_{\kk'_1+\o_i p_F,\o_i}$ It is possible to prove the following lemmas. \vskip1cm {\bf \rm LEMMA 1} {\it Assume that $\sum_{i=1}^n \e_i \o_i p_F+2mp\npt=0$ and $m\not=0$. Then \equ(bbb) is identically vanishing unless} $$|m|\geq C_1[\g^{-h\over\t}-n {p\over p_F}]\Eq(3.21hh)$$ \vskip.5cm {\bf \rm PROOF} Remembering the compact support of the Grassmanian operator $\psi^{\e_i \le h}_{\kk'_i+\o_i p_F,\o_i}$ we can write $$a_0 n\g^h\geq |\sum_{i=1}^n \e_i\kk'_i|\geq |2mp+\sum_{i=1}^n\e_i \o_i p_F|\ge C_0( 2|m| p+n p_F)^{-\t}$$ from which \equ(3.21hh) holds. \vskip1cm {\bf \rm LEMMA 1} {\it Assume that $\sum_{i=1}^n \e_i \o_i p_F+2mp\npt=0$ and $m=0$. Then \equ(bbb) is identically vanishing unless $f\geq \bar h$, if $\bar h$ is a suitable constant.} \vskip.5cm {\bf \rm PROOF} Using again compact support of the Grassmanian operator we can write $$a_0 n\g^h\geq |\sum_{i=1}^n \e_i \o_i p_F|\geq 2p_F$$ \vskip2.truecm \centerline{\titolo 3. Analiticity of the effective potential} \*\numsec=3\numfor=1 It is convenient to introduce the fields $\psi^\e_{\xx,\o}$, defined as $$\psi^\e_{\xx,\o}=\int d\kk' e^{-i\e\kk'\xx}\psi^\e_{\kk'+\o p_F,\o}$$ Moreover we define: $$W_{n,m}^h(\xx_1,...,\xx_n;\{\o\})\equiv\bar W_{n,m}^h=\int \prod_{i=1}^n d\kk'_i \d(\sum_{i=1}^n(\kk'_i+\o_i p_F)+2m p) \bar W_{n,m}^h(\kk_1,...,\kk_n;\{\o\})$$ Note that, if $\sum_{i=1}^n\o_i p_F+2m p=0$ then $\bar W_{n,m}^h$ is a translation invariant function. We can write the analogue of \Eq(loc1) \Eq(loc2) in coordinate space: \vskip.5cm 1)if $n>4$ or if the kernels $\bar W_{n,m}^h$ are not traslation invariant $$\RR \int \prod_{i=1}^n d\xx_i \prod_{i=1}^n\psi^{\e_i}_{\xx_i,\o_i}\bar W^h_{n.m}= \int \prod_{i=1}^n d\xx_i \prod_{i=1}^n\psi^{\e_i}_{\xx_i,\o_i}\bar W^h_{n.m}$$ \vskip.5cm 2) if $n=4$ and $\bar W^h_{4,m}$ is traslation invariant %The $\RR$ operator, when its action is not trivial, %in the coordinate space can be written (simply %performing the Fourier transform). %In the case of eq(??), with $(\o_1+\o_2-\o_3-\o_4)=0$, $n=0$, and calling %$W_{4,0}(\xx_1-\xx_2,\xx_2-\xx_4,\xx_3-\xx_4;\{\o\})$ the Fourier transform of %$W_{4,0}(\kk'_1+\o_1 p_F,...,\kk'_4+\o_4 p_F)\d(\kk'_1+\kk'_2-\kk'_3-\kk'_4)$ %one obtains $$\RR \int \prod_{i=1}^4 d\xx_i\prod_{i=1}^4\psi^{\e_i}_{\xx_1,\o_i} (W_{4,m}^h(\xx_1-\xx_4,\xx_2-\xx_4,\xx_3-\xx_4;\{\o\})= \; , \Eq(z.1) $$ $$\int \prod_{i=1}^4 d\xx_i\prod_{i=1}^4\psi^{\e_i}_{\xx_1,\o_i}\{ W_{4,m}^h(\xx_1-\xx_4,\xx_2-\xx_4,\xx_3-\xx_4;\{\o\})-\d(x_1-x_2)\d(x_2-x_3)\d(x_3-x_4) \int d{\bf t}_1 d{\bf t}_2 d{\bf t}_3 W_{4,m}^h({\bf t}_1,{\bf t}_2,{\bf t}_3;\{\o\})\}$$ \vskip.5cm 3)in $n=2$ and $\bar W^h_{2,m}$ is traslation invariant $$\RR \int \prod_{i=1}^2 d\xx_i\prod_{i=1}^2\psi^{\e_i}_{\xx_i,\o_i} W_{2,m}^h(\xx_1-\xx_2;\{\o\})=\int \prod_{i=1}^2 d\xx_i\prod_{i=1}^2\psi^{\e_i}_{\xx_1,\o_i} [W^h_{2,0}^h(\xx_1-\xx_2;\{\o\})-\; , \Eq(z.2)$$ $$\d(\xx_1-\xx_2)\int d\vec t W_{2,m}^h(\vec t;\{\o\})- \partial_{x_2^0} \d(\xx_1-\xx_2) \int d\vec t t_0 W_{2,m}^h(\vec t;\{\o\})-\bar\partial_{x_2}\d(\xx_1-\xx_2) \int d \vec t \o t W_{2,m}^h(\vec t;\{\o\})]$$ where $\bar \partial_{x_2}\d(x_1-x_2)$ is the distribution $\sum_{k}\o E(k+\o p_F)e^{i k(x_1-x_2)}$. It is convenient to write $\o E(k+\o p_F)=2 p_F k+\bar f(k)$ with $|\bar f(k)|\le C k^2$. Then $\bar \partial_{x_2}\d(x_1-x_2)=\partial_{x_2}\d(x_1-x_2)+\d^2\d(x_1-x_2)$ and, in the distribution sense, $(x_1-x_2)^a \d^2\d(x_1-x_2)=0$ if $a>2$ \vskip1cm The following identitys will be useful in the following \vskip.5cm $$(x_i-x_j)^a\RR \int \prod_{i=1}^4 d\xx_i\prod_{i=1}^4\psi^{\e_i}_{\xx_1,\o_i} (W_{4,0}^h(\xx_1-\xx_4,\xx_2-\xx_4,\xx_3-\xx_4;\{\o\})=\Eq(z.3)$$ $$\int \prod_{i=1}^4 d\xx_i\prod_{i=1}^4\psi^{\e_i}_{\xx_1,\o_i} (W_{4,0}^h(\xx_1-\xx_4,\xx_2-\xx_4,\xx_3-\xx_4;\{\o\})$$ %The following trivial identity are valid in the distribution sense: %$$(x_i-x_j)^a \RR(W_{4,0}(\xx_1-\xx_2,\xx_2-\xx_4,\xx_3-\xx_4;\{\o\})= %(x_i-x_j)^a W_{4,0}(\xx_1-\xx_2,\xx_2-\xx_4,\xx_3-\xx_4;\{\o\}) \; , \Eq(z.3)$$ if $a$ is any integer $\geq 1$ and $i,j$ can take any value between $1$ and $4$. \vskip.5cm $$(\xx_1-\xx_2)^a \RR \int \prod_{i=1}^2 d\xx_i\prod_{i=1}^2\psi^{\e_i}_{\xx_i,\o_i} W_{2,0}^h(\xx_1-\xx_2;\{\o\})=\; , \Eq(z.4)$$ $$(\xx_1-\xx_2)^a\int \prod_{i=1}^2 d\xx_i\prod_{i=1}^2\psi^{\e_i}_{\xx_i,\o_i} W_{2,0}^h(\xx_1-\xx_2;\{\o\})+(\xx_1-\xx_2)^a \d^2\d(x_1-x_2) \int d \vec t \o t W_{2,0}^h(\vec t;\{\o\})$$ if $a$ is any integer $\geq 2$ \vskip.5cm $$(x_1-x_2) \RR \int \prod_{i=1}^2 d\xx_i\prod_{i=1}^2\psi^{\e_i}_{\xx_i,\o_i} W_{2,0}^h(\xx_1-\xx_2;\{\o\})=\; , \Eq(z.5)$$ $$(x_1-x_2)\int \prod_{i=1}^2 d\xx_i\prod_{i=1}^2\psi^{\e_i}_{\xx_i,\o_i} W_{2,0}^h(\xx_1-\xx_2;\{\o\})-\d(x_1-x_2)\int d\vec t t W^h(t;\{\o\})$$ $$-(\xx_1-\xx_2) \d^2\d(x_1-x_2) \int d \vec t \o t W_{2,0}^h(\vec t;\{\o\})$$ and a similar equation if $(x_1-x_2)$ is replaced by $(x_1^0-x_2^0)$. \vskip.5cm %W_{2,0}(\xx_1-\xx_2;\{\o\})=(\xx_1-\xx_2)^a W_{2,0}(\xx_1-\xx_2;\{\o\}) %\; , \Eq(z.4)$$ %if $a$ is any integer value $\geq 2$; and %$$(x_1-x_2)\RR W_{2,0}(\xx_1-\xx_2;\{\o\})=(x_1-x_2) %W_{2,0}(\xx_1-\xx_2;\{\o\})-\d(\xx_1-\xx_2) %\int d\vec t t W_{2,0}(\vec t;\{\o\}) \; , \Eq(z.5)$$ %$$(x_1^0-x_2^0)\RR W_{2,0}(\xx_1-\xx_2;\{\o\})=(x_1-x_2) %W_{2,0}(\xx_1-\xx_2;\{\o\})-\d(\xx_1-\xx_2)\int d\vec t t^0 %W_{2,0}(\vec t;\{\o\}) \; , \Eq(z.6)$$ One can integrate the $\d$ functions in \equ(z.1) so obtaining $$\int \prod_{i=1}^4 d\xx_i\prod_{i=1}^4\psi^{\e_i}_{\xx_1,\o_i} \RR(W_{4,m}^h(\xx_1-\xx_4,\xx_2-\xx_4,\xx_3-\xx_4;\{\o\})=\; , \Eq(z.7)$$ $$\int \prod_{i=1}^4 d\xx_i W_{4,m}(\xx_1-\xx_2,\xx_2-\xx_4,\xx_3-\xx_4;\{\o\})$$ $$[\psi^+_{\xx_1,\o_1}\psi^+_{\xx_2,\o_2}\psi^-_{\xx_3,\o_3}\psi^-_{\xx_4,\o_4}- \psi^+_{\xx_1,\o_1}\psi^+_{\xx_1,\o_2}\psi^-_{\xx_1,\o_3}\psi^-_{\xx_1,\o_4}]$$ The square brakets in the above equation can be written as $$\psi^+_{\xx_1,\o_1} D^+_{\xx_{2,1},\o_2}\psi^-_{\xx_3,\o_3}\psi^-_{\xx_4,\o_4} +\psi^+_{\xx_1,\o_1}\psi^+_{\xx_1,\o_2} D^-_{\xx_{3,1},\o_3}\psi^-_{\xx_4,\o_4}+ \psi^+_{\xx_1,\o_1}\psi^+_{\xx_1,\o_2}\psi^-_{\xx_1,\o_3} D^-_{\xx_{1,4},\o_4}$$ so that the effect of $\RR$ is essentially to change the coordinate of the $\psi$ fields and to replace a field $\psi_{x_i,\o}^\e$ with $D_{x_{ij}}^\e$ where $$D_{x_{ij}}^\e=\psi^\e_{x_i,\o_i}-\psi^\e_{x_j,\o_i}=(x_i-x_j)\int_0^1 du\partial \psi^\e_{x_{ji}(u),\o_i}\; , \Eq(z.8)$$ with $x_{ji}(u)=u x_j+(1-u) x_i$. %We will see that in the estimates the presence of a field %$D_{x_{ij}}^\e$ instead of a field $\psi^e$ produces a gain. In the same time integrating the $\d$'s in \Eq(z.2) one can see that the effect of $\RR$ is to replace a field $\psi$ with a field $A$ wich can be written as: $$(x_2-x_1)^2\int_0^1 dt_1 \int_{0}^{t_1} dt_2\partial^2\psi_{x_{21}(t_2)}+(x_2-x_1)\D\psi_{x_2}\; , \Eq(z.9)$$ with $\D\psi_{x_2}=\int d\kk \bar f(k) e^{i \kk' x} \psi_{\kk'+\o p_F}$. Note the two different ways to write the $\RR$ operation: in \equ(z.1),\equ(z.2) the $\RR$ acts on the kernels of the effective potential \ie it consists in replacing $W^h_{n,m}$ with $\RR W^h_{n,m}$ (the definition of $\RR W^h_{n,m}$ is evident from \equ(z.1),\equ(z.2)); on the other hand writing as in \equ(z.7) the $\RR$ operation act on the $\psi$ fields, leaving the kernel untouched. \vskip.5cm >From the construction of the preceding section we have that $$V^h(\sqrt{Z_h}\psi^{\le h})=\sum_{n=1}^{\io}{1\over n!} E^T_{h+1}(\hat V^{h+1} (\sqrt{Z_h}\psi^{\le h+1})$$ and $$\hat V^h(\sqrt{Z_{h-1}} \psi^{\le h})=\LL \hat V^h+\RR \hat V^h=\LL^* V^h(\sqrt{Z_{h-1}}\sqrt{Z_h\over Z_{h-1}} \psi^{\le h}) +\RR V^h(\sqrt{Z_{h-1} \sqrt{Z_h\over Z_{h-1}} \psi^{\le h})\Eq(z.99)$$ where $\LL^* V^h$ differs from $\LL V^h$ only becouse it does not contain anymore the terms $F_\s$ and $F_\d$. Iterating the above equation we obtain that the effective potential can be written in term of a tree expansion in the following way. $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ We call $\t_{n,k}$ the set of all the labeled trees with $n$ end-points $\t\in\t_{n,k}$ that can be constructed as follows. We draw on the $(x,y)$ plane vertical lines at $x=k,k+1,...,0,1$. Let $r$ ({\it the root}) be a point on the line $x=k$. Starting from $r$ we draw an horizontal line leading to a point $v_0$ ( the {\it first vertex} of $\t$) on the line $x=k_{v_0}=k+1$. Choose $s_{v_0}\ge 0$ and draw $s_{v_0}$ lines (called {\it tree lines}) from $v_0$ to $s_{v_0}$ points $v_0^1,...,v_0^{s_{v_0}}$ on the lines $k_{v_0^1},..., k_{v_0^{s_{v_0}}}$ with $k_{v_0^i}>k_{v_0}$ \ie the lines cannot go back. We repeat the above procedure with the points ${v_0^i}$ and go on recursively. The intersections of the vertical lines $x=h_v$ with the tree lines are called vertices $v$ with scale $h_v$. >From a vertices $v$ start $s_v$ tree lines: if $s_v=0$ the vertices is called {\it end-point}, if $s_v=1$ is called {\it trivial vertex} and if $s_v>1$ is called {\it non trivial vertex}. The vertices on $\t$ are naturally (partially) ordered. Giving to the tree an orientation from left to rigth, we write $v_1k}|v_h|\le\bar\e$ and $\sup_{h>k}|{Z_h\over Z_{h-1}}|\le\bare^{c_1\bar\e^2}$, then $$\int d\xx^{P_{v_0}}\sum_{\t\in\t_{n,k}}|W^k(\t,P_{v_0},\xx^{P_{v_0}})| (1+\g^k d(P_{v_0})^N\le \L \g^{-D(P_{v_0})}(C_N\bar\e)^N$$ where $N$ is a positive integer, $x^{P_{v^0}}$ is the set of points associated to $P_{v_0}$, $c_2,C_N$ are constants, $d(P_{v_0})$ is the length of the shortest tree connecting the set of points $\xx^{P_{v_0}}$ and $$D(P_{v_0})=-2+\sum_{f\in P_{v_0}}(1/2+m_f)$$ where $m_f$ is the order of the derivative applied to the fields of label $f$} \vskip.5cm In order to write in an explicit form \equ(z11) it is convenient to start studying $\RR \hat V^h$: $$\RR \hat V^k(\t,\sqrt{Z_{k-1}}\psi^{\le k})={1\over s_{v_0}!}\sum_{P_{v_0}}(\sqrt{Z_{k-1}})^{|P_{v_0}|} \RR\{\tilde\psi^k(P_{v_0}) \sum_{ P_{v_0^1},..,P_{v_0^s_{v_0}} } \sqrt{Z_{k}\over Z_{k-1}}^{|P_{v_0}|}\sum_{ Q_{v_0^1},..,Q_{v_0^s_{v_0}} } $$ $$\E_{k+1}^T[\int d\xx_{v_0^1} \tilde\psi^{k+1}(P_{v_0^1})\bar W^{k+1}(\t,P_{v_0^1},\xx_{v_0^1}),...,\int d\xx_{v_0^{s_{v_0}}} \tilde\psi^{k+1}(P_{v_0^{s_{v_0}}})\bar W^{k+1}(\t,P_{v_0^{s_{v_0}}},\xx_{v_0^{s_{v_0}}})\}\Eq(bab)$$ >From \equ(z.7)-\equ(z.9) we know that the effect of $\RR$ is simply to replace $\tilde\psi^k(P_{v_0})$ with $$\sum^*_{i,j,b}(\xx_i-\xx_j)_b^{a_{P_{v_0}}} \hat\psi^k(P_{v_0})\Eq(bab1)$$ where $b=0,1$ is an index distinguishing the space ot time component of $\xx$ (in the following this index will be omitted), $\xx_i,\xx_j$ are teo coordinates in $\xx^{P_{v_0}}$ and \vskip.5cm 1)$a=0$ and \hat\psi^k(P_{v_0})=\tilde\psi^k(P_{v_0})$ if $|P_{v_0}|>4$ or the field index do not verify verify the Kronecker $\d$ in \equ(loc1), \equ(loc2) \vskip.5cm 2)if $|P_{v_0}|=4$ and the field index verify the Kronecker $\d$ in \equ(loc1) then $a=1$ and $\hat\psi^k(P_{v_0})$ differs from $\tilde\psi^k(P_{v_0})$ becouse a field $\int_0^1 \partial\psi^\e_{x_{ij}(u)}$ replaces a $\psi$ field, and the remaning $\psi$ fields are applied on different coordinates among $\xx^{P_{v_0}}$ \vskip.5cm 3)if $|P_{v_0}|=2$ and the field index verify the Kronecker $\d$ in \equ(loc2), $a=2$ and $\hat\psi^k(P_{v_0})$ differs from $\tilde\psi^k(P_{v_0})$ becouse a field $\int_0^1 dt_1 \int_0^{t_1} dt_2 \partial^2\psi_{x_{ij}(t_2)}$ or a ${\D\psi\over (\xx_1-\xx_2)$ replaces a $\psi$ field, and the remaning $\psi$ fields are applied on different coordinates among $\xx^{P_{v_0}}$. %$\x_i,\xx_j$ are two coordinates among %$\cup_{f\in P_{v_0}}\xx(f)$ \vskip.5cm Let we call: $$\hat\psi(P)=\prod_{f\in P}\partial^{q(f)}_{\xx(f)}\psi_{\xx(f)}^{\e(f)}\Eq(bab7)$$ where $q=0,1,2,3$, $\partial^0=1$, $\partial^1=\partial_\xx$, $\partial^2_\xx=\partial_\xx \partial_\xx$, $\partial^3={1\over (\xx_1-\xx_2)}\D$. We remember the well known expansion of truncated expectation in term of interpolating parameters $s_t$, $t=1,..,k-1$: $$\E^T_{h}(\hat\psi(P_1),...,\hat\psi(P_4))=\sum_{T}\prod_{l\in T} \partial^{q_l}_{x_l} \partial^{q_l}_{y_l} g^h(x_l-y_l)\int dP_{T}(s) det G^T(s)\Eq(bab2)$$ where $T$ is a set of lines forming an {\it anchored tree graph} between the cluster of vertices from which the fields labeled with $P_1,..,P_k$ emerge: this means that $T$ is a set of lines connecting two points in different clusters, which becomes a tree graph if one identifies all the points in the same cluster; if $l\in T$ $x_l,y_l$ are the end-points of the line and are such that $x_l=x_{ij}$ or $y_l=x_{i'j'}$, where $x_{ij}$ denotes the coordinate of the $i$-th field of the monomial $\tilde\psi(P_j)$. If $f$ is a field variable such that $x(f)=x_l$, $q(f)=q_l$ then $\partial^{q_l}_{x_l}=\partial^{q(\bar f)}_{x(f)}$. $G^T(s)$ is a $(n-k+1) \times (n-k+1)$ matrix (if $n$ is the total number of fields), whose elements are $G^T_{jij'i'}=S_{jj'} \partial^{q_{ij}}\partial^{q_{i'j'}} g^h(x_{ij}-x_{i'j'})$ with $x_{ij}-x_{i'j'}$ non belonging to $T$, $S_{jj'}=\prod_{i=j}^{j'-1} s_t$ and $dP_T(s)$ is a normalized measure which depends on $s_t$ and $T$ (for the explicit formula of $dP_T$ and for its derivation, one can look for istance [Le],[BGPS]). %Insterting this expression in \equ(bab) we obtain, with the above notations: We write in an explicit way the action of $\RR$ in \equ(bab) obtaining $$\RR \hat V^k(\t,\sqrt{Z_{k-1}}\psi^{\le k})={1\over s_{v_0}!}\sum_{P_{v_0}} \sqrt{Z_{k}}^{|P_{v_0}|} \hat\psi^k(P_{v_0}) \sum_{ P_{v_0^1},..,P_{v_0^s_{v_0}} } \sqrt{Z_{k}\over Z_{k-1}}^{|P_{v_0}|}\sum_{ Q_{v_0^1},..,Q_{v_0^s_{v_0}} } \int d\xx_{v_0}\Eq(bab3)$$ $$[\sum^*_{ij}(\xx_i-\yy_j)^{a(P_{v_0})} \E^T_{k+1}(\tilde\psi^{k+1}(P_{v_0^1}),...,\tilde\psi^{k+1}(P_{v_0^{s_{v_0}}}) ) ]\bar W^{k+1}(\t_1,P_{v_0^1},\xx_{v_0^1})...\bar W^{k+1}(\t_{s_{v_0}},P_{v_0^{s_{v_0}}},\xx_{v_0^{s_{v_0}}})]$$ if $x_i,y_j$ are two points among $x^{P_{v_0}}$ and $\sum_{\a,\b}^*$ is present only if $a(P_{v_0})\not=0$. We can write, for any anchored tree graph $T_{v_0}$ connecting the clusters $L_{v_0^1}...L_{v_0^{s_{v_0}}}$: $$(\xx_i-\yy_j)=\sum_{l\in T}(\xx_{l}-\yy_{l})+\sum^*_{iji'j'}(\xx_{ij}-\yy_{i'j'})\Eq(bab7)$$ where $\sum^*_{iji'j'}$ is a sum such that $\xx_{ij},\yy_{i'j'}$ ore not the end-points of a line $l\in T_{v_0}$. So we can write: $$\sum^*_{ij}(\xx_\a-\yy_\b)^{a(P_{v_0})}\E^T_{k+1}(\tilde\psi^{k+1}(P_{v_0^1}),... ) ]\bar W(\t,P_{v_0^1},\xx_{v_0^1})...\bar W(\t,P_{v_0^{s_{v_0}}},\xx_{v_0^{s_{v_0}}})]=$$ $$\sum^*_{\{a\}} \sum_{T_{v_0}}\{\prod_{l\in T_{v_0}} (\xx_l-\yy_l)^{a_l} g^{k+1}(\xx_l-\yy_l) [\int dP_{T}(s) det G^T(s)]$$ $$|\xx^{P_{v_0^1}}|^{a_{v_0^1}}\bar W^{k+1}(\t,P_{v_0^1},\xx_{v_0^1})...|\xx_{v_0^{s_{v_0}}}|^{a_{v_0^{s_{v_0}}}}\bar W^{k+1}(\t,P_{v_0^{s_{v_0}}},\xx_{v_0^{s_{v_0}}})]\Eq(zzz)$$ %Replacing this formula in eq.(!!) we obtain %$\RR V^k(\t,\sqrt{Z_k}\psi^{\le k})={1\over s_{v_0}!}\sum_{P_{v_0}}\sqrt{Z_{k}}^{|P_{v_0}|} %\hat\psi^k(P_{v_0}) \sum_{ P_{v_0^1},..,P_{v_0^s_{v_0}} } %\sqrt{Z_{k+1}\over Z_k}^{|P_{v_0}|}\sum_{ Q_{v_0^1},..,Q_{v_0^s_{v_0}} } \int d\xx_{v_0}$$ %$$\sum_{T}\prod_{l\in T} \sum^* %|x_l-y_l|*{a_l}g^{k+1}(x_l-y_l)\int dP_{T}(s) det G^T(s) |\xx^{P_{v_0^1}}|^{a_{v_0^1}}\bar %W(\t,P_{v_0^1},\xx^{P_{v_0^1}})...|\xx^{P_{v_0^{s_{v_0}}}|^{a_{v_0^{s_{v_0}}}}\bar %W(\t,P_{v_0^{s_{v_0}}},\xx^{P_{v_0^{s_{v_0}}}})]$$ where $|\xx^{P_{v_0^i}}|$ is the difference among two coordinates in $\xx^{P_{v_0^i}}$, and $\sum^*$ is over the indices $a_l+a_{v_0^1}+..$ with the constraint that $a_l+a_{v_0^1}+...+ a_{v_0^{s_{v_0}}}=a(P_{v_0})$. We are now in position to iterate the above procedure studying each $|\xx^{P_{v_0^i}}|^{a_{v_0^i}}\bar W(\t,P_{v_0^i},\xx^{P_{v_0^i}})$. We can distinguish several cases \vskip.5cm 1)if $\t^i$ is a trivial tree, $a=0$ and $\bar W^{k+1}=\vec v_{k+1}$ and the iterative procedure stops. \vskip.5cm 2)if $\t^i$ is not a trivial tree, there are the following possibilityes: \vskip.5cm 2a)$a_{v_0^i}=0$. In this case we repeat word by word the analysis for $W^k$ \equ(bab),\equ(bab3), with the trivial substitution $k\to k+1$, $v_0\to v_0^i$. Note that if $\RR\not=0$ the effect of the renormalization is of replacing in the truncated expectation in \equ(bab3) $\tilde\psi(P_{v_0^i})$ with $\hat\psi(P_{v_0^i})$, defined in an analogous way as $\hat\psi(P_{v_0})$, and to produce a factor $(\xx_i-\xx_j)^{a_{P_{v_0^i}}}$ which is written as in \equ(bab7). %, differing from $\tilde\psi(P_{v_0^i})$ because, if $\RR$ %is given by eq.() one field $\psi^\e$ is replaced with $\int dt_1 %\partial\psi^\e_{x_{ij}(t_1)}$ and if $\RR$ %is given by eq.() one field $\psi^\e$ is replaced with $\int_0^1 dt_1\int_0^{t_1} dt_2 %\partial^2\psi^\e_{x_{ij}(t_1)}$. \vskip.5cm 2b)if $a_{v_0^i}\not =0$ we have to use \equ(z.3) \equ(z.4) \equ(z.5) \equ(z.6) and we preceed as in \equ(bab),\equ(bab3). Note that the propagators belonging $T_{v_0^i}$ are of the form $(\xx_l-\xx_l)^a g\xx_l-\xx_l)$ with $a<2$. %write the analogue of eq.(). \vskip1cm Iterating this procedure we find at the end %$$\sum^* \sum_T\{\prod_{l\in T_{v_0^i}} (\xx_l-\yy_l)^{a_l} g^{k+1}(\xx_l-\yy_l) %[\int dP_{T}(s) det G^T(s)] %|\xx^{P_{v_0^1}}|^{a_{v_0^1}}\bar %W(\t,P_{v_0^1},\xx^{P_{v_0^1}})...|\xx^{P_{v_0^{s_{v_0}}}|^{a_{v_0^ %\vskip.5cm %2b)$a_{v_0^i}\not 0$. In this case we use eq %We can iterate this procedure, by noting that: %\vskip.5cm %1)in the case in which $\bar %W(\t,P_{v_0^1},\xx^{P_{v_0^1}})=\RR W(\t,P_{v_0^1},\xx^{P_{v_0^1}})$ %we can use eq(!!). %\vskip.5cm %2)the effect of the renormalization of the subtrees $\t_1,..,\t_{s_{v_0}}$ by %using eq.(!!) has the effect that $x_{ij},x_{i'j'}$ in the truncated extepctation %$\Epsilon^T_{k+1}$ can be possibly replaced by eq(!!). %\vskip.5cm %Iterating the above procedure and taking into account the above remarks we can write $$\hat V^k(\t,\sqrt{Z_{k-1}}\psi^{\le k})=\sum_{\{ P_{v} \}}\{\int dt\}\int d\xx_{v^0} \prod_{v not e.p.} |{Z_{h_v-1}\over Z_{h_v-2}}|^{|P_v|/2}{1\over s_v!}$$ $$\sum_{\{q\},\{a \}}^*\tilde\E^T_{h_v}(\hat\psi(P_{v^1}/Q_{v^1}),... ,\hat\psi(P_{v^{s_v}}/Q_{v^{s_v}}) [\prod_{i\in V_1} v_{h_i}][\prod_{i\in V_2} \hat\ph_{n_i} e^{in_ip\xx_i}\Eq(pal)$$ where: \vskip.5cm 1)$\hat\psi(P_{v}/Q_{v})=\prod_{f\in P_{v}/Q_{v}}\partial^{a_f}\psi^{\e(f)}_{x'_{ij}(t)}$ where $a_f=0,1,2,3$ with the same conventions as in \equ(bab7). %if $|P_{v}|=2,4$ \ie if the renormalization act on the cluster $v$. Moreover $x_{ij}(t)=\sum_{i'}\e_{i'j}(t) x_{i'j}$ with $\sum_{i'}\e_{i'j(t)}=1$ and $\e_{i'j}(0)=\d_{i,i'}$, if $x_{i',j}$ are the coordinates contained in the cluster $L_v$. Moreover $$\tilde\E^T_{h}(\hat\psi(P_1),...,\hat\psi(P_k))=\sum_{T} \prod_{l\in T} \partial^{q_l}_{x_l} \partial^{q_l}_{y_l} (x_l-y_l)^{a_l}g^h(x_l-y_l) \int dP_{T}(s) det {G}^T(s)$$ %where $T$ is a set of lines forming %an {\it anchored tree graph} between the cluster of vertices %from which the fileds labeled with $P_1,..,P_k$ %emerge: this means that $T$ is a set of lines connecting two points %in different clusters, which becomes a tree graph if one identifies %all the points in the same cluster; %if $l\in T$ $x_l,y_l$ are the end-points of the line and are %such that $x_l=x_{ij}$ or $y_l=x_{i'j'}$, where $x_{ij}$ denotes the coordinate of the $i$-th field %of the monomial $\tilde\psi(P_j)$. If $y_l=x_{i'j'}=x(\bar f)$ then %with $a_l\le 2$, %$a_l\not=0$ only if $l$ it is contained on a cluster in which the %renormalization act non trivially. Moreover $\sum_{l\in T}a_l\le 2$. where $\sum^*_{\{a \},\{q \}}$ are constrained sums with the with the following constraints. If $v$ is such that $|P_v|=4$ and the Kronecker $\d$ of \equ(loc1) is verified %\vskip.5cm %3)$\sum_{z}^*$ is a sum over the derivatives choices in the monomial $\hat\psi$ %and $\sum_{a}^*$ over the order of zero. The constraints on these sums are that %if $L_v$ is a cluster on which $\RR$ act as in eq(!!) then $\hat\psi(P_v)$ contains a field $\partial\psi_{x'_{ij}}$ and there is a line $\bar l\in T_{\bar v}$, with $\bar v>v$, with $a_{\bar l}=1$ and for any $l\in T_{\hat v}$, $l\not=\bar l$, $\bar v\ge \hat v>v$ $a_l=0$. If $v$ is such that $|P_v|=2$ and the Kronecker $\d$ of \equ(loc2) is verified then $\hat\psi(P_v)$ contains a field $\partial^2\psi_{x'_{ij}}$ or $\D\psi$ and there are two lines $l_1\in T_{\bar v_1}$, $l_2\in T_{\bar v_2}$ with $\bar v_1>v$, $\bar v_2>v$ with $a_l=1$ and for any $l\in T_{\hat v}$, $l\not=l_1,l_2$ $\bar v_1\ge \hat v>v$, $\bar v_1\ge \hat v>v$ one have $a_l=0$. %in a %cluster $L_{\bar v}$ contained in $L_v$ such that $a_l=1$; morover for any $l\in L_{v_i}$, %with $L_{v_i}$ containing $_{\bar v}$ and contained in $L_v$ $a_l=0$. If %$L_v$ is a cluster on which $\RR$ act as in eq(!!) then $\hat\psi(P_v)$ %contains a field $\partial^2\psi_{x'_{ij}}$ and there are two lines $l$ %(possibly coingiding) %in a %cluster $L_{\bar v_1}$ and $L_{\bar v_1}$ %contained in $v$ such that $a_l=1$; morover for any $l\in L_{v_i}$, %with $L_{v_i}$ containing $L_{\bar v_1}$ or $L_{\bar v_2}$ %and contained in $L_{v}$ $a_l=0$. If $v$ is a cluster on which $\RR=0$ %then $\hat\psi(P_v)=\tilde\psi(P_v)$. %\vskip.5cm %3a)if $i\in P_v$ $z_i=0$ if the renormalization acts trivially on $v$; if it act as in %eq() then $z_i=1$ and if t act as in %eq() then $z_i=2$. %\vskip.5cm %3b) $\sum_{l\in T}a_l\le 2$ and $a_l\not=0$ only if $l$ %it is contained on a cluster in which the %renormalization act non trivially. If $l$ is contained in a cluster $\bar v$ %and $\hat v$ is the first cluster containing \vskip.5cm 3)$\{\int dt\}$ is a product of integral over the interpolation parameters. \vskip.5cm 4) $V_1$ is the set of the end points associated to $\t$ to which is associated a term of the relevant part of the effective potential, while $V_2\cup V_1$ is the set of all the end points. %$[\prod_i \sum_{n_i}\hat\ph_{n_1} %e^{inp\xx}$ is a product over the end points associated with scale %$0$ to which is associated $\RR V^0$. \vskip1cm Remembering the definition of $\hat \t$ we have that $$\hat V^k(\hat\t,\sqrt{Z_{k-1}}\psi^{\le k})= \sum_{\{ P_{v} \}}\{\int dt\}\int d\xx_{v^0}[\prod_{i\in V_2}\sum_{n_i}] %[\prod_{i\in V_1} v_{h_i}][\prod_{i\in V_2}\sum_{n_i} %\hat\ph_{n_i} %e^{in_ip\xx_i}] \prod_{v not e.p.} |{Z_{h_v-1}\over Z_{h_v-2}}|^{|P_v|/2}{1\over s_v!} \sum_{\{q \},{a \}}^*$$ $$\tilde\E^T_{h_v}(\hat\psi(P_{v^1}/Q_{v^1}),... ,\hat\psi(P_{v^{s_v}}/Q_{v^{s_v}})$$ $$[\prod_{i\in V_1} v_{h_i}][\prod_{i\in V_2} \hat\ph_{n_i} e^{in_ip\xx_i}] \Eq(pal1)$$ Given a cluster $L_v$ (with scale $h_v$), we define $$N_v=\sum_{i\in L_v} n_i$$ where $i$ are the end-points contained in $L_v$, and $n_i=\pm 1$ if to $i$ is associated a $F_\t$ or a $F_\iota$ term. By definition $N_v=N_{v^1}+...+N_{v^{s_v}}$. It is convenient then to rewrite the above expression as $$\hat V^k(\hat\t,\sqrt{Z_{k-1}}\psi^{\le k})= \sum_{\{ P_{v} \}}[\sum_{\{N_v\}}\prod_{v {\rm not} e.p.}\d_{N_v,N_{v^1}+...+N_{v^{s_v}}}] \{\int dt\}\int d\xx_{v^0} %[\prod_{i\in V_1} v_{h_i}] %[\sum_{\{N_v\}}\prod_{v {\rm not} e.p.}\d_{N_v,N_{v^1}+...+N_{v^{s_v}}} %\prod_{i\in V_2} \hat\ph_{n_i} %e^{in_ip\xx_i}] \prod_{v not e.p.} |{Z_{h_v-1}\over Z_{h_v-2}}|^{|P_v|/2}{1\over s_v!}$$ $$\sum_{z}^*\tilde\E^T_{h_v}(\hat\psi(P_{v^1}/Q_{v^1}),... ,\hat\psi(P_{v^{s_v}}/Q_{v^{s_v}})$$ $$[\prod_{i\in V_1} v_{h_i}]\prod_{i\in V_2} \hat\ph_{n_i} e^{in_ip\xx_i}] \Eq(pal2)$$ where $\sum_{\{N_v\}}$ is over the $N_v$ of all the clusters, including the end points. By lemma 1 we know that $N_v\geq C(P_v)\g^{-h_v\over\t}$ so that $$\hat V^k(\hat\t,\sqrt{Z_{k-1}}\psi^{\le k})= \sum_{\{ P_{v} \}}[\sum_{\{N_v\geq C(P_v) \g^{-h_v\over\t}\}}\prod_{v {\rm not} e.p.}\d_{N_v,N_{v^1}+...+N_{v^{s_v}}}] \{\int dt\}\int d\xx_{v^0} %[\prod_{i\in V_1} v_{h_i}] %[\sum_{\{N_v\}}\prod_{v {\rm not} e.p.}\d_{N_v,N_{v^1}+...+N_{v^{s_v}}} %\prod_{i\in V_2} \hat\ph_{n_i} %e^{in_ip\xx_i}] \prod_{v not e.p.} |{Z_{h_v-1}\over Z_{h_v-2}}|^{|P_v|/2}{1\over s_v!}$$ $$\sum_{z}^*\tilde\E^T_{h_v}(\hat\psi(P_{v^1}/Q_{v^1}),... ,\hat\psi(P_{v^{s_v}}/Q_{v^{s_v}})$$ $$[\prod_{i\in V_1} v_{h_i}]\prod_{i\in V_2} \hat\ph_{n_i} e^{in_ip\xx_i}] \Eq(pal3)$$ We can bound $det G^T$ by the Grahm-Hadamard inequality ([Le],[BGPS]) finding $$|det G|\le \g^{{h\over 2}\sum_i \sum_{j=0}^2 (2j+1) |P_i^j|-h(k-1-s)} C^{\sum_i |P_i|-(k-1)}$$ where $P^j$ is the subset of $P$ of the fields with a derivative of order $j$ ($\D$ is of order $2$). Once that we have bounded the determinant, we have to make the integration over the coordinates and over the interpolation variables. It is convenient to change variables from $\{\xx\}$ to $\{\rr\}$, where $\{\rr\}$ is the collection of the difference $\xx_l-\yy_l$ appearing in the factors $\prod_l |\xx_l-\yy_l^a g^(\xx_l-\yy_l)$. Note that $\xx_l\equiv x_{ij}(t)$ so that the determinant of the Jacobian of this trasformation is a function of $t$, and one can worry about its integrability. However it is possible to show (see [BM], App. 3) that such determinant is exactly $1$. %integrating over the difference of $T$ %and making the change of variable $\xx_l-\yy_l\to \g^h(\xx_l-\yy_l)$, if $h$ %is the scale of the cluster to which $l$ belongs, we find $$|\hat V^k(\hat\t,\sqrt{Z_{k-1}}\psi^{\le k})|\le \g^{-k D(p_{v_0}} \sum_{\{ P_{v} \}}[\sum_{\{N_v\geq C(P_v)\g^{-h_v\over\t} \}} \prod_{v {\rm not} e.p.}\d_{N_v,N_{v^1}+...+N_{v^{s_v}}}] [\prod_{v not e.p. } |{Z_{h_v-1}\over Z_{h_v-2}}|^{|P_v|\over 2} $$ $$C^{|Q_v|-|P_v|}d\{d\xx_l\} J(\t,P_{v_0},x_{v_0})g^{-[D(P_v)+z_v(N_v,P_v)](h_v-h_{v'}} \prod_{i\in V_1} v_{h_i}]\prod_{i\in V_2} |\hat\ph_{n_i}|]$$ where $$\int d\{d\rr_l\} J(\t,P_{v_0},x_{v_0})=\int d\{d\rr_l\} \prod_{v not e.p.}{1\over s_v!}\sum_{T_v} \int dt \prod_{l\in T} \partial_{\rr_l}^{q_l} |\rrl|^a \bar g^{h_v}(\rr_l)$$ and $D(P_v)=-2+\sum_{f\in P_v}(1/2+m_f)$, with $m_f$ being the order of derivatives applied to the field of label $f$, $v'$ is the vertex preceding $v$ on the tree (so that $h_{v'}=h_v-1$). The presence $z_v(N_v,P_v)$ is due to the renormalization procedure and it is defined as \vskip1cm 1)$z_v(N_v,P_v)=1$ if $|P_v|=4$, $\sum_{f\in P_v} m_f=0$ and $\sum_i\e_i\o_i p_F+2 N_v p=0$ \vskip.5cm 2)$z_v(N_v,P_v)=1$ if $|P_v|=2$, $\sum_{f\in P_v} m_f=1$ and $\sum_i\e_i\o_i p_F+2 N_v p=0$ \vskip.5cm 3)$z_v(N_v,P_v)=2$ if $|P_v|=2$, $\sum_{f\in P_v} m_f=0$ and $\sum_i\e_i\o_i p_F+2 N_v p=0$ \vskip1cm Integrating we find %$$|\hat V^k(\hat\t,\sqrt{Z_{k-1}}\psi^{\le k})|\le \g^{-k D(p_{v_0}} %\sum_{\{ P_{v} \}}[\sum_{\{N_v\geq C(P_v)\g^{-h_v\over\t} \}} %\prod_{v {\rm not} e.p.}\d_{N_v,N_{v^1}+...+N_{v^{s_v}}}] %[\prod_{v not e.p. } |{Z_{h_v-1}\over Z_{h_v-2}}|^{|P_v|\over 2} $$ %$$\g^{-[D(P_v)+z_v(N_v,P_v)](h_v-h_{v'}} C^{\sum_i |P_{v^i}|-|P_v|} %\prod_{i\in V_1} v_{h_i}]\prod_{i\in V_2} |\hat\ph_{n_i}|]$$ %where $D(P_v)=-2+\sum_{f\in P_v}(1/2+m_f)$, with $m_f$ being the order of derivatives %applied to the field of label $f$, $v'$ is the vertex preceding $v$ on the tree (so that %$h_{v'}=h_v-1$). The presence $z_v(N_v,P_v)$ %is due to the renormalization procedure and it is defined as %\vskip1cm %1)$z_v(N_v,P_v)=1$ if $|P_v|=4$, $\sum_{f\in P_v} m_f=0$ and $\sum_i\e_i\o_i p_F+2 N_v p=0$ %\vskip.5cm %2)$z_v(N_v,P_v)=1$ if $|P_v|=2$, $\sum_{f\in P_v} m_f=1$ and $\sum_i\e_i\o_i p_F+2 N_v p=0$ %\vskip.5cm %3)$z_v(N_v,P_v)=2$ if $|P_v|=2$, $\sum_{f\in P_v} m_f=0$ and $\sum_i\e_i\o_i p_F+2 N_v p=0$ %\vskip1cm We can write then $$\prod_{v not e.p.} \g^{-[D(P_v)+z_v(N_v,P_v)](h_v-h_{v'}}\le \prod_{v not e.p.} \g^{-|P_v|\over 6}[\prod_{v\in T_4}\g^{h_v-h_{v'}}] [\prod_{v\in T_2}\g^{h_v-h_{v'}}][\prod_{v\in T_3}\g^{2(h_v-h_{v'})}]$$ where \vskip1cm 1)$T_4$ is the set of clusters with $|P_v|=4$, $\sum_{f\in P_v} m_f=0$ and $\sum_i\e_i\o_i p_F+2 N_v p\not =0$ \vskip.5cm 2)$T_2$ is the set of clusters with $|P_v|=2$, $\sum_{f\in P_v} m_f=1$ and $\sum_i\e_i\o_i p_F+2 N_v p\not =0$ \vskip.5cm 3)$T_3$ is the set of clusters with $|P_v|=2$, $\sum_{f\in P_v} m_f=0$ and $\sum_i\e_i\o_i p_F+2 N_v p\not =0$ \vskip.5cm Let we call {\it hard vertices} $\bar v$ the vertices such that there exists a $v$ following $\bar v$ such that for any $\bar v\le \hat v\le v$ it holds $|P_{\hat v}|=|P_{\bar v}||P_{v}|$ and $|P_{\bar v'}|\not=|P_{\bar v}|$, $|P_{\tilde v}|\not=|P_{v}|$ if $\tilde v'=v$. The set of such vertices is called $\TT$. Given $\t\in \t_{n,k}$ the number of hard vertices is bounded dy $C n$, if $C$ is a suitable constant. In fact in each graph associated to the tree the clusters associated to hard vertices must differ at least for a line, and the number of lines is of course $O(n)$. To an hard vertex $\bar v$ we associate a {\it depth}, defined in the following way: if $\bar v$ is the first hard vertex preceding an end point on the tree then $D_{\bar v}=1$, otherwise $D_{\bar v}=1+max_{\bar v''}\{D_{\bar v''}\}$, where $\bar v''$ are hard vertices following $bar v$ on the tree and such that there are no other hard vertices between $\bar v$ and anyone of the $\bar v''$. Note also that $D_{\bar v}\le -h_{\bar v}+2$. We can write $$\prod_{v not e.p.} \g^{-[D(P_v)+z_v(N_v,P_v)](h_v-h_{v'}}\le \prod_{v not e.p.} \g^{-|P_v|\over 6}[\prod_{\bar v\in \TT_4}\g^{-h_{\bar v'}}] [\prod_{\bar v \in \TT_2}\g^{-h_{\bar v'}}][\prod_{\bar v\in \TT_3} \g^{-2h_{\bar v'})}]$$ where $\bar v'$ is a the vertex preceding $\bar v$ and $\TT_i$ is the intersection between $\TT$ and $T_i$ . We write $T_i=T'_i\bigcup T''_i$ with $T'_i$ such that $N_v\not=0$ and $T''_i$ such that $N_v=0$. By Lemma 2 it follows that $$\prod_{\bar v\in T''_i}\g^{-h_{\bar v'}}\le C^n$$ On the other hand it is easy to show that $$\prod_i |\hat\ph_{n_i}|\le e^{\x n\over 2} \prod_i e^{-\x |n_i|/2}\prod_{\bar v\in\bar \TT} e^{-\x |N_{\bar v}|\over 2^{D_{\bar v}+1} }\Eq(fon)$$ In fact let us consider a hard vertex $\bar v_{1}$ not followed by any other hard vertex. Then we can write $$\prod_{i\in L_{\bar v_{1}}}e^{-\x |n_i|\over 2}\le e^{-\x |N_{\bar v_{1}} |\over 4} e^{-\x |N_{\bar v_{1}} |\over 4}$$ where the product is over the end-points contained in the cluster $\bar v_{1}$. So for the hard vertex $\bar v_1$ we have found the factor $e^{-\x |N_{\bar v_{1}} |\over 2^{D_{\bar v_1}+1}}$ and we have an extra factor $e^{-\x |N_{\bar v_{1}} |\over 4}$. Let us consider now a vertex $\bar v_{2}$ with depth $2$ \ie followed only by end points and hard vertex $v_{i}$ with depth $1$; we can write $$\prod_{i\in L_{\bar v_2}}e^{-\x |n_i|\over 2}\prod_{v_1\atop L_{v_1}\i L_{v_2}} e^{-\x |N_{\bar v_{1}} |\over 4}\le e^{-\x |N_{\bar v_{1}} |\over 8} e^{-\x |N_{\bar v_{1}} |\over 8}$$ so again the the factor $e^{-\x |N_{\bar v_{1}} |\over 2^{D_{\bar v_1}+1}}$ and an extra factor $e^{-\x |N_{\bar v_{1}} |\over 8}$. Proceding in this way we have \equ(fon), noting that to the end-points of kind $\iota,\t$ it is not associated any factor $e^{-\x}$ (this explains the factor $e^{{\x n\over 2}$ in front of \equ(fon)). Using Lemma 1 and the fact that $D_{\bar v}\le -h_{\bar v}+2$ we have $$\prod_i |\hat \ph_{n_i}|\le \prod_{\bar v\in \TT_1\cup \TT_2\cup \TT_3} e^{-\x C_2 \g^{-h_{\bar v}\over \t}\over 2^{-h_{\bar v}+3}}$$ At the end, summing over $N_v$, we can write $$|\hat V^k(\hat\t,\sqrt{Z_{k-1}}\psi^{\le k})|\le C^n\e^n \g^{-k D(P_{v_0})}\sum_{\{P_v\}}\prod_{v not e. p.}\g^{-|P_v|\over 8}$$ $$\prod_{\bar v\in\TT_1}\g^{-h_{\bar v}}e^{-\x C_2 \g^{-h_{\bar v}\over \t}\over 2^{-h_{\bar v}+3}} \prod_{\bar v\in\TT_2}\g^{-h_{\bar v}}e^{-\x C_2 \g^{-h_{\bar v}\over \t}\over 2^{-h_{\bar v}+3}} \prod_{\bar v\in\TT_3}\g^{-2h_{\bar v}}e^{-\x C_2 \g^{-h_{\bar v}\over \t}\over 2^{-h_{\bar v}+3}}$$ Choosing $\g$ so that $\g^{1\over\t}/2>1$ and remembering that the ard verices are $\le C_1 n$ we have that $$prod_{\bar v\in\TT_i}\g^{-2h_{\bar v}}e^{-\x C_2 \g^{-h_{\bar v}\over \t}\over 2^{-h_{\bar v}+3}} \le C_2^n$$ %$$\sum_{r=0}^\io \g^{2r} e^{-\x C_2 \g^{r\over \t}\over 2^{3+3}}\le C$ By a standard calculation $$\sum_{\t\in\t_{k,n}}\sum_{\{\P_v}}\prod_{v not e.p.} \g^{-|P_v|\over 8}\le C_3^n$$ and this completes the proof of Theorem 2. \vskip4.cm \pagina \centerline{\titolo References} \* \halign{\hbox to 1.2truecm {[#]\hss} & \vtop{\advance\hsize by -1.25 truecm \0#}\cr A& {S. Aubry in Polarons and Bipolarons in hich $T_c$ superconductors and related materials}, editors Salye E, Alexandrov A., Liang W AAR& {S. Aubry, G. Abramovici, J. Raimbaut: Chaotic polaronic and bipolaronic states in the adiabatic Holstein model, {\it J. Stat. Phys.} {\bf 67}, 675--780 (1992). }\cr % BLT& {J. Belissard, R. Lima, D. Testard: A metal-insulator transition for almost Mathieu model, {\it Comm. Math. Phys.} {\bf 88}, 207--234 (1983). }\cr % D& {H. Davenport: {\sl The Higher Arithmetic}, Dover, New York, 1983.}\cr % DS& {E.I. Dinaburg, Ya.G. Sinai: On the one dimensional Schroedinger equation with a quasiperiodic potential, {\it Funct. Anal. and its Appl.} {\bf 9}, 279--289 (1975). }\cr % E& {L.H. Eliasson: Floquet solutions for the one dimensional quasi periodic Schroedinger equation, {\it Comm. Math. Phys.} {\bf 146}, 447--482 (1992). }\cr % G& {G. Gallavotti: Twistless KAM tori, {\it Comm. Math. Phys.} {\bf 164}, 145--156 (1994). }\cr % GM& {G. Gentile, V. Mastropietro: Methods for the analysis of the Lindstedt series for KAM tori and renormalizability in classical mechanics. A review with some applications, {\it Rev. Math. Phys.} {\bf 8}, 393--444 (1996). }\cr % H& {T. Holstein: Studies of polaron motion, part 1. The molecular-crystal model. {\it Ann. Phys.} {\bf 8}, 325--342, (1959). }\cr % JM& {R.A. Johnson, J. Moser: The rotation number for almost periodic potentials, {\it Commun. Math. Phys.} {\bf 84}, 403--438 (1982).}\cr % KL& {T. Kennedy, E.H. Lieb: An itinerant electron model with crystalline or magnetic long range order, {\it Physica A} {\bf 138}, 320--358 (1986). }\cr % %L& {E.H. Lieb: %A model for crystallization: a variation %of the Hubbard model, %{\it Physica A} {\bf 140}, 240--250 (1986). }\cr % LMR& {Lee P.A., Rice P.M., Anderson P.W., Solid State Comm. 14,703,1974. }\cr % L& Lee P.A. Nature 291,11-12,1981. }\cr % LM& {J. L. Lebowitz, N. Macris: Peierls instability and low temperature phases of the Static Holstein Model: rigorous results, {\it J. Stat. Phys.} {\bf 76}, 91--123 (1994). }\cr % MP& {J. Moser, J. P\"oschel: An extension of a result by Dinaburg and Sinai on quasi periodic potentials, {\it Comment. Math. Helv.} {\bf 59}, 39--85 (1984). }\cr % NO& {J.W. Negele, H. Orland: {\sl Quantum many-particle systems}, Addison-Wesley, New York, 1988. }\cr % PF& {L. Pastur, A. Figotin: {\sl Spectra of random and almost periodic operators}, Springer, Berlin, 1991. }\cr % P& {R.E. Peierls: {\sl Quantum theory of solids}, Clarendon, Oxford, 1955. }\cr % %T& {E.C. Titchmarsh: %{\sl Eigenfunctions expansions associated with second %order differential equations}, %Clarendon, Oxford, 1955. }\cr % %To& {M. Toda: %{\sl Theory of nonlinear lattices}, Springer, Berlin, 1958. }\cr } % \* \ciao {\bf BIBLIOGRAPHY} \vskip.5cm [P] Peierls,R.E. "Quantum theory of solids", Oxford University press,London, 1955 \vskip.5cm [F] Frohlich H, Proc. R. Soc., A223,296,1954 \vskip.5cm [LRA] Lee P.A., Rice P.M., Anderson P.W., Solid State Comm. 14,703,1974 \vskip [L] Lee P.A. Nature 291,11-12,1981 \vskip [A] Aubry S. in {\it Polarons and Bipolarons in hich $T_c$ superconductors and related materials}, editors Salye E, Alexandrov A., Liang W \ciao \ciao This imply the analiticity in terms of the effective potential if $|v_h|\le \e$ and $e^{-c_1\e^2}\le {Z_h\over Z_{h-1}}\le e^{-c_2\e^2}$$ \ciao $D_{\bar v}$ is the {\it depth} of the cluster $L_{\bar v}$ and $D_{\bar v}\le -h_{\bar v}+2$. We obtain $$C^n\e^n \g^{-k D(p_{v_0} \sum_{\{P_v\}}\prod_{v not e.p.} |{Z_{h_v-1}\over Z_{h_v-2}}|^{|P_v|\over 2} \g^{-|P_v|\over 6}[\prod_{\bar v\atop v\in T'}\g^{-2h_{\bar v}}e^{-\x { \g^{-h_{\bar v}\over\t}\over 2^{-h_{\bar v}} }]$$ \ciao Estimating the determinant by the Grahm-Hadamard inequality it follows $$\le\bar\e^n\sum_{\{ P_{v_0} \}}\{\int dt\} \prod_{v not e.p.} |{Z_{h_v}\over Z_{h_v-1}}|^{|P_v|/2} C^{|Q_v|-|P_v|} J(\t,P_{v_0},x_{v_0})\g^{h_v/2 \sum_{j=0}^2 (2j+1)\sum_i(|P^j_{v^i}|-|P^j_{v^i}|)}$$ \ciao %$G'^T(s)$ is a %$(n-k+1) %\times (n-k+1)$ matrix (if $n$ is the total number of fields), whose elements are %$G^T_{jij'i'}=S_{jj'} \partial^{q_{ij}}\partial^{q_{i'j'}} %g^h(x_{ij}-x_{i'j'})$ with $x_{ij}-x_{i'j'}$ non belonging %to $T$, $\partial^{q_{ij}}=\partial^{q(\bar f)}_{x(f)}$ if $x_{i'j'}=x(\bar f)$, %$S_{jj'}=\prod_{i=j}^{j'-1} s_t$ and $dP_T(s)$ is a normalized measure which depends on %$s_t$ and $T$ (for the explicit formula of $dP_T$ and for its derivation, one can look for istance %[BGPS]) (see the Appendix). where $\{\partial\}$ generated by the $\RR$ operation, and $\tilde\Epsilon^T_h$ is given by eq.(!!) with the following differences: \vskip.5cm 1)the points $x_{ij}$ can be interpolated points \vskip.5cm 2)$g(x_l-y_l)$ is replaced by $|x'_l-y'_l| g(x'_l-y'_l)$ \vskip.5cm \ciao Integrating the $\d$'s due to the first $\RR$ we obtain a factor $(x_i-x_j)^a$ which can be written in the following way. Let be a generic tree $T$ connecting the cluster of coordinates in $\t_1,..,\t_{s_{v_0}}$. So by definition $$(x-y)^a=(\sum (x_{ij}-x_{i',j'})^a$$ where $x_{ij}-x_{i',j'}$ are defined in the following way: 1)$x_{ij},x_{i',j'}$ both belongs to the tree 2)or at least one it not belongs to the tree. So insterting in the above expression, and remembering that $$\e^T_h=\sum_T \prod_{l\in T} g(l) det G$$ we have that or $x_{ij}-x_{i',j'}$ coincides with $l$ or not; in this case we use eq.(!!) to conclude that $$(x-y) \RR W()=W$$ if $p_v=4$, and so one. We obtain them .... Now we iterate proceding in the same way for $\RR V$, and at the end we find: \ciao with suitable constants $v_{-1}$, and, by following the same procedure which led from \equ(3.9) to \equ(3.12), we have % $$ \eqalign{ \int P(d\psi^{(\le -1)}) \, e^{\VV^{(-1)}(\psi^{(\le -1)})} & ={1\over \NN_1} \int \tilde P (d\psi^{(\le -1)}) \, e^{\tilde \VV^{(-1)}(\psi^{(\le -1)})} \cr & = {1\over \NN_1}\int P(d\psi^{(\le -2)}) \int \tilde P(d\psi^{(-1)}) \, e^{\tilde \VV^{(-1)} (\psi^{(\le -1)})} \cr} \Eq(3.22) $$ % where, up to a constant, % $$ \eqalign{ \tilde P(d\psi^{(\le -1)}) &= \prod_{\kk'}\prod_{\o=\pm1} d\psi^{(\le -1)+}_{\kk'+\o\pp_F,\o} d\psi^{(\le -1)-}_{\kk'+\o\pp_F,\o} \cr \exp \Big\{ &-\sum_{\o=\pm1} {1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}}\, Z_{-2}(\kk') C_{-1}(\kk') \, \Big[ \Big(-ik_0-(\cos k'-1)\cos p_F +\o v_0\sin k' \Big) \cr & \qquad\psi^{(\le-1)+}_{\kk'+\o\pp_F,\o} \psi^{(\le-1)-}_{\kk'+\o\pp_F,\o} - \sigma_{-2}(\kk') \, \psi^{(\le-1)+}_{\kk'+\o\pp_F,\o} \psi^{(\le-1)-}_{\kk'-\o\pp_F,-\o} \Big] \Big\} \; , \cr} \Eq(3.23) $$ % with $Z_{-2}(\kk')\sigma_{-2}(\kk')$ $=$ $Z_{-1}(\sigma_{-1}(\kk')+C_{-1}^{-1}(\kk') s_{-1}$ and $ % $$ \LL\tilde \VV^{(-1)}= \g^{-1} \nu_{-1} F_\nu^{(-1)} \; . \Eq(3.24) $$ The above procedure can be iterated, and at each step one has to perform the integration % $$ \eqalignno{ &\quad\int P(d\psi^{(\le h)}) \, e^{\VV^{(h)}(\psi^{(\le h)})} ={1\over \NN_h} \int \tilde P(d\psi^{(\le h)})\,e^{\tilde \VV^{(h)}(\psi^{(\le h)})} = &\eq(3.25)\cr & ={1\over \NN_h} \int P(d\psi^{(\le h-1)})\int \tilde P(d\psi^{(h)}) \, e^{\tilde \VV^{(h)}(\psi^{(\le h)})} = {1\over \NN_h} \int P(d\psi^{(\le h-1)}) e^{\VV^{(h-1)}(\psi^{(\le h-1)}) +\tilde E_h}\; , \cr}$$ % which gives $\sigma_{h-1}(\kk')$ $=$ $\sigma_h(\kk')+C_{h}^{-1}(\kk') s_{h}$, and defines the propagator % $$ g^{(h)}(\xx;\yy) = \sum_{\o,\o'=\pm1} e^{-i(\o x - \o' y)p_F}\, g^{(h)}_{\o,\o'}(\xx;\yy) \; , \Eq(3.26) $$ % with % $$ \eqalign{ g^{(h)}_{\o,\o'}(\xx;\yy) & \= \int \tilde P(d\psi^{(h)})\, \psi^{(h)-}_{\xx,\o}\psi^{(h)+}_{\yy,\o'} \cr & = {1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \, e^{-i\kk'\cdot(\xx-\yy)}f_h(\kk')[T_{h}^{-1}(\kk')]_{\o,\o'} \; , \cr} \Eq(3.27) $$ % and % $$ \cases{ [T_{h}(\kk')]_{1,1} = \left(-ik_0-(\cos k'-1)\cos p_F+ v_0\sin k' \right) \; , & \cr [T_{h}(\kk')]_{1,2} = [T_{h}(\kk')]_{2,1} = - \sigma_{h}(\kk') \; , & \cr [T_{h}(\kk')]_{2,2} = \left(-ik_0-(\cos k'-1)\cos p_F-v_0\sin k'\right) \; , \cr} \Eq(3.28) $$ % so that % $$ T_{h}^{-1}(\kk')= {1\over A_{h}(\kk') } \left( \matrix{ [\t_{h}(\kk')]_{1,1} & [\t_{h}(\kk')]_{1,2} \cr [\t_{h}(\kk')]_{2,1} & [\t_{h}(\kk')]_{2,2} \cr} \right) \; , \Eq(3.29) $$ % where % $$ \cases{ [\t_{h}(\kk')]_{1,1} = [-ik_0-(\cos k'-1) \cos p_F-v_0\sin k'] \; , & \cr [\t_{h}(\kk')]_{1,2} = [\t_{h}(\kk')]_{2,1} = \sigma_{h}(\kk') \; , & \cr [\t_{h}(\kk')]_{2,2} = [-ik_0-(\cos k'-1)\cos p_F + v_0 \sin k' ] \; . & \cr} \Eq(3.30) $$ % and % $$A_{h}(\kk') = \left[-ik_0-(\cos k'-1)\cos p_F\right]^2 - \left(v_0\sin k'\right)^2 - \s_{h}(\kk')^2 \; .\Eq(3.31) $$ % We can define also % $$ \eqalign{ g^{(\le h)}_{\o,\o'}(\xx;\yy) & \= \int \tilde P(d\psi^{(\le h)})\, \psi^{(\le h)-}_{\xx,\o}\psi^{(\le h)+}_{\yy,\o'} \cr & = {1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \, e^{-i\kk'\cdot(\xx-\yy)}C_h^{-1}(\kk')[T_{h}^{-1}(\kk')]_{\o,\o'} \; , \cr} \Eq(3.32) $$ % where the last identity follows from \equ(3.23) (with $h$ in place of $-1$) and \equ(3.25). Set % $$ \tilde g^{(h)}_{\o,\o'}(\kk') = f_h(\kk')[T_{h}^{-1}(\kk')]_{\o,\o'} \; , \qquad \tilde g^{(\le h)}_{\o,\o'}(\kk') = C_h^{-1}(\kk')[T_{h}^{-1}(\kk')]_{\o,\o'} \; , \Eq(3.33) $$ % so that % $$ \eqalign{ g^{(h)}_{\o,\o'}(\xx;\yy) & = {1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \, e^{-i\kk'\cdot(\xx-\yy)} \, \tilde g^{(h)}_{\o,\o'}(\kk')\; , \cr % g^{(\le h)}_{\o,\o'}(\xx;\yy) & = {1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \, e^{-i\kk'\cdot(\xx-\yy)} \, \tilde g^{(\le h)}_{\o,\o'}(\kk')\; . \cr} \Eq(3.34) $$ The localized part of the effective potential will be written as % $$ \LL \tilde \VV^{(h)}=\g^h \n_h F_\n^{(h)} \; , \Eq(3.35) $$ % which defines the {\sl running coupling constants} $\n_h$. Moreover % $$\s_h(\kk')=\sum_{j=h}^{0} C_{j}^{-1}(\kk') \, s_j \; . \Eq(3.36) $$ % Note that, thanks to the definition of $\c(\kk')$, see \equ(2.3), if $f_h(\kk') \not= 0$, we have % $$ \sigma_h(\kk')= C_h^{-1}(\kk') \, s_h + \sum_{j=h+1}^{0} s_j \; . \Eq(3.37) $$ % Hence, by \equ(2.5) and the second equation in \equ(3.8), $\s_h(\kk')$ is a smooth function on $\TTT^1\times \RRR$, such that $\s_h(\kk')= \sum_{j=h}^0 s_j$ for $0\le |\kk'|\le t_0\g^h$ and $\s_h(\kk')= \sum_{j=h+1}^0 s_j$ for $|\kk'|= a_0\g^h$; we define $\sigma_h\equiv \sum_{j=h}^0 s_j$. The r.h.s of \equ(3.9) can be written as % $$ {1 \over \NN_0}\int P(d\psi^{(\le -1)}) \int \tilde P(d\psi^{(0)}) \, e^{\tilde \VV^{(0)}(\psi^{(\le 0)})} \; , \Eq(3.12) $$ % where $ P(d\psi^{(\le -1)})$ and $\tilde P(d\psi^{(0)})$ are given by \equ(3.10) with $C_0(\kk')$ replaced with $C_{-1}(\kk')$ and $f_0^{-1}(\kk')$ respectively, and $\psi^{(\le 0)}$ replaced with $\psi^{(\le -1)}$ and $\psi^{(0)}$ respectively. The Grassmanian integration $\tilde P(d\psi^{(\le 0)})$ has propagator % $$ g^{(0)}(\xx;\yy) = \sum_{\o,\o'=\pm1} e^{-i(\o x - \o' y)p_F}\, g^{(0)}_{\o,\o'}(\xx;\yy) \; , \Eq(3.13) $$ % if % $$ g^{(0)}_{\o,\o'}(\xx;\yy)\=\int \tilde P(d\psi^{(0)})\, \psi^{(0)-}_{\xx,\o}\psi^{(0)+}_{\yy,\o'} \Eq(3.14) $$ % is given by % $$ g^{(0)}_{\o,\o'}(\xx;\yy)={1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \, e^{-i\kk'\cdot(\xx-\yy)}f_0(\kk')[T_{0}^{-1}(\kk')]_{\o,\o'} \; , \Eq(3.15) $$ % where the $2\times2$ matrix $T_{0}(\kk')$ has elements % $$ \cases{ [T_{0}(\kk')]_{1,1} = \left(-ik_0-(\cos k'-1)\cos p_F+ v_0\sin k' \right) \; , & \cr [T_{0}(\kk')]_{1,2} = [T_{0}(\kk')]_{2,1} = - \sigma_{0}(\kk') \; , & \cr [T_{0}(\kk')]_{2,2} = \left(-ik_0-(\cos k'-1)\cos p_F-v_0\sin k'\right) \; , \cr} \Eq(3.16) $$ % which is well defined on the support of $f_0(\kk')$, so that, if we set % $$ A_{0}(\kk') = \det T_0(\kk') = [ -ik_0-(\cos k'-1)\cos p_F ]^2 - (v_0\sin k')^2 - [\sigma_{0}(\kk')]^2 \; ,\Eq(3.17) $$ % then % $$ T_{0}^{-1}(\kk')= {1\over A_{0}(\kk') } \left( \matrix{ [\t_{0}(\kk')]_{1,1} & [\t_{0}(\kk')]_{1,2} \cr [\t_{0}(\kk')]_{2,1} & [\t_{0}(\kk')]_{2,2} \cr} \right) \; , \Eq(3.18) $$ % with % $$ \cases{ [\t_{0}(\kk')]_{1,1} = \left[-ik_0-(\cos k'-1) \cos p_F-v_0\sin k'\right] \; , & \cr [\t_{0}(\kk')]_{1,2} = [\t_{0}(\kk')]_{2,1} = \sigma_{0}(\kk') \; , & \cr [\t_{0}(\kk')]_{2,2} = \left[-ik_0-(\cos k'-1)\cos p_F + v_0 \sin k' \right] \; . & \cr} \Eq(3.19) $$ % We perform the integration % $$ \int \tilde P(d\psi^{(0)}) \, e^{\tilde \VV^{(0)}(\psi^{(\le 0)})} \= e^{\VV^{-1}(\psi^{(\le -1)}) + \tilde E_0} \; , \Eq(3.20) $$ % where $\tilde E_0 = \log \int \tilde P(d\psi^{(0)}) \, \exp \{\tilde \VV^{(0)}(\psi^{(0)}) \}$. We can write % $$ \LL \VV^{(-1)}= \g^{-1}\n_{-1} F_\nu^{(-1)}+s_{-1} F_\sigma^{(-1)} \; , \Eq(3.21) $$ % with suitable constants $\n_{-1}$ and $s_{-1}$, and, by following the same procedure which led from \equ(3.9) to \equ(3.12), we have % $$ \eqalign{ \int P(d\psi^{(\le -1)}) \, e^{\VV^{(-1)}(\psi^{(\le -1)})} & ={1\over \NN_1} \int \tilde P (d\psi^{(\le -1)}) \, e^{\tilde \VV^{(-1)}(\psi^{(\le -1)})} \cr & = {1\over \NN_1}\int P(d\psi^{(\le -2)}) \int \tilde P(d\psi^{(-1)}) \, e^{\tilde \VV^{(-1)} (\psi^{(\le -1)})} \cr} \Eq(3.22) $$ % where, up to a constant, % $$ \eqalign{ \tilde P(d\psi^{(\le -1)}) &= \prod_{\kk'}\prod_{\o=\pm1} d\psi^{(\le -1)+}_{\kk'+\o\pp_F,\o} d\psi^{(\le -1)-}_{\kk'+\o\pp_F,\o} \cr \exp \Big\{ &-\sum_{\o=\pm1} {1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}}\, C_{-1}(\kk') \, \Big[ \Big(-ik_0-(\cos k'-1)\cos p_F +\o v_0\sin k' \Big) \cr & \qquad\psi^{(\le-1)+}_{\kk'+\o\pp_F,\o} \psi^{(\le-1)-}_{\kk'+\o\pp_F,\o} - \sigma_{-1}(\kk') \, \psi^{(\le-1)+}_{\kk'+\o\pp_F,\o} \psi^{(\le-1)-}_{\kk'-\o\pp_F,-\o} \Big] \Big\} \; , \cr} \Eq(3.23) $$ % with $\sigma_{-1}(\kk')$ $=$ $\sigma_0(\kk')+C_{-1}^{-1}(\kk') s_{-1}$ and % $$ \LL\tilde \VV^{(-1)}= \g^{-1} \n_{-1} F_\nu^{(-1)} \; . \Eq(3.24) $$ The above procedure can be iterated, and at each step one has to perform the integration % $$ \eqalignno{ &\quad\int P(d\psi^{(\le h)}) \, e^{\VV^{(h)}(\psi^{(\le h)})} ={1\over \NN_h} \int \tilde P(d\psi^{(\le h)})\,e^{\tilde \VV^{(h)}(\psi^{(\le h)})} = &\eq(3.25)\cr & ={1\over \NN_h} \int P(d\psi^{(\le h-1)})\int \tilde P(d\psi^{(h)}) \, e^{\tilde \VV^{(h)}(\psi^{(\le h)})} = {1\over \NN_h} \int P(d\psi^{(\le h-1)}) e^{\VV^{(h-1)}(\psi^{(\le h-1)}) +\tilde E_h}\; , \cr}$$ % which gives $\sigma_{h-1}(\kk')$ $=$ $\sigma_h(\kk')+C_{h}^{-1}(\kk') s_{h}$, and defines the propagator % $$ g^{(h)}(\xx;\yy) = \sum_{\o,\o'=\pm1} e^{-i(\o x - \o' y)p_F}\, g^{(h)}_{\o,\o'}(\xx;\yy) \; , \Eq(3.26) $$ % with % $$ \eqalign{ g^{(h)}_{\o,\o'}(\xx;\yy) & \= \int \tilde P(d\psi^{(h)})\, \psi^{(h)-}_{\xx,\o}\psi^{(h)+}_{\yy,\o'} \cr & = {1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \, e^{-i\kk'\cdot(\xx-\yy)}f_h(\kk')[T_{h}^{-1}(\kk')]_{\o,\o'} \; , \cr} \Eq(3.27) $$ % and % $$ \cases{ [T_{h}(\kk')]_{1,1} = \left(-ik_0-(\cos k'-1)\cos p_F+ v_0\sin k' \right) \; , & \cr [T_{h}(\kk')]_{1,2} = [T_{h}(\kk')]_{2,1} = - \sigma_{h}(\kk') \; , & \cr [T_{h}(\kk')]_{2,2} = \left(-ik_0-(\cos k'-1)\cos p_F-v_0\sin k'\right) \; , \cr} \Eq(3.28) $$ % so that % $$ T_{h}^{-1}(\kk')= {1\over A_{h}(\kk') } \left( \matrix{ [\t_{h}(\kk')]_{1,1} & [\t_{h}(\kk')]_{1,2} \cr [\t_{h}(\kk')]_{2,1} & [\t_{h}(\kk')]_{2,2} \cr} \right) \; , \Eq(3.29) $$ % where % $$ \cases{ [\t_{h}(\kk')]_{1,1} = [-ik_0-(\cos k'-1) \cos p_F-v_0\sin k'] \; , & \cr [\t_{h}(\kk')]_{1,2} = [\t_{h}(\kk')]_{2,1} = \sigma_{h}(\kk') \; , & \cr [\t_{h}(\kk')]_{2,2} = [-ik_0-(\cos k'-1)\cos p_F + v_0 \sin k' ] \; . & \cr} \Eq(3.30) $$ % and % $$A_{h}(\kk') = \left[-ik_0-(\cos k'-1)\cos p_F\right]^2 - \left(v_0\sin k'\right)^2 - \s_{h}(\kk')^2 \; .\Eq(3.31) $$ % We can define also % $$ \eqalign{ g^{(\le h)}_{\o,\o'}(\xx;\yy) & \= \int \tilde P(d\psi^{(\le h)})\, \psi^{(\le h)-}_{\xx,\o}\psi^{(\le h)+}_{\yy,\o'} \cr & = {1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \, e^{-i\kk'\cdot(\xx-\yy)}C_h^{-1}(\kk')[T_{h}^{-1}(\kk')]_{\o,\o'} \; , \cr} \Eq(3.32) $$ % where the last identity follows from \equ(3.23) (with $h$ in place of $-1$) and \equ(3.25). Set % $$ \tilde g^{(h)}_{\o,\o'}(\kk') = f_h(\kk')[T_{h}^{-1}(\kk')]_{\o,\o'} \; , \qquad \tilde g^{(\le h)}_{\o,\o'}(\kk') = C_h^{-1}(\kk')[T_{h}^{-1}(\kk')]_{\o,\o'} \; , \Eq(3.33) $$ % so that % $$ \eqalign{ g^{(h)}_{\o,\o'}(\xx;\yy) & = {1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \, e^{-i\kk'\cdot(\xx-\yy)} \, \tilde g^{(h)}_{\o,\o'}(\kk')\; , \cr % g^{(\le h)}_{\o,\o'}(\xx;\yy) & = {1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \, e^{-i\kk'\cdot(\xx-\yy)} \, \tilde g^{(\le h)}_{\o,\o'}(\kk')\; . \cr} \Eq(3.34) $$ The localized part of the effective potential will be written as % $$ \LL \tilde \VV^{(h)}=\g^h \n_h F_\n^{(h)} \; , \Eq(3.35) $$ % which defines the {\sl running coupling constants} $\n_h$. Moreover % $$\s_h(\kk')=\sum_{j=h}^{0} C_{j}^{-1}(\kk') \, s_j \; . \Eq(3.36) $$ % Note that, thanks to the definition of $\c(\kk')$, see \equ(2.3), if $f_h(\kk') \not= 0$, we have % $$ \sigma_h(\kk')= C_h^{-1}(\kk') \, s_h + \sum_{j=h+1}^{0} s_j \; . \Eq(3.37) $$ % ssible Hence, by \equ(2.5) and the second equation in \equ(3.8), $\s_h(\kk')$ is a smooth function on $\TTT^1\times \RRR$, such that $\s_h(\kk')= \sum_{j=h}^0 s_j$ for $0\le |\kk'|\le t_0\g^h$ and $\s_h(\kk')= \sum_{j=h+1}^0 s_j$ for $|\kk'|= a_0\g^h$; we define $\sigma_h\equiv \sum_{j=h}^0 s_j$. \ciao