The paper is written in Plain Tex and contains 11 figures, inserted in a postscript file, to be found between the lines FIGURES: and ENDFIGURES: below, before the body of the paper. This file should be printed separately by a postscript laser printer. In the printed paper there will appear some empty space in place of the figures. 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\newdimen\gwidth \def\BOZZA{ \def\alato(##1){ {\vtop to \profonditastruttura{\baselineskip \profonditastruttura\vss \rlap{\kern-\hsize\kern-1.2truecm{$\scriptstyle##1$}}}}} \def\galato(##1){ \gwidth=\hsize \divide\gwidth by 2 {\vtop to \profonditastruttura{\baselineskip \profonditastruttura\vss \rlap{\kern-\gwidth\kern-1.2truecm{$\scriptstyle##1$}}}}} } \def\alato(#1){} \def\galato(#1){} \def\veroparagrafo{\number\numsec}\def\veraformula{\number\numfor} \def\verafigura{\number\numfig} \def\geq(#1){\getichetta(#1)\galato(#1)} \def\Eq(#1){\eqno{\etichetta(#1)\alato(#1)}} \def\eq(#1){\etichetta(#1)\alato(#1)} \def\Eqa(#1){\eqno{\etichettaa(#1)\alato(#1)}} \def\eqa(#1){\etichettaa(#1)\alato(#1)} \def\eqv(#1){\senondefinito{fu#1}$\clubsuit$#1\write16{No translation for #1}% \else\csname fu#1\endcsname\fi} %\def\equ(#1){\senondefinito{e#1}\eqv(#1)\else\csname e#1\endcsname\fi} \def\equ{} \let\EQS=\Eq\let\EQ=\Eq \let\eqs=\eq \let\Eqas=\Eqa \let\eqas=\eqa \def\include#1{ \openin13=#1.aux \ifeof13 \relax \else \input #1.aux \closein13 \fi} %\openin14=\jobname.aux \ifeof14 \relax \else %\input \jobname.aux \closein14 \fi %\openout15=\jobname.aux \vglue2truecm \nopagenumbers\numfig=1 {}\noindent {\bf Beta function and Schwinger functions for a many fermions system in one dimension. Anomaly of the Fermi surface.} \vglue1.5truecm {\bf G. Benfatto\footnote{${}^1$} {\arm Dipartimento di Matematica, Universit\`a di Roma ``Tor Vergata'', 00133 Roma, Italia.},} {\bf G. Gallavotti\footnote{${}^2$}{\vtop{\hsize=15.9truecm\arm \baselineskip=12pt \\Dipartimento di Fisica, Universit\`a di Roma ``La Sapienza'', P. Moro 5, 00185 Roma, Italia; and Rutgers University, Mathematics Dept., Hill Center, New Brunswick, N.J. 08903, USA.}},} {\bf A. Procacci\footnote{${}^3$}{\arm Dipartimento di Fisica, Universit\`a di Roma ``La Sapienza''.},} {\bf B. Scoppola\footnote{${}^4$}{\arm Dipartimento di Matematica, Universit\`a di Roma ``La Sapienza''.}} \vglue1.5truecm {\bf Abstract}: {\it We present a rigorous discussion of the analyticity properties of the beta function and of the effective potential for the theory of the ground state of a one dimensional system of many spinless fermions. We show that their analyticity domain as a function of the running couplings is a polydisk with positive radius bounded below, uniformly in all the cut offs (infrared and ultraviolet) necessary to give a meaning to the formal Schwinger functions. We also prove the vanishing of the scale independent part of the beta function showing that this implies the analyticity of the effective potential and of the Schwinger functions in terms of the bare coupling. Finally we show that the pair Schwinger function has an anomalous long distance behaviour.} \vglue 1.5truecm \item{\S1 } Introduction. \item{\S2 } Functional integral representation of fermionic correlation functions. \item{\S3 } Ultraviolet limit for the effective potential. \item{\S4 } The effective potential in the infrared region. Failure of normal scaling. \item{\S5 } The effective potential in the infrared region. Running couplings and anomalous scaling. \item{\S6 } The two point Schwinger function. \item{\S7 } The vanishing of the beta function and completion of the theory of spinless Fermi systems. {\parindent=55pt \item{Appendix 1 } Bounds on the free propagators. \item{Appendix 2 } The Gramm-Hadamard (and related) inequalities. \item{Appendix 3 } The bound \equ(5.60). \item{Appendix 4 } Simplified beta functional.} \pagina \footline={\hss\tenrm\folio\hss} \pageno=1 \vglue1.truecm {\it\S1 Introduction.} \vglue1.truecm\numsec=1\numfor=1 In this paper we study a system of interacting one dimensional fermions. The Hamiltonian for $n$ spinless particles in a periodic box of length $L$ will be: % $$H=\sum_{i=1}^n({-\D_{\xx_i}\over 2m}-\m )+ 2\l\sum_{i0$ is the particles mass, $\m$ is the chemical potential, $2\l \bar v(\rr)$ is the pair potential, which we suppose bounded, smooth, even in $\rr$ and with finite range $p_0^{-1}$. Physically one defines the Fermi momentum $p_F$ so that the ground state energy of $H$ has the minimum at $n=2p_FL/2\pi$ when $\m=p_F^2/2m$, while the mass of the particles is defined by computing the minimum energy increase obtained by adding one particle to the ground state. Usually one requires that $p_F$ has a given value and that the minimum energy increase has the form: % $$e(\kk_0)=(\kk_0^2-p_F^2)/2m\Eq(1.2)$$ % where $\kk_0$ is the smallest $\kk$ of the form $2\pi s L^{-1}$, $s$ integer, larger than $p_F$; this, however, cannot be imposed on \equ(1.1) as there are not free parameters. Hence we shall study: % $$H=\sum_{i=1}^n({-\D_{\xx_i}\over 2m}-\m )+ \a\sum_{i=1}^n({-\D_{\xx_i}\over 2m}-\m )+ \n n+ 2\l \sum_{i0$, that $p_F$ and $L$ are so related that $2\pi L^{-1} (n_F+1/2)=p_F$ with $n_F$ integer; this implies, in particular, that no particle can have momentum $\pm p_F$ and that $\kk_0 = p_F+{\p\over L}$. It is very useful to write the Hamiltonian $H$ in second quantization, \ie in terms of creation and annihilation operators $a^+_{k}, a^-_{ k}$. Defining: % $$\ps{\pm}{ \xx}=L^{-1/2}\sum_{ \kk}e^{\pm i \kk\cdot \xx} a^{\pm}_{ \kk}\Eq(1.4)$$ % we have: % $$\eqalign{ T&=\sum_{\V k}({{\V k}^2\over2m}-\m)a^+_{\V k}a^-_{\V k} = \int_L\,d\V x\, ({1\over2m}\Dp\ps+{\V x}\Dp\ps-{\V x}-\m\ps+{\V x}\ps-{\V x})\cr N&=\sum_{\V k}a^+_{\V k}a^-_{\V k} = \int_L\,d\V x\,\ps+{\V x} \ps-{\V x}\cr \bar V&= \l \int_{L\times L} d\V xd\V y\,\bar v(\V x-\V y)\ \ps+{\V x}\ps+{\V y} \ps-{\V y}\ps-{\V x}\cr}\Eq(1.5)$$ % Let us denote $E^{(n)}$ the ground state energy of the system with $n$ particles and let us define $N=2n_F+1$. By first order perturbation theory it is easy to see that: % $$\eqalignno{ E^{(N+1)}-E^{(N)} &= (1+\a) e(p_F+{\p\over L}) + \n + 2\l \fra{2\p}L \sum_{\kk: e(\kk)<0} [\hat v(0)-\hat v(p_F+{\p\over L}-\kk)] & \eq(1.6) \cr E^{(N-1)}-E^{(N)} &= -(1+\a) e(p_F-{\p\over L}) - \n - 2\l \fra{2\p}L \sum_{\kk: e(\kk)<0} [\hat v(0)-\hat v(p_F-{\p\over L}-\kk)] & \eq(1.7) \cr}$$ % where $2\p \hat v(\kk) = \int d\xx e^{-i\kk\xx} \bar v(\xx)$. The conditions that the system has $p_F$ as Fermi momentum and $m$ as mass in presence of interaction can be translated in the conditions: % $$E^{(N\pm 1)}-E^{(N)} = \pm e(p_F\pm{\p\over L}) \Eq(1.8)$$ % which imply, by using $\hat v(p_F \pm {\p\over L}-\kk) = \hat v(p_F-\kk) \pm {\p\over L} \hat v'(p_F-\kk) + O(L^{-2})$, that: % $$\eqalign{ &\n + 2\l \int_{e(\kk)<0} d\kk [\hat v(0)-\hat v(p_F-\kk)] + O(L^{-1}) = 0 \cr &\a {p_F\over m} - 2\l \int_{e(\kk)<0} d\kk \hat v'(p_F-\kk) + O(L^{-1}) = 0 \cr} \Eq(1.9)$$ Recursively one can determine the higher order corrections to $\a,\n$. So far with formal perturbation theory. If one, however, attempts at estimating the remainders one meets serious difficulties. Unless one is willing to take $\l$ so small to be of order much smaller than $L^{-1}$, say of $O(\eta L^{-1})$ for some small $\h$. In the latter case one can easily check that there are no convergence problems for the perturbation expansions (and in fact the first order is dominant), as it is physically obvious. For $\eta$ small enough and $L$ fixed the perturbation theory converges, the ground state is unique and separated by a gap of order $L^{-1} p_F/m$ from the first excited state. One possible approach to the theory of low temperature Fermi gases, that we shall follow, is to study the above perturbation expansions as well as the expansions of the other interesting quantities (like the system reduced density matrices or the Schwinger functions, see \S2), and to show that they can be resummed so that, after resummation, they admit analytic continuation in $\l$ up to $\l$'s of size of order 1 uniformly in $L$ and $\b$. If this goal is achieved, it is clear that we have constructed objects of interest for low temperature physics: they can be interpreted as Gibbs states of the system provided they verify the necessary positivity properties. The latter are, essentially, automatically verified as we know that for $L,\b>0$ fixed none of the correlations functions has a singularity for $\l,\a,\n$ real (small or large). In this paper we study a resummation algorithm, generated by the application of the renormalization group methods to the study of the above series. We show that the resummation can be described in terms of stability properties of a well defined dynamical system. We call {\it beta function} the functions defining the dynamical system iteration map $B_h$: the latter operates on a three dimensional set of parameters called the running couplings denoted by $\undr $. Each triple $\undr_0$ of initial data generates, for $h=0,-1,-2,\ldots$ a trajectory $\undr_{h-1}= \undr_h + B_h(\undr _h,\undr _{h+1}, \ldots,\undr _0)$ which, under the condition that $|\undr_h|$ remains small, provides a set of parameters in terms of which the relevant dynamical quantities (Schwinger functions) can be expanded in a {\it convergent power series}. The reason we call the above a {\it resummation} is because the expansion constructed is not in power series of $\undr _0$: if we express $\undr _h$ in powers of $\undr _0$ it may well be that the convergence radius of the expansion shrinks to zero as $h\rightarrow -\infty$. Our main results are: {\it\item{1) }the existence and boundedness and analyticity of the functions $B_h(\undr _h,\ldots, \undr _0)$ as functions of their arguments (regarded as independent arguments), if they are small enough: $|\undr _h|\le \e$, for all $h\le 0$. \item{2) }We also show that $B_h(\undr ,\undr ,\ldots, \undr )\equiv \b_h(\undr )$ is the sum of two parts $\b_h(\undr )=\b(\undr )+\hat{\b}_h(\undr )$ with $\hat{\b}_h(\undr )\to 0$, for $h\to -\io$ and for $\vert \undr \vert \le\e$, exponentially fast, and with $\b(\undr )$ ({\it ``scaling part of the beta function''}) which we show (in \S7) to be zero. \item{3) }We deduce from 1),2) an expansion in powers of $\{\undr_h\}_{h\le0}$, convergent if $|\undr_h|\le \e$ for all $h\le0$, of the pair (and higher) Schwinger functions. The expansion implies, if $|\undr_h|\le \e$ for all $h\le0$, that the pair Schwinger function approaches $0$ as its argument $\x\to\io$ faster than the free Schwinger function does, and we compute exactly how fast (\ie we compute the {\sl anomaly exponent}).} Some support to the validity of the vanishing of $\b(\V r)$ in 2) above was given in [BG], [BGM], by reducing it to the proof of a similar conjecture for the Luttinger model. In [BGM] the proof of the conjecture was reduced to a property of the Schwinger functions which is implied by the results in 3) above, plus the independence of the exact solubility of the Luttinger model from the cut offs necessary to define it. Thus we showed that the exact solubility of the latter model would allow us to establish a rigorous proof of the conjecture if we knew suitable uniformity properties on the Luttinger model running constants defined in a way entirely analogous to the one followed for our problem, see [BGM]. The above scheme of proof is discussed in \S7 and, using the new results derived in the previous sections (\S3$\div$6), it is completed. The discussion of 1) requires the solution of two distinct problems. The first is an ultraviolet problem, which should be considered trivial as it is technically similar (but easier) to the theory of the ultraviolet stability of the Gross Neveu model in field theory, whose solution is well known [GK]: we perform it in detail (as our formalism differs from that of [GK]), but we find no unexpected difficulty (\S3). The second problem is an infrared problem: this presents new difficulties as it requires the discussion of a {\it anomalous dimension} (physically this means that the perturbed system has correlations which decay at $\io$ faster than the free ones). To our knowledge this is the first example of a rigorous theory of the beta function of an anomalous renormalization group flow and, technically, represents the major part of this work. The results of \S7 also imply the existence of a one parameter family of non trivial fixed points of our renormalization group transformation: this can be regarded as the origin of the anomalous dimension: however we only allude (\S5) to such a corollary as it not essential for our work. In the next section we set up the formalism in a self consistent way trying to discuss the rigour issues growing out of the functional integral representation of the Schwinger functions that we plan to use in the rest of the paper. It is useful to state our main result in a form independent on the subsidiary concepts (like running coupling, beta function, {\it etc.}) and based solely on the hamiltonian \equ(1.3) and on the standard notion of pair Schwinger function, $S(x)$, of the model (introduced formally in the next section); it can be summarized in the following theorem: \vskip0.5truecm \\{\bf Theorem:\nobreak \it Given a pair potential $\l \bar v(\V x-\V y)$, with $\bar v$ smooth and with short range $p_0^{-1}$, one can find analytic functions $\a(\l),\n(\l)$, holomorphic near $\l=0$ and of order $\l$, such that the one dimensional spinless Fermi gas with hamiltonian: % $$\sum_{i=1}^N\Big({-\D_{\V x_i}\over 2m(\l)}-{p_F^2\over 2 m(\l)}+\n(\l)\Big)+2\l\sum_{i0$, admits a zero temperature Gibbs state (defined as the $T\to0$ limit of a $T>0$ Gibbs state) with a Euclidean pair Schwinger function $S(x-y)$ verifying, for $|x-y|p_0$ large, the relation: % $$S(x-y)= (1+ A_0(\l)){S_0(x-y)\over(p_0|x-y|)^{2\h(\l)}}+ A_1(\l){1\over(p_0|x-y|)^{1+2\h(\l)}}\Eq(1.11)$$ % with $\h(\l)$, $A_i(\l)$ analytic near $\l=0$, $\h(\l)=O(\l^2)$, $A_i(\l)=O(\l)$, $A_0$ independent of $x,y$ and with $S_0$ being the pair Schwinger function for the free gas with Fermi momentum $p_F$ and mass $m$.} \vskip0.5truecm Note that $S_0(x-y)$ tends to zero with oscillations on scale $p_F^{-1}$ and speed $|x-y|^{-1}$, so that the first term in \equ(1.11) dominates over the second ``when non zero''. The theorem was proposed by Tomonaga who developed theoretical arguments for its validity, [T]; on the basis of Tomonaga's work Luttinger proposed a model which, if Tomonaga's ideas were correct, should behave in the same way as the system \equ(1.1) that we are considering, [L]. The model differs from \equ(1.1) in two respects: first there are two spinless particles and second the kinetic energy is linear in the momentum. Luttinger also gave arguments to suggest that the model might be exactly soluble. The model was solved exactly, later, by Mattis and Lieb, proving that indeed it did behave as expected on the basis of its heuristic equivalence to the Tomonaga's theory of the model \equ(1.1), [ML]. \vskip1.truecm {\bf Acknowledgements:} GG is indebted to Giovanni Felder for introducing him to the anomalous dimension in $\f^4$ field theory. We thank G. Gentile, V. Mastropietro and W. Metzner for many useful discussions. We acknowledge support from Ministero della Ricerca Scientifica, from Gruppo Nazionale della Fisica Matematica, from C.N.R (BS), from Rutgers University (GG and BS) and from Institut des Hautes \'Etudes Scientifiques, Paris (GG). \vskip2truecm \vglue1.truecm {\it\S2 Functional integral representation of fermionic correlation functions} \vglue1.truecm\numsec=2\numfor=1 The Schwinger functions of a Hamiltonian $H$, like \equ(1.3) are defined by: % $$S(\xx_1,t_1,\s_1,\ldots,\xx_s,t_s,\s_s)= {Tr(e^{-(\b -t_1)H}\ps{\s_1}{\xx_1}\ldots e^{-(t_{s-1}-t_s)H} \ps{\s_s}{\xx_s}e^{-t_sH})\over Tr\, e^{-\b H}}\Eq(2.1)$$ % for $\b>t_1>t_2>\ldots>t_s>0, \quad \ps{\s}{\xx},\s=\pm$, being field operators on the Fock space of a fermion system confined in a box of size $L$, with periodic boundary conditions, and at temperature $\b^{-1}>0$. At fixed $\b,L$ the \equ(2.1) are, by inspection, real analytic in $\l,\a,\n$: their holomorphy domain has complex size which, for the time being, is totally out of control and it may shrink to 0 as $\b\rightarrow\infty$ or $L\rightarrow\infty$. If we are willing to take $\l,\n,\a$ of $O(\eta L^{-1})$ with $\eta$ small, it is not difficult to see that we have in fact uniformity in $\b$ as $\b\rightarrow\infty$. The basic reason is that, if $\l,\n,\a$ are so small, we see by perturbation theory that the lowest eigenvalue of $H$ is separated by a gap from the next. Hence the limit as $\b\rightarrow\infty$ is simply expressed in terms of the expectation value in the ground state $|0\rangle_{\l,\n,\a}$ (which is also analytic in such small $\l,\n,\a$), as: % $$S(\xx_1,t_1,\s_1,\ldots,\xx_s,t_s,\s_s)=\, _{\l,\n,\a}\kern-2pt\langle0|\ps{\s_1}{\xx_1}\ldots e^{-(t_{s-1}-t_s)H} \ps{\s_s}{\xx_s}|0\rangle_{\l,\n,\a}\Eq(2.2)$$ % This is manifestly analytic in $\l,\n,\a$. Knowing the above analyticity property we can find the expansion coefficients in powers of $\l,\n,\a$. The classical calculation is as follows. We define the imaginary time fields (see \equ(1.4)) as: % $$ \ps{\pm}{\xx,t}=\,L^{-1/2}\sum_{ \kk}\,e^{\pm i \kk \xx\pm e( k)t} a^{\pm}_{ \kk}\equiv e^{tT}\ps{\pm}{ \xx}e^{-tT}\Eq(2.3)$$ % Then by using the representation (where $V\equiv \bar V+\n N+\a T$, see \equ(1.3)): $$e^{-tH}=\lim_{n\to\infty}\bigl(e^{-tT/n}(1-{tV\over n})\bigr)^n \Eq(2.4)$$ % we find that the numerator of \equ(2.1) becomes: % $$\sum\pm\ig\hbox{\rm Tr}\bigl\{e^{-\b T}V(t'_1)\ldots V(t'_{p_1-1}) \ps{\s_1}{\V x_1,t'_{p_1}}\ldots\ps{\s_s}{\V x_s,t'_{p_1+\ldots+p_s}} \ldots V(t'_{p_1+\ldots+p_{s+1}})\bigr\}\,d{\underline{t}'}\Eq(2.5)$$ % where $V(t)=e^{tT}Ve^{-tT}$ and the sum runs over integers $p_1,p_2,\ldots$ while the integral is over all the $t'_j$ variables with $j\ne p_1,p_1+p_2,\ldots,p_1+p_2+\ldots+p_s$ and $t'_{p_1},t'_{p_1+p_2},\ldots,t'_{p_1+p_2+\ldots+p_s}$ are fixed to be $t_1>t_2>\ldots>t_s\ge0$, respectively; finally the $t'$ variables are constrained to decrease in their index $j$, and the sign $\pm$ is $+$ if the number of $V$ factors is even and $-$ otherwise. Since the product of $V$'s is an integral of a sum of products of $\ps{\pm}{\xx,t}$ operators and since the $T$ is a quadratic hamiltonian in the $\ps{\pm}{}$ operators, the Wick's theorem holds for evaluating $\hbox{Tr} (\exp-\b T(\cdot))/\hbox{Tr}(\exp-\b T)$ (see, for example, [NO]) and therefore it will be possible to express the various terms in \equ(2.5) as suitable integrals of sums of products of expressions like: $$\eqalign{ g_+(\xxi,\t)=&\hbox{Tr}\,e^{-\b T}\ps-{ \xx,t}\ps+{ \xx',t'} /\hbox{Tr}\,e^{-\b T}\cr g_-(\xxi,\t)=&\hbox{Tr}\,e^{-\b T}\ps+{ \xx,t}\ps-{ \xx',t'} /\hbox{Tr}\,e^{-\b T}\cr}\Eq(2.6)$$ % if $\xxi= \xx- \xx',\ \t=t-t'>0$, which we combine to form a single function: % $$g(\xxi,\t)=\cases{g_+(\xxi,\t)&if $\t>0$\cr -g_-(-\xxi,-\t)&if $\t\le0$\cr}\Eq(2.7)$$ % Then it is easy to see, from Wick's theorem, that the generic term in \equ(2.5) can be expressed graphically as follows. One lays down graph elements like: \insertplot{300pt}{100pt}{fig0}{fig0} \\symbolizing respectively: % $$\eqalign{ &-\l \bar v( \xx_1- \xx_2)\ps+{ \xx_1,t}\ps+{ \xx_2,t} \ps-{ \xx_2,t}\ps-{ \xx_1,t}\cr &-(\nu-\m\a) \ps+{ \xx,t}\ps-{ \xx,t}\cr &(\a /2m) \ps+{\xx,t} (-\D) \ps-{\xx,t}\cr &\ps+{\xx,t}\qquad\hbox{\rm and}\qquad\ps-{\xx,t}\cr}\Eq(2.8)$$ % One should then draw $n+s$ such elements so that the first $n$ have a shape of one of the first three forms with labels $(\yy_i,t_i)$ attached arbitrarily to the vertices (``free labels'') and the last $s$ have a shape of the last two forms (representing respectively $\ps-{\xx,t}$ or $\ps+{\xx,t}$) and carry ``external labels'' $(\xx_1,t_1),\ldots,(\xx_s,t_s)$. Then one considers all {\it Feynman graphs}, that is all possible ways of joining together lines in pairs so that no unpaired line is left over and so that only lines with consistent orientations are allowed to form a pair. To each graph we assign a sign $\s=\pm$ obtained by considering the permutation necessary to bring next to each other the pairs of operators which, in the given graph, are paired (one says also {\it contracted}), with the $\ps-{}$ to the left of the associated $\ps+{}$, and then setting $\s=(-1)^\p$ if $\p$ is the permutation parity. To each graph we assign a {\it value} which is the integral over the free vertices of the product of the sign factor times the product of factors $g(\xxi,\t)$ (or of some of its derivatives) for every line with an arrow pointing from $(\xx_1,t_1)$ to $(\xx_2,t_2)$ with $\xxi=( \xx_2- \xx_1),\ \t=t_2-t_1$, times a factor \ \ $-\l \bar v(\xx_1- \xx_2)$ for every wiggly line joining $(\xx_1,t)$ to $(\xx_2,t)$, times a factor \ \ $-(\nu-\m\a)$ or $-\a/2m$ for every vertex of the type with only two lines. The {\it propagator} function $g$ is given by \equ(2.7) and can be represented as: % $$g( \xxi,\t)=L^{-1}\sum_{\kk} e^{-i \kk\xxi}\bigl\{ {e^{-\t e(\kk)}\over1+e^{-\b e(\kk)}}\chi(\t>0) -{e^{-(\b+\t)e(\kk)}\over1+e^{-\b e(\kk)}}\chi(\t\le0)\bigr\}\Eq(2.9)$$ where $\chi(``condition'')=1$ if $``condition''$ is verified and $\chi=0$ otherwise. This can be written: $$g=\lim_{K\rightarrow\infty}{1\over \b L} \sum_{k_0,\kk\atop e^{-ik_0\b}=-1,e^{i\kk L}=1}{e^{-i(k_0(\t+0^-)+ \kk\xxi)} \over-ik_0+e(\kk)}\D({\sqrt{k_0^2+\kk^2}\over K})\Eq(2.10)$$ % with the sum running over the $k_0,\kk$ verifying $e^{-i k_0 \b}=-1,\,e^{-i \kk L}=+1$; and $\D$ is a cutoff function like one of the following: % $$\D_s(x)=\chi (x<1),\qquad\D_{\a}(x)=(1+{x^2\over\a})^{-\a}, \qquad \D_{\infty}(x)=e^{-x^2}\Eq(2.11)$$ % The \equ(2.10) can be proved, in the case of the first regularization $\D=\D_s$ ("sharp momentum regularization"), by remarking that, if $\t>0$ and $\kk$ is fixed in the r.h.s. of \equ(2.10), the sum over $k_0$ has a limit, for $K\to\i$, equal to: % $${1\over 2\pi} \oint{e^{-iz\t}\over (-iz+e(\kk))(1+e^{-iz\b})}dz \Eq(2.12)$$ % with the contour running parallel to the real axis (nearer than $|e(\kk)|\ge e_{min}\sim {p_F\over m}{\pi\over L}$) and going from $-\infty$ to $+\infty$ if Im$z<0$ and from $+\infty$ to $-\infty$ if Im$z>0$. Using that $\b>\t>0$ we easily see that \equ (2.10) implies \equ (2.7). If $\t<0$ (but $\b>|\t|$ so that $\b+\t>0$) we see from \equ(2.10) that the sum has value $-1$ times the value when $\t$ is replaced by $\b+\t$ (because $e^{i\b k_0}\equiv -1$). Hence for such values of $\t$ the value of $g$ is given by $-1$ times the value of \equ(2.12) with $\t$ replaced by $\b+\t$, and \equ(2.9) follows also for $\t<0$. The cutoff $\D_{\a}$ can be treated in the same way (if $\a$=positive integer as we suppose): one finds instead of \equ (2.10) a complex integral that can be, essentially, explicitly evaluated and one can therefore estimate easily the difference between \equ(2.10) and \equ(2.9) as $K\rightarrow\infty$ The gaussian cut off $\D_{\infty}(x)$ cannot be treated by using complex integrals because $\D_{\infty}$ has bad behaviour at $\pm i\infty$. But $\D_{\infty}(x)-\D_{\a}(x)= O(x^4)$ as $x\rightarrow 0$ and this, together with the fact that we know that \equ(2.10) holds with the regularization $\D_{\a}$, easily implies the validity of \equ(2.10) with the gaussian cut off as well. Therefore we can compute the coefficients of the perturbation theory for the Schwinger functions by the above graphical algorithms and by using propagators with one of the above cut offs and then removing it. The above discussion suggests the following definition: \vskip0.3truecm% {\bf Definition}: {\it suppose that, for $\l$ in a small neighbourhood $D$ of the origin in the complex plane, and for $\a,\n$ suitably chosen as analytic functions of $\l$ in $D$, the perturbation series for the Schwinger functions can be shown to admit an analytic continuation to the domain $D$, extending on the real axis to $\l$'s of $O(1)$, i.e. $\b,L$-independent, and suppose that the limits as $\b\rightarrow\infty$, $L\rightarrow\infty$ of the Schwinger functions exist in $D$. Then we say that the limit as $L\rightarrow\infty$ of the Schwinger functions defines a Gibbs state for our system with Fermi momentum $p_F$ and particles mass $m$. The limit of the Schwinger functions as $\b\rightarrow\infty$ will be called a ground state with Fermi momentum $p_F$ and particles mass $m$.} \vskip0.3truecm Note that such a definition would certainly not be adequate for $d=3$ (because changing the sign of $\l$ destroys the stability of the Hamiltonian, see [R], [Th], and the system collapses) and probably not even for $d$=2 (although in this case the sign of $\l$ does not affect stability, if $\l$ is small enough). Hence, for $d>1$, we would replace the requirement that $D$ is a neighbourhood of the origin by the requirement that it is a domain in the right half plane. This shows that one can conceive a purely perturbative approach to the low temperature Fermi systems. One starts with some expressions of the perturbation expansion for the Schwinger functions depending on various parameters to be eventually sent to $\infty$ (e.g. $\b$ or $L$ or others that will be introduced later). At fixed values of the parameters the expansions should be obviously convergent for small $\l,\a,\n$. Then one proves uniform analyticity in a region $D$ of complex $\l$, where $\a,\n$ are suitably chosen as a function of $\l$ (analytic in the same domain) and thus one defines, by removing the cutoffs, a Gibbs state in the above sense. As long as other cut offs, besides $\b,L$, are removed first, the already remarked and obvious analyticity in $\l,\a,\n$ at fixed $\b,L$ guarantees that the functions obtained in this way do have the required positivity property necessary to interpret them as Schwinger functions for a Gibbs state (namely the reflection positivity). In fact the series expansions for real $\l,\a,\n$ must coincide with the non perturbative definitions of the same expressions by analyticity and the latter, of course, have the reflection positivity property. The most convenient representation of the Schwinger functions, for the above purposes, is the {\it Euclidean functional integral representation.} Such a representation is set up with the help of two extra regularization parameters that we call $R,U$, with $R\le U$, and of a family of Grassmanian variables. Here the Grassmanian variables will be denoted $A^{h\s}_{k\o},\e^{\s}_{k}$ and they bear labels $h\in{\bf Z}, \o =\pm 1,\s =\pm 1, k=(k_0,\kk)$ such that: % $$e^{-ik_0\b}=-1,\qquad e^{i\kk L}=+1\Eq(2.13)$$ % They must verify anticommutation rules: % $$\{A,A'\}=0,\quad \{A,\e\}=0,\quad \{\e,\e'\}=0\Eq(2.14)$$ % It is most convenient to think of the $A,\e$ as concrete objects by using a representation on a Hilbert space ${\bf h}$. The best Hilbert space is probably the countable tensor product of two dimensional spaces ${\bf C}^2$: ${\bf h}= \otimes_{j=1}^\io{\bf C}^2$. Then we order (absolutely arbitrarily) the variables labels, by replacing each of them with an integer label $j=1,2,\ldots$ and set the $j-$th Grassmanian variable to be: $$[\otimes_{i0$; therefore the particle fields will be defined in terms of suitable $A_k^{h\s}$ variables in a different way. However, in order to simplify the notation, we nevertheless proceed in a symmetric way in the ultraviolet and infrared region; it will be clear that our definitions would work also for the representation of the field used in the following sections. The Grassmanian or fermionic functional integral is then defined as a linear functional on the operators on ${\bf h}$, in the algebra generated by the Grassmanian variables. The integration rule is simply the Wick rule based on the following ``propagator'': % $$\ig A^{h-}_{k\o}A^{h'+}_{k'\o'}dP=\d_{hh'}\d_{kk'}\d_{\o\o'}\Eq(2.19)$$ % while all the integrals of $A^+A^+$ and $A^-A^-$ vanish. This means that the integral of an arbitrary monomial in the $A^+$ and $A^-$ is obtained as a sum over the pairings of the factors into pairs with non zero propagator of the product of the propagators corresponding to the pairs times a sign $\pm$ equal to the parity of the permutation necessary to bring the considered pairs next to each other. %%%!!! va bene???? The above rule is just a linear functional and we may have problems in the integration of expressions which are not finite linear combination of products of $A$: but of course this is precisely the kind of operation that we shall wish to do. Therefore it is convenient to define a class of operators on ${\bf h}$ on which we can operate the functional integral ``absent-mindedly''. It will be the class of {\it integrable operators}. \vskip0.3truecm {\bf Definition}: {\it An operator $O(\psi,\f)$ is said to be integrable if it has the form: % $$\eqalign{ O(\psi,\f) &= \sum_{n,m,\undo,\undo'}\ig \left( \prod_{i=1}^n dx_i dy_i \right) \left( \prod_{j=1}^m du_i dv_i \right) O_{n,m}(\undx,\undy,\undu,\undv,\undo,\undo') \cdot \cr & \cdot\, [{\cal D}_1 \ps{+}{u_1\o_1}\ldots{\cal D}_{2m}\ps{-}{v_m\o'_m}]\, \f^+_{x_1} \ldots \f^-_{y_n} \cr} \Eq(2.20)$$ % where the $O_{n,m}(\ldots)$ are the "kernels of $O(\psi,\f)$" and $\psi^{\pm}$ are quasi particle operators on various scales between two scales $R,U$, for all $n$, and ${\cal D}_1\ldots {\cal D}_{2m}$ are differential operators with constant coefficients (possibly dependent on $h,\o$), and with order bounded by some $N$, for all $n$. Furthermore the $O_{n,m}$ should be measures (i.e. $\d$ functions are allowed) and: % $$\eqalign{ |O(\psi,\f)|_b &\equiv \sum_{n,m,\undo,\undo'} b^{n+m} \ig \left( \prod_{i=1}^n dx_i dy_i \right) \left( \prod_{j=1}^m du_i dv_i \right) \cdot\cr & |O_{n,m}(\undx,\undy,\undu,\undv,\undo,\undo')| < \io \qquad \forall b>0 \cr} \Eq(2.21)$$ % Then we define (consistently with \equ(2.19), as it is possible to check): % $$\eqalign{ \ig P(d\psi)O(\psi,\f) &= \sum_{n,m,\o,\o'}\ig \left( \prod_{i=1}^n dx_i dy_i \right) \left( \prod_{j=1}^m du_i dv_i \right) O_{n,m}(\undx,\undy,\undu,\undv,\undo,\undo') \cdot \cr & \cdot\, \DD_1\,\DD_2\ldots\DD_{2m} \det \left[ g^{[R,U]}_{\o_i\o_j'}(u_i - v_j) \right] \f^+_{x_1} \ldots \f^-_{y_n} \cr} \Eq(2.22)$$ % where the {\sl propagator} $g^{[R,U]}(x-y)$ is $\sum_{h=R}^U g^{(h)}(x-y)$, with: % $$g^{(h)}_{\o\o'}(x-y)={\d_{\o\o'}\over\b L}\sum_ke^{-i[k(x-y)- p_F\o(\xx-\yy)] } {e^{-\g^{-2h} \b(k)}-e^{-\g^{-2h+2} \b(k)} \over -ik_0+e(\kk)} \c(\o\g^{-h}\kk)\Eq(2.23)$$ % } \vskip0.3truecm % {\bf Remark:} The r.h.s. of \equ(2.22) is a well defined operator, thanks to \equ(2.21), as a consequence of the Gramm-Hadamard inequality (see appendix 2): % $$|\DD_1\ldots\DD_{2m}\,\det \left[g^{[R,U]}_{\o_i\o_j'}(u_i-v_j)\right] | \le B_{R,U}^m\Eq(2.24)$$ % Furthermore the definition is meaningful since the representation \equ(2.20) is unique if the kernels: % $$\DD_1\,\DD_2\ldots\DD_{2m} O_{n,m}(\undx,\undy,\undu,\undv,\undo,\undo') \Eq(2.25)$$ % are antisymmetric in the permutation between themselves of the $(u_i,\o_i)$, of the $(v_i,\o_i')$, of the $x_i$ and of the $y_i$. \vskip0.3truecm Several easy theorems follow. For instance, if $O$ is integrable also $\exp\, O$ is integrable: this is a key property that overcompensates the fact that the fermionic integration is not a positive functional in the sense of measure theory (and makes the world of fermionic integration look like a fairy tale compared to that of measure theory). Also, if $O(\psi,\f)$ is integrable and if we write $\psi^{[R,U]}=\psi_1+\psi_2$ with $\psi_1=\psi^{[R,U_1]}$ and $\psi_2=\psi^{[U_1,U]}$, then $O(\psi_1+\psi_2,\f)= \sum O_1(\psi_1,\f)O_2(\psi_2,\f)$ and $O_i$ are integrable; moreover: $$\ig P(d\psi)O(\psi,\f) =\sum \ig P(d\psi_1)O_1(\psi_1,\f) \ig P(d\psi_2)O_2(\psi_2,\f) \Eq(2.26)$$ i.e. "Fubini's theorem" holds. %Finally, the coefficients of $\f^+_{x_1} \ldots \f^-_{y_n}$ in \equ(2.20), %which are also integrable, will be denoted: % %$${\d^{2n}\,O(\psi,\f)\over %\d\f^{+}_{x_1}\ldots\d\f^{-}_{y_n}} %\Big|_{\f=0}\Eq(2.29)$$ % %and called the {\it functional derivatives} of $O$. The above obvious remarks constitute the theory of non commutative or fermionic Grassmanian integration. Its interest lies in the fact that it is easy to see that the coefficients of the perturbation expansion of the Schwinger functions are generated by: % $$q_{R,U}(\f)=\log\ig P(d\psi^{[R,U]}) e^{-V(\psi)+\ig dx(\f^+_x\ps{-}{x}+\ps{+}{x}\f^-_x)} \Eq(2.27)$$ % via: % $$S^T(x_1,\s_1,...,x_n,\s_n)=\lim_{U\rightarrow\infty \atop R\rightarrow -\infty} {\d^{2n}q_{R,U}(\f)\over \d\f^{+}_{x_1}\ldots\d\f^{-}_{y_n}} \Big|_{\f=0}\Eq(2.28)$$ % Hence we shall confine ourselves to studying $q_{R,U}(\f)$ and reorganizing the expansion of $S^T$ in powers of $\l$ (with $\n,\a$ also expanded in terms of $\l$) so that the expansion have analyticity properties in $\l$ uniform in $R,U$ as well as in $L,\b$. We shall also use the expansion to infer the long distance behaviour of $S^T(x_1,\s_1,...,x_n,\s_n)$ (long means $O(L)$ in space and $O(\b)$ in time). \vskip.3truecm {\bf Remark:} $q_{R,U}(\f)$ has an expression like \equ(2.20) (with $n=0$), whose kernels are the {\it functional derivatives} appearing in the r.h.s. of \equ(2.28). Furthermore one can define the $|q_{R,U}(\f)|_b$ norm as in \equ(2.21) and it is possible to see (using \equ(2.24) and some standard procedure to bound the truncated expectations, see last part of Appendix 3) that this norm is finite for $b\le b_0$, with $b_0$ depending on the strength of the interaction; this is sufficient to define $q_{R,U}(\f)$ as a bounded operator. \bigskip In order to simplify the notation, in the following sections we shall consider, for the propagator, only the limiting case $L=\b=\i$, by interpreting the functional integrals as a formal tool to represent in a convenient way the expansions of the Schwinger functions in powers of $\l$, $\a$, $\n$. It will be clear that all our results are valid also for $L$, $\b$ finite and that one can take the limit $L,\b\to\i$ without any further problem. Moreover we shall change the meaning of the symbol $\ps{\s}{x}$ (see \equ(2.3)), which from now on will denote the formal limit $R\to -\i,U\to +\i$ of the grassmanian field $\ps{[R,U]\s}{x}$ defined in \equ(2.18). Then we can write the generating functional of the Schwinger functions, in the limit where all the cut off are removed, as: $$q(\f)=\log\ig P(d\psi^) e^{-V(\psi)+\ig dx(\f^+_x\ps{-}{x}+\ps{+}{x}\f^-_x)} \Eq(2.29)$$ and we can say that $P(d\psi)$ is {\it grassmanian gaussian measure} with propagator: % $$g(x-y) = \int P(d\psi) \ps{-}{x}\ps{+}{y}= {\ii} {dk_0 d\kk \over (2\pi)^2} {e^{-i[k_0((x_0-y_0)+0^-)+\kk(\xx-\yy)]} \over -ik_0+ e(\kk)} \Eq(2.30)$$ % where the $0^-$ in the exponential means that $g(0,\xx)$ must be interpreted as \hfill\break $\lim_{x_0\to 0^-} g(x_0,\xx)$. Moreover, if $\L=L \times [0,\b]$: % $$\eqalign{ & V(\psi) = \l \ig_{\L\times\L} dx\,dy \,v(x-y) \ps{+}{x}\ps{+}{y}\ps{-}{y}\ps{-}{x} + (\n-\m\a) \ig_\L dx \,\ps{+}{x}\ps{-}{x} + \a\ig_\L dx\, \ps{+}{x}(-\D)\ps{-}{x}\cr & v(x-y) \equiv \d(x_0-y_0) {\bar v}(\xx-\yy)\cr}\Eq(2.31)$$ where $\D=\dpr_\xx^2$ \ is the Laplacian in the space variables. A very convenient object which is related to $q(\f)$ is the {\it effective potential} defined by: % $$e^{-V_{eff}(\f)} = {1\over \NN} \ig P(d\psi) e^{-V(\psi+\f)} \Eq(2.32)$$ where $\NN$ is a normalization constant chosen so that $V_{eff}(0)=0$ The relation is, if $(g\f)^-=g*\f^-$ and $(g\f)^+=\f^+*g'$, where the $*$ denotes convolution and $g'(x)=g(-x)$, the following: % $$ -V_{eff}(g\f)+(\f^+,g\f^-)=q(\f)\Eq(2.33)$$ The above relations are formally trivial if one treats $\ii P(d\psi)\cdot$ as an ordinary integral with respect to a grassmanian measure proportional to: % $$ d\ps+{}d\ps-{}e^{-\ii[\ps+x(\dpr_t+(-\D+p_F^2)/2m]\ps-x dx}\Eq(2.34)$$ % and proceeding to the {\it change of variables} $\psi+g\f=\tilde\psi$. The formal argument on the change of variables is meaningless as presented; however if one writes the above calculations (\ie the change of variables) as relations between the power series in the fermion fields defining the fermionic integrals, one sees that they are indeed valid. The equation \equ(2.33) should allow us, in principle, to reduce the study of the Schwinger functions to that of the effective potential. However, because of the anomalous large distance behaviour, this is not so simple, in the sense that it is not possible to use directly \equ(2.33), see [BGM]. In any event, the analysis of the effective potential will play an essential role; therefore, in the following three sections, we shall analyze the integral in the r.h.s. of \equ(2.32) by an iterative procedure, based on the scale decomposition \equ(2.17) of the field. This will allow us to define the effective potential on scale $\g^{-h}$, whose properties will be used in \S6 to study the pair Schwinger function, by an expansion that will take the place of the relation \equ(2.33). The same technique could be used also to study the other Schwinger functions, but we shall not do it explicitly. \vskip2truecm \vglue1.truecm {\it\S3 Ultraviolet limit for the effective potential} \vglue1.truecm\numsec=3\numfor=1 In this section we shall begin the analysis of the effective potential defined in equation \equ(2.32), by studying the ultraviolet problem. We start by decomposing $g(x)$ in its u.v. (ultraviolet) and its i.r. (infrared) part: $$g(x)=g_{u.v.}(x) +g_{i.r.}(x) \Eq(3.1)$$ with: % $$g_{u.v.}(x) = {\ii} {dk_0 d\kk\over (2\pi)^2} {1-e^{-(k_0^2+e(\kk)^2)p_0^{-2}} \over -ik_0+ e(\kk)} e^{-i(k_0(x_0+0^-)+\kk\xx)}\Eq(3.2)$$ % where $p_0^{-1}$ is the range of the potential, see \equ(1.1) and \equ(3.24) below. It is easy to see that: % $$g(x) = \theta(x_0) e^{x_0p_F^2\over 2m} \left( {m\over 2\p x_0} \right)^{1/2} e^{-{m\xx^2\over 2x_0}} - \int_{-p_F}^{p_F} {d\kk \over 2\p} e^{-i\kk\xx- x_0 {\kk^2-p_F^2 \over 2m} } \Eq(3.3)$$ % where $\theta(x_0)$ is the step function. Hence we can write: % $$g_{u.v.}(x)= G(x)+R(x)\Eq(3.4)$$ % with: % $$G(x) = h(\xx) h(x_0) \theta(x_0) e^{x_0p_F^2\over 2m} \left( {m\over 2\p x_0} \right)^{1/2} e^{-{m\xx^2\over 2x_0}} \Eq(3.5)$$ % $$\eqalign{ R(x) &= [1-h(\xx) h(x_0)]g_{u.v.}(x) -h(\xx) h(x_0) g_{i.r.}(x) -\cr & - h(\xx) h(x_0) \int_{-p_F}^{p_F} {d\kk \over 2\p} e^{-i\kk\xx- x_0 {\kk^2-p_F^2 \over 2m} } }\Eq(3.6)$$ % where $h(t)$, $t\in {\bf R}^1$, is a smooth function of compact support such that $h(t)=1$, if $|t|\le 1$, and $h(t)=0$, if $|t|\ge \g$, $\g$ being any number greater than $1$, fixed once and for all. It is easy to show that $R(x)$ is a smooth function on ${\bf R}^2$, such that, for suitable $A,\bar\k$: % $$|R(x)|\le Ae^{-\bar\k |x|}\Eq(3.7)$$ The equations \equ(3.1), \equ(3.4) and \equ(2.32) imply that: $$e^{-V_{eff}(\f)} = {\bar{\NN}^{(0)}\over \NN} \ig P^{(i.r.)}(d\psi^{(i.r.)}) e^{-\bar{V}^{(0)}(\psi^{(i.r.)} + \f)} \Eq(3.8)$$ where $$e^{-\bar{V}^{(0)}(\f)} = { \NN^{(0)} \over \bar{\NN}^{(0)}} \int P^{(R)}(d\psi) e^{-V^{(0)}(\psi+\f)} \Eq(3.9)$$ and % $$e^{-V^{(0)}(\f)} = {1\over \NN^{(0)}} \ig P^{(G)}(d\psi) e^{-V(\psi+\f)} \Eq(3.10)$$ % with $\NN^{(0)}$, $\bar\NN^{(0)}$ defined so that $V^{(0)}(0)=\bar V^{(0)}(0)= 0$, and $P^{(i.r.)}(d\psi)$, $P^{(R)}(d\psi)$, $P^{(G)}(d\psi)$ are the grassmanian integrations with propagator $g_{i.r.}(x)$, $R(x)$ and $G(x)$, respectively. In order to give a meaning to \equ(3.10), we now introduce an u.v. cutoff by replacing $G$ with: % $$G_N(x) = \theta_N(x_0) h(\xx) e^{x_0p_F^2\over 2m} \left( {m\over 2\p x_0} \right)^{d/2} e^{-{m\xx^2\over 2x_0}} \Eq(3.11)$$ where $\theta_N(t)$ is a smooth function with support in the interval $[\g^{-N},\g]$ and $N$ is a large positive integer. Note that the cut off is different from that introduced in \S2, which has allowed us to present in a symmetric way the ultraviolet and the infrared problems. However, one can check that the results of this section do not depend on the choice of the cut off; in fact, one could add the new cut off to the previous one, parametrized by $U$, and note that all bounds are uniform in $U$. Furthermore, in this section we shall use only the particle field representation of the grassmanian integrations, see \equ(2.18). It is convenient to define more precisely $\theta_N(t)$ in the following way: % $$\theta_N(t) = \sum_{i=1}^N f(\g^i t)\Eq(3.12)$$ % where: % $$f(t) = [h(t/\g) -h(t)]\, \theta(t) \Eq(3.13)$$ % is a smooth function with support on $[1,\g^2]$. The function $\theta_N(t)$ has the claimed support properties and : % $$\theta(t) h(t) = \lim_{N\to\i} \theta_N(t)\Eq(3.14)$$ % It is worth remarking that: % $$\lim_{N\to\i} G_N(x) = G(x),\qquad {\rm for\ all}\quad x\in {\bf R}^2 \Eq(3.15)$$ % because in the discontinuity point $x_0=0$, by definition, $G(0,x_1) = \lim_{x_0\to 0^-} G(x_0,x_1) = 0 = G_N(0,x_1)$. Two other consequences immediately follow from \equ(3.15): 1) in \equ(3.10) we can suppose that the potential \equ(2.31) is Wick ordered w.r.t. $G_N$, since only products of fields at coinciding times appear in it; 2) all Feynman graphs with closed fermion loops in the perturbative expansion of $V^{(0)}(\f)$ vanish; furthermore, because of the $\d(x_0-y_0)$ in \equ(2.31), also the loops containing some lines $v(x-y)$ are forbidden, if the directions of the fermionic lines are compatible. Then we define: $$V^{(0)}(\f) = \lim_{N\to\i} \log {1\over \NN^{(0)}} \ig P^{(\le N)}(d\psi^{(\le N)}) e^{-V(\psi^{(\le N)}+\f)} \Eq(3.16)$$ where $P^{(\le N)}(d\psi^{(\le N)})$ is the grassmanian integration with propagator $G_N$. We want to prove that the limit exists and that it is an analytic function of $z= (\l,\n,\a)$ in a neighbourhood of $z=0$, in the sense that the kernels $O_n$ of the operator $O=V^{(0)}(\f)$, defined as in \equ(2.20) (without the sum over $\o,\o'$), are analytic functions verifying, in their holomorphy domain, bounds like \equ(2.21). We shall also prove that $V^{(0)}(\f)$ has some ``exponential decay'' properties (\ie its kernels decay exponentially fast as the arguments separate to $\io$). The extension of these results to $\bar V^{(0)}(\f)$ will be trivial. More precisely we shall prove the following theorem: \vskip0.5truecm {\bf Theorem 1}: {\it There exist $\e>0$ and $D>0$ such that $\bar{V}^{(0)}(\f)$ can be written, for $|z|\le \e$, if $z=(\a,\n,\l)$, in the following way: % $$\eqalignno{ \bar V^{(0)} (\psi) & = \l \int dx\,dy \, v(x-y) \ps{+}{x}\ps{+}{y}\ps{-}{y}\ps{-}{x} + 2\l \int dx\,dy \, v(x-y) R(x-y) \ps{+}{x}\ps{-}{y} + \cr & + ( \n- 4\p\l \hat v(0) R(0) ) \int dx \,\ps{+}{x}\ps{-}{x} + \a\int dx\, \ps{+}{x}({-\D-p_F^2\over 2m})\ps{-}{x} +\cr &+ \int dx\,dy\, \ps{+}{x} \D\ps{-}{y} \tilde{W}_2(z,x-y) +&\eq(3.17)\cr &+\sum_{n=1}^{\infty}\sum_{n_1,n_2\atop n_1+n_2=2n} \int dx_1\ldots dx_{2n} \ps{+}{x_1}\ldots\ps{+}{x_n}\ps{-}{x_{n+1}}\ldots\ps{-}{x_{2n-n_2}} \,\cdot\cr &\cdot\, \D\ps{-}{x_{2n-n_2+1}}\ldots\D\ps{-}{x_{2n}} W_{n_1n_2}(z,x_1\ldots x_{2n})\cr}$$ % where the {\it kernels} $W_{n_1n_2}$ are products of suitable delta functions by smooth functions, which are analytic in $z$ if $|z|\le\e$, and satisfy, uniformly in $N$, the following estimate: % $$\int dx_1\ldots dx_{2n} |W_{n_1n_2}(z,x_1\ldots x_{2n})| e^{{\k\over 2} d^{(0)}(x_1\ldots x_{2n})} \le |\L| (D |z|)^{\max \{2,n-1\} } \Eq(3.18)$$ % while $\tilde{W}_2(z,x)$ singles out some "special" contributions (see discussion after \equ(3.40) below) and satisfies (uniformly in $N$): % $$\int dx\, |\tilde{W}_2(z,x)| |x| e^{{\k\over 2}|x|} \le (D|z|)^2 \Eq(3.19)$$ % $$\int dx\, \tilde{W}_2(z,x) = 0 \Eq(3.20)$$ % The r.h.s. of \equ(3.18) is summable in $n$, for $|z|$ small enough and we shall take this property as definition of analyticity around $z=0$ for a function of the field of the general form \equ(3.17), see also \S2, \equ(2.20), \equ(2.21).} % \vskip0.5truecm We shall study the integral in \equ(3.16) by decomposing the grassmanian integration $P^{(\le N)}(d\psi^{(\le N)})$ in the product of the independent integrations $P^{(h)}(d\psi^{(h)})$, $h=1,\dots,N$, with propagator: % $$C_h(x)= f(\g^h x_0) h(\xx) e^{x_0p_F^2\over 2m} \left( {m\over 2\p x_0} \right)^{1/2} e^{-{m\xx^2\over 2x_0}} = \g^{h/2} \bar C_h(\g^h x_0,\g^{h/2}\xx) \Eq(3.21)$$ % where $\bar C_h(x)$ is a smooth function such that, for suitable $A$ and $\bar \k$: % $$|\bar C_h(x)| \le A e^{-{\bar\k}|x|} \quad,\quad \forall \,h\ge 1 \Eq(3.22)$$ % and $\bar\k$ can be taken to be the same as in \equ(3.7). In fact, by \equ(3.12) % $$G_N(x) = \sum_{h=1}^N C_h(x)\Eq(3.23)$$ % We shall assume that $A$ is chosen so that also the following bound is satisfied: % $$|\bar v (\xx-\yy)| \le A e^{-p_0|\xx-\yy|} \Eq(3.24)$$ % for a suitable $p_0$; we shall call $p_0^{-1}$ the {\it range} of the potential $\bar v$, see \equ(1.1). We shall integrate iteratively the fields $\psi^{(h)}$ in \equ(3.16), by studying the properties of the {\it effective potential on scale $\g^{-k}$}, defined by: % $$V^{(k)}(\f) = \lim_{N\to\i} \log {1\over \NN^{(k)}} \ig P^{(k+1)}(d\psi^{(k+1)}) \dots P^{(N)}(d\psi^{(N)}) e^{-V(\psi^{(k+1)} +\dots \psi^{(N)} +\f)} \Eq(3.25)$$ % so that: % $$e^{-V^{(0)}(\f)} = {\NN^{(k)}\over \NN^{(0)}} \ig P^{(\le k)}(d\psi^{(\le k)}) e^{-V^{(k)}(\psi^{(\le k)} +\f)} \Eq(3.26)$$ An essential role in our analysis will be plaid by the tree expansion (see Ref. [G]), with which we assume that the reader is familiar. We start with some definitions and notations. \insertplot{300pt}{150pt}{fig1}{fig1} 1) Let us consider the family of all trees which can be constructed by joining a point $r$, the {\it root}, with an ordered set of $n\ge 1$ points, the {\it endpoints} of the {\it unlabeled tree} (see Fig. 2). Two unlabeled trees that can be superposed by a suitable continuous deformation, so that the endpoints with the same index coincide, will be said to have the same {\it topological structure} and they will be regarded as equivalent. The unlabeled trees are partially ordered from the root to the endpoints in the natural way (we shall use the symbol $<$ to denote the order); $n$ will be called the {\it order} of the unlabeled tree. We shall consider also the {\it labeled trees} (which in general will be simply called trees in the following); they are defined by associating some {\it labels} with the unlabeled trees, as explained in the following items. We shall denote $\TT_n$ the set of labeled trees of order $n$. 2) Given $\t\in \TT_n$, we associate with each endpoint one of the three terms of \equ(2.31), which we denote $\tilde V_\a$, $\a$ being a suitable label, and which we represent pictorially by the following {\it graph elements}: \insertplot{300pt}{100pt}{fig2}{fig2} \\We shall say that the three different graph elements are of type $4,2,2'$, respectively, and we shall call {\it space vertices} the corresponding integration variables (a more appropriate name would be ``space inverse-temperature vertices'', but this is too long). 3) We introduce a family of vertical lines, labeled by a {\it frequency index} $h$, which takes all the integer values between $k$ and $N+1$; the vertical lines are ordered from left to right as the frequency index increases. Furthermore the root of the labeled tree must belong to the line with index $k$, the endpoints must belong to the line index $N+1$ and, finally, any branch point must belong to a vertical line with index larger than $k$ and smaller than $N+1$. We call {\it non trivial vertices} of $\t$ its branch points (this set is empty if $n=1$ and, in this case, there is only one unlabeled tree); we call {\it trivial vertices} the points where the branches {\it connecting two non trivial vertices} intersect the family of vertical lines; finally, we call vertices the trivial or non trivial vertices and the endpoints (see the dots in Fig. 2). Note that there are no vertices on the endbranches of the tree except the endpoints. Given a vertex $v$, we denote $h_v$ the frequency index of the vertical line containing it; note that: % $$h_{v'}v_0\atop\vnotep} \sum_{P_{v}}\right] \,\cdot \cr &\quad \cdot\, \left\{ \prod_\vnotep {1\over s_v!} \ET_{h_v} \left( \tilde{\psi}^{(h_v)} (P_{v^1}\backslash Q_{v^1}),\ldots,\tilde{\psi}^{(\le h_v)} (P_{v^{s_v}}\backslash Q_{v^{s_v}}) \right) \right.\,\cdot&\eq(3.37)\cr &\quad \cdot\, \left. (-\a)^{n_2'} (-\n)^{n_2} \prod_{i=1}^{n_4} [ -\l v(x_{2i-1}-x_{2i})]\right\}\cr} $$ % where: \vglue2.pt \item{1) }$v^1,\ldots,v^{s_v}$ are the vertices immediately following $v$; \item{2) } If $v$ is a trivial or non trivial vertex $P_v=\bigcup_j Q_{v^j}$ and $Q_{v^i}=P_{v}\bigcap P_{v^i}$, then $P_{v}$ is a subset of the set $I_v$ of $n_{\t^{(v)}}$ fields in $\t^{(v)}$; if $v$ is an endpoint of the tree, $P_v$ coincides with the set of fields appearing in the corresponding graph element. \vglue2.pt If we expanded the expectations in \equ(3.37) by Wick's theorem, we could represent the r.h.s. as a sum of {\it Feynman graphs} in the usual way (see, however, comments after \equ(3.44) below). Such graphs have {\it internal lines} with {\it propagator} $C_{h_v}$ (and we shall say that they have frequency $h_v$), if they are {\it generated} in $v$ by the operation $\ET_{h_v}$; the {\it external lines} are associated with the fields appearing in $\tilde{\psi}^{(\le k)}(P_{v_0})$. Furthermore, if $\GG_\t$ is the set of all Feynman graphs associated to $\t$, given $g\in\GG_\t$, it is natural to associate a {\it subgraph} $g_v$ to the vertex $v$; the internal lines of $g_v$ are the lines generated in all vertices $\ge v$, while the external lines are those associated with the fields appearing in $\tilde{\psi}^{(\le h_v-1)}(P_v)$. If we insert \equ(3.37) in \equ(3.31), we obtain a rather explicit expression for the {\it kernel} $W^{(k)}$. It is an expression that we shall use to prove that the effective potential is an analytic function of $z\equiv (\l,\a,\n)$ around $z=0$ (in the sense of the theorem that we are proving), uniformly in $N$, and that it decays exponentially on scale $\g^{-k}$, as the distance between the space vertices $\undx^{(P_{v_0})}$ goes to infinity. This will be the interpretation of the following {\it ultraviolet bound} stating that, for all $N,n,\t,P_{v_0}$: % $$\int d \undx^{(P_{v_0})} \c_\t (\undx^{(P_{v_0})}) |W^{(k)}(N,\t,P_{v_0},\undx^{(P_{v_0})})| e^{{\k\over 2}d^{(k)} (P_{v_0})}\le (C|z|)^n|\L| \Eq(3.38)$$ % where (here and always in the following) $C$ denotes a suitable positive constant and $\k$ is the minimum between $\bar\k$ and $p_0$ (see \equ(3.24), \equ(3.7), \equ(3.22) ); furthermore $d^{(k)} (P_{v_0})$ and $\c_\t(\undx^{(P_{v_0})})$ are defined in the following way. Let $\bf T$ be the set of all connected tree graphs joining the $m(P_{v_0})= |\undx^{(P_{v_0})}|$ space vertices; if ${\bf b}\in {\bf T}$, we call $b^{(1)},\ldots, b^{(m(P_{v_0})-1)}$ its bonds and $b^{(i)}_j$, $j=0,1$, the two components of $b^{(i)}$ ($0$ is the index of the time component); then: % $$d^{(k)} (P_{v_0}) \equiv \min_{{\bf b}\in {\bf T}} \sum_{i=1}^{m(P_{v_0})-1}(\g^k| b^{(i)}_0| +| b^{(i)}_1|)\Eq(3.39)$$ % Let $\TT^*_n\subset \TT_n$ be the family of trees satisfying one of the following two conditions: a) the graph elements associated with the endpoints of $\t$ are all of type $2'$ and, as a consequence, $\tilde{\psi}^{(\le k)}(P_{v_0}) = \ps{+(\le k)}{x_1} \D \ps{-(\le k)}{x_n}$; b) there are $(n-1)$ graph elements of type $2'$, while the other one is of type $2$ and its $\psi^-$ line is an external line, so that $\tilde{\psi}^{(\le k)}(P_{v_0}) = \ps{+(\le k)}{x_1} \ps{-(\le k)}{x_n}$. \\We define: % $$\c_\t(\undx^{(P_{v_0})}) = \cases{\g^k|t_n-t_1| + \g^{k/2}|\xx_n-\xx_1| & if $\t\in\TT_n^*$\cr 1 & otherwise\cr} \Eq(3.40)$$ % Note that, if $\t\in\TT_n^*$, the corresponding graph expansion of $V^{(k)}(N,\t,\{h_v\},P_{v_0},\undx)$ contains only chains connecting $x_1$ to $x_n$, see Fig. 4. \insertplot{300pt}{100pt}{fig3}{fig3} \\It is easy to see that their contribution has a singularity, as $|x_1-x_n|\to0$, whose $L_1$ norm is logarithmically divergent when $N\to\i$; the $\c_\t$ factor in \equ(3.38) is introduced to deal with the singularity, (see below). The contribution to the effective potential of such trees can be easily summed; the result can be expressed in terms of the same two Feynman graphs of Fig. 4, where now the lines represent the full propagator $\sum_{h=k+1}^N C_h$. Let us consider, for example, the chain of item a) for $k=0$; it is easy to see, by explicit calculation, that such graphs behave, when $|x_1-x_n|$ is small, in the limit $N\to\i$, as: % $$\a^n \th(t_n-t_1) {(t_n-t_1)^{n-2} \over (n-2)!} {\dpr^{n-1} \over \dpr t_n^{n-1} } e^{ -{ m (\xx_1 -\xx_n)^2 \over 2(t_n-t_1)}} \left( {m\over t_n-t_1} \right)^{1/2}\Eq(3.41)$$ % which is not $L_1$. The origin of this singularity can be easily understood. Suppose, in fact, that there is an infrared cutoff on scale $1$, so that the full propagator coincides with $G(x)$. Hence the contribution of the chain to the two points Schwinger function $S_2(x-y)$ is obtained by substituting the two external lines with the full propagators $G(x-x_n)$ and $G(x_1-y)$ and one finds that the leading contribution for $|x-y|\to 0$ behaves as: % $$\a^n \th(t-t') {(t-t')^n \over n!} {\dpr^n \over \dpr t^n } e^{ -{ m (\xx -\yy)^2 \over 2(t-t')}} \left( {m\over t-t'} \right)^{1/2}\Eq(3.42)$$ % The latter expression can be summed over $n$ and we get a function with the same behaviour of $G(x-y)$ with the substitution $m \to m/(1+\a)$; this result should have been expected, since the term proportional to $\a$ in the interaction could be absorbed in the free grassmanian integration producing exactly such change in the {\it bare} mass of the particles. The proof of equation \equ(3.38) will make use of the fermionic nature of the fields and of the explicit form of the propagator defined in section 2. We shall need the following results for the fermionic expectations: % $${1\over s!}\Big|\ET_h\big(\tilde{\psi}^{(h)} (P_1),\ldots, \tilde{\psi}^{(h)} (P_s)\big)\Big|\le \g^{{h\over4}\sum_j | P_j^1|}\g^{{5\over4}h\sum_j |P_j^2|} C^{\sum_j | P_j|} {1\over s!} \sum_T e^{-\k d^{(h)}_T (P_1,\ldots, P_s)}\Eq(3.43)$$ % where $| P|=| P^1| +| P^2|$ is the number of elements in $P$, $| P^1|$ is the number of fields $\psi^{(\cdot)}$, $| P^2|$ is the number of fields $\Delta\psi^{(\cdot)}$. Furthermore $T$ is an {\it anchored tree graph} between the clusters of space vertices from which the fields labeled by $P_{1},\dots ,P_{s}$ 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 $b^1,\ldots,b^s$ are the lines belonging to $T$ we define: % $$ d^{(h)}_T (P_1\ldots P_s) = \sum_{j=1}^s(\g^h |b^j_0| +\g^{h/2} |b^j_1|)\Eq(3.44)$$ % Note that, if $s=1$, the sum over $T$ is void and must be understood as a trivial factor $1$. The proof of the bounds \equ(3.43) is in Appendix 2; here we want to stress the absence of factorials in the number of fields, which is essentially linked to the fact that {\it we do not expand the l.h.s. in Feynman graphs}. With the aid of \equ(3.43) we can bound \equ(3.37) as follows: % $$\eqalignno{ &|V^{(k)}(N,\t,\{h_v\},P_{v_0},\undx)| \le \left\{ \prod_{v>v_0\atop \vnotep} \sum_{P_v} \right\} \,\cdot \cr &\cdot\, \prod_\vnotep \prod_j\left[\g^{{h_v\over4}[| P_{v^j}^1|- |Q^1_{v^j}|+5| P_{v^j}^2|-5| Q_{v^j}^2}|]\right]C^{\sum_j( | P_{v^j}| -| Q_{v^j}|)} \,\cdot&\eq(3.45)\cr &\cdot\, \left\{ \prod_\vnotep {1\over s_v!} \sum_{T_v} e^{-\k d^{(h_v)}_{T_v} (P_{v^1},\ldots,P_{v^{s_v}})} \right\} |\a|^{n_2'}|\n|^{n_2} \prod_{i=1}^{n_4}|\l v(x_{2i-1}-x_{2i})|\cr }$$ % where $Q^i_{v^j}=P_{v}^{i}\bigcap P^i_{v^j}$, $i=1,2$ and $j=1,\ldots, s_{v}$. Now we have to integrate the expression \equ(3.45) multiplied by the weight $e^{{\k\over 2}d^{(k)} (P_{v_0})}$. It is clear that in the r.h.s. of \equ(3.45) $\undx$ appears only in the last line; therefore we have to evaluate the expression: % $$\int \left\{ \prod_\vnotep {1\over s_v!} \sum_{T_v} e^{-\k d^{(h_v)}_{T_v} (P_{v^1},\ldots,P_{v^{s_v}})} \right\} \prod_{i=1}^{n_4}|\l v(x_{2i-1}-x_{2i})| e^{{\k\over 2}d^{(k)} (P_{v_0})}d\underline x\Eq(3.46)$$ % Here we have to use the properties of $v(x-y)$: in fact a global tree graph (on all the scales) requires in general also the $v$'s to insure the connection. The property of $v$ that we need is (see \equ(3.24) and \equ(2.31)): % $$|\l v(x-y)|\le |\l| A e^{-p_0 |\xx-\yy|} \d(t-t')\Eq(3.47)$$ % where $x=(t,\xx), y=(t',\yy)$. The latter inequality and a standard estimation of the integral allow to bound \equ(3.46) by: % $$C^n |\L|\prod_{v\ge v_0}\g^{-{3\over 2}h_v(s_v-1)} |\l|^{n_4}\Eq(3.48)$$ % By \equ(3.48) and \equ(3.45) we have: % $$\eqalign{ &{1\over |\L|} \int d\undx |V^{(k)}(N,\t,P_{v_0},\undx)| e^{{\k\over 2}d^{(k)} (P_{v_0})} \le C^n\,|\l|^{n_4} |\a|^{n_2'} |\n|^{n_2} \,\cdot \cr &\quad \cdot\, \left\{ \prod_{v>v_0\atop\vnotep} \sum_{P_v} \right\} \prod_\vnotep [\g^{{h_v\over4}[\sum_j| P_{v^j}^1|- | P_v^1|+ 5\sum_j| P_{v^j}^2|-5| P_v^2|]} \g^{-{3\over2}h_v(s_v-1)}] \cr} \Eq(3.49) $$ % and we note that: % $$\eqalign{ &\prod_\vnotep \g^{{h_v\over4}[\sum_j| P_{v^j}^1|- | P_v^1|+ 5\sum_j| P_{v^j}^2|-5| P_v^2|]}=\cr &=\left[\prod_\vnotep [\g^{{1\over 4}[6n_v^{2'}+4n_v^4 +2n_v^2-| P_v^1|-5| P_v^2|]}\right] \g^{{k\over4}[6n_{2'}+4n_4 +2n_2-| P_{v_0}^1|-5| P_{v_0}^2|]}\cr}\Eq(3.50)$$ % and: % $$\prod_\vnotep \g^{-{3\over 2}h_v(s_v-1)}=\left[\prod_\vnotep \g^{-{3\over 2} (n_v-1)}\right]\g^{-{3\over 2}k(n-1)}\Eq(3.51)$$ % Therefore we can rewrite the last factor of \equ(3.49) as: % $$ \left[\prod_\vnotep \g^{-{1\over 4}[|P_v^1|+5| P_v^2|+2n_v^4+4n_v^2-6] }\right] \g^{-k/4[2n_4 +4n_2+| P_{v_0}^1|+5| P_{v_0}^2|-6]}\Eq(3.52)$$ % where $n_v$ is the number of endpoints which follow $v$ in the tree, while $n_v^4,n_v^2,n_v^{2'}$ are the numbers of endpoints of type 4,2,2' which follow $v$. where $n_v$ is the number of endpoints which follow $v$ in the tree, while $n_v^4,n_v^2,n_v^{2'}$ are the numbers of endpoints of type 4,2,2' which follow $v$. Let us observe now that: $$[| P_v^1|+5| P_v^2|+2n_v^4+4n_v^2-6] > 0\Eq(3.53)$$ % (hence $\ge 1$), except in the following cases, that we discuss separately. 1) $| P_v^1|=2,\quad n_v^4=2,\quad| P_v^2|=n_v^2=0$.\par The only possible Feynman graphs associated with $\t_v$ are, in this case: \insertplot{300pt}{100pt}{fig4}{fig4} \\where the dots on the inner lines and on the external outgoing lines represent insertions of type $2'$ graph elements. However, their contribution is exactly zero by the remark 2) after \equ(3.15), which is valid also for Feynman graphs with propagators of different frequencies. 2) $| P_v^1|=2,\quad n_v^4=1,\quad| P_v^2|=n_v^2=0$.\par This is the case of the graphs: \insertplot{150pt}{120pt}{fig5}{fig5} \\which vanish for the same reason of the case 1). 3) $| P_v^1|=1,\quad | P_v^2|=1,\quad n_v^4=n_v^2=0$.\par This is the case of the trees, whose graph elements are all of type $2'$, so that only the chains of Fig. 7 are allowed. \insertplot{300pt}{50pt}{fig6}{fig6} If $v\not= v_0$, one of the two lines external with respect to $v$ is internal to the non trivial vertex $v'$ preceding $v$. To be definite, let us suppose that this is the case for the line emerging from $x_m$ (the other case can be treated in the same way); then all terms contributing to the expansion in Feynman graphs of $W^{(k)}(N,\t,P_{v_0},\undx^{(P_{v_0})})$ contain a factor of the type: % $$\int dx_2\ldots dx_m \D_{x_2} C_{h_1}(x_1-x_2) \ldots \D_{x_m} C_{h_{m-1}}(x_{m-1}-x_m) (\D_y)^\r C_{h_{v'}}(x_m-y) \Eq(3.54)$$ % where $\r=0$ or $\r=1$. Let us suppose first that all the lines have the same frequency, that is $h_i = h_v$, for $i=1,\ldots,m-1$. Then, since $\int dx_m \D C_h (x_{m-1}-x_m)=0$, we can substitute in \equ(3.54) $C_{h_{v'}} (x_m-y)$ with: % $$C_{h_{v'}} (x_m-y) - C_{h_{v'}} (x_{m-1}-y) = (x_m-x_{m-1}) \int_0^1dt\partial C_{h_{v'}}(x_m-y-t(x_m-x_{m-1})) \Eq(3.55)$$ % and it is easy to see that such substitution allows us to improve the bound by a factor $\g^{-(h_v-h_{v'})/2}$. If the lines have different frequencies, \ie if there are other non trivial vertices following $v$, %(for the graphs that we are considering, no internal line %can be associated to a trivial vertex), % Questo commento mi pare molto oscuro: in realta' si dovrebbe dire % che le linee di frequenza corrispondente ad un vertice banale % sono interne al vertice stesso we have to apply the previous argument iteratively starting from the higher vertices. The only change is that some covariance in \equ(3.54) is substituted by its gradient calculated at an interpolated point as in the r.h.s. of \equ(3.55); it is easy to see that the improvement for each non trivial vertex is always the same, \ie $\g^{-1/2}$ raised to a power equal to the difference between the frequency of the vertex and that of the preceding non trivial one. Furthermore, there is at most a factor $|x-x'|$ for each line connecting $x$ and $x'$ and each covariance must be interpolated at most two times; so no dangerous factorials appear. Of course, in order to improve the bound, we have to expand in Feynman graphs the subtree starting in the vertex $v$ and {\it extract} the propagator $C_{h_{v'}} (x_m-y)$ from the truncated expectation associated with $v'$. One could be afraid that this destroys the good combinatorial properties of \equ(3.43), but this is not the case. In fact the subtree starting from $v$ belongs to $\TT_m^*$ and it is easy to see that its expansion in Feynman graphs contains exactly $s_v!$ terms, which is compensated by making use of the $1/s_v!$ factors of \equ(3.37); so there is no combinatorial problem here. The problem of the extraction of $C_{h_{v'}} (x_m-y)$ from the truncated expectation is not really present, since each term contributing to the r.h.s. of \equ(3.43) has a factor equal to one of the external propagators of $v$ (see the proof of \equ(3.43) in appendix 2). We have still to consider the case $v=v_0$, but now $\t\in\TT_n^*$ and we can use the factor $\c_\t (\undx^{(P_{v_0})})$ in \equ(3.38) to improve the bound by a factor $\g^{-(h_v-k)/2}$. 4) $| P_v^1|=2,\quad | P_v^2|=0,\quad n_v^4=0, \quad n_v^2=1$\par This is the case of the tree with an arbitrary number of type $2'$ graph elements and one of type $2$. The same considerations of the case 3) apply, so that again we can improve the bound by a factor $\g^{-(h_v-h_{v'})/2}$. We can summarize the discussion above, by saying that the last line of \equ(3.49) can be replaced by the expression: % $$\left[ \prod_\vnotep \g^{-{1\over 4} D_v} \c(D_v>0) \right] \g^{-{k \over 4} D_{v_0}} \Eq(3.56)$$ % where $\c(D_v>0)$ is the characteristic function of the set $\{D_v>0\}$ (it recalls us that the graphs of items 1 and 2 above are not allowed) and % $$D_v=|P_v^1|+5|P_v^2|+2n_v^4+4n_v^2-6+2\d_{|P_v^1|,1} \d_{|P_v^2|,1}\d_{n_v^4,0}\d_{n_v^2,0}+2\d_{|P_v^1|,2} \d_{|P_v^2|,0}\d_{n_v^4,0}\d_{n_v^2,1} \Eq(3.57)$$ The above discussion shows the essentially trivial renormalizability of this model. In fact, since the number of unlabeled trees with $n$ endpoints can be bounded by $4^n$, in order to prove the bound \equ(3.38) it is sufficient to control the multiple sums in \equ(3.49) and the sum over the labeled trees with a fixed topological structure. This can be easily done by using the factors $\g^{-{1\over 4}D_v}$ of \equ(3.56). We first observe that, given an unlabeled tree $\tilde\t$, there are only $3^n$ corresponding families of labeled trees differing for the choice of the graph elements associated with the endpoints; hence it is sufficient to consider only one of such families, say $\tilde\TT$. The trees $\t\in\tilde \TT$ can be distinguished by fixing the frequencies indices of the non trivial vertices, which we shall denote $\tilde v$. We can write: % $$\prod_\vnotep \g^{-{1\over 4} D_v} \le \left[ \prod_{\tilde v} \g^{-{1\over 8}(h_{\tilde v}-h{\tilde v'})} \right] \left[ \prod_\vnotep \g^{-{1\over 8} D_v} \right] \Eq(3.58)$$ % where $\tilde v'$ is the non trivial vertex immediately preceding $\tilde v$ or the root, if there is no such vertex. The sum over the set $\tilde\TT$ of the first factor in the r.h.s. of \equ(3.58) can be bounded in a trivial way by a factor $C^n$. Furthermore, by \equ(3.57), if $D_v>0$: % $$D_v\ge \max \{1,|P_v|-2\} \ge {|P_v|\over 3} \Eq(3.59)$$ % Hence, in order to complete the proof of \equ(3.38), it is sufficient to prove that: % $$\prod_\vnotep \sum_{P_v} \g^{-{|P_v|\over 24}} \equiv S(P_{v_0},\t,n)\le C^n\Eq(3.60)$$ % where the sums over the sets $P_v$ are constrained by the condition that $P_v=\bigcup_j Q_{v^j}$ with $Q_{v^j}$ a subset, possibly empty, of $P_{v^j}$; furthermore $P_v$ is a fixed set with four or two elements, if $v$ is an endpoint, and we have eliminated the constraint that $P_{v_0}$ is a fixed subset of the fields associated with the tree graph elements. The latter estimate, evident for large $\g$, can be proved in the general case $\g>1$ in the following way. We note that: % $$S(P_{v_0},\t,n) \le \prod_\vnotep \sum_{p_v} \g^{-{p_v\over 24}} C_v \Eq(3.61)$$ % where $C_v$ counts the number of ways of choosing a subset $P_v$ with $p_v$ elements, satisfying the constraints; hence it can be easily bounded by a binomial coefficient and we obtain: % $$S(P_{v_0},\t,n) \le \prod_\vnotep \sum_{p_v} \g^{-{p_v\over 24}} {\sum_{j=1}^{s_v} p_{v^j}\choose p_v}\Eq(3.62)$$ Set $\bar\g=\g^{1\over 24}$ and let us denote with $\cal P$ a path from the root of the tree to an endpoint and with $l(\cal P)$ the number of vertices lying on $\cal P$. It is easy to show, by simply performing the sums in \equ(3.62) one after the other starting from $v_0$, that: % $$S(P_{v_0},\t,n) \le \prod_{\cal P} \left( \sum_{n=0}^{l(\cal P)}\bar\g^{-n}\right)^4 \le \Big({1\over 1- \bar\g^{-1}}\Big)^{4n}\Eq(3.63)$$ The bound \equ(3.38) implies that we can sum, for $|z|$ small enough, say $|z|\le \e$, and uniformly in $N$, the terms in the effective potential, which have the same dependence on the field (\ie that have the same set of labels $\{\s_f, f\in P_{v_0} \}$). In fact, we have still to bound only the sum of all trees of order $n$ satisfying that condition: as mentioned above this gives simply another factor $\le 4^n$, as the trees are ``topological trees'', see item 1) after Fig. 2. \vskip.5truecm We can now integrate also the field fluctuations associated with the regular part $R(x)$ of the u.v. covariance, see \equ(3.4) and \equ(3.9). The regularity of the propagator $R$ makes this a trivial repetition of, say, the last integration lowering the u.v. cut off from $h=1$ to $h=0$ and we do not have to perform it in detail. The bounds of this section imply that $\bar{V}^{(0)}(\f)$ can be written, for $|z|\le \e$, as in \equ(3.17) and that a similar expression is valid for the effective potential on scale $\g^{-k}$. Furthermore the kernel $\tilde{W}_2(z,x)$ singles out the contributions coming from the trees in $\TT_n^*$ (see discussion after \equ(3.40) ) and therefore satisfies (uniformly in $N$) the bound \equ(3.19) and the equation \equ(3.20). >From the considerations of \S2 it is almost obvious that the effective potentials can be given the expression \equ(3.17) for $|z|0$ and of order $O(1)$ and have a uniform exponential decay ($\L,N$-independent): see \equ(3.18),\equ(3.19). This means that we can sum the coefficients of give order in $z$ and that their sum admits good exponential bounds. Note that this {\it is not sufficient} to guarantee the integrability in the sense of \S2 of $\exp \bar V^{(0)}(\psi^{(i.r.)} +\f)$ with respect to the i.r. part of the grassmanian fields for $|z|<\e$. We shall proceed by imagining that we have a u.v. cut off $N$ and perform the integrations down to the infrared cut off $R$: and we shall see that it is possible to perform a resummation of perturbation theory permitting to express the effective potentials as uniformly convergent power series in a sequence of constants $\undr_h$, called the {\it running couplings}, which are themselves expressed as sums of series in the initial couplings $z$. The series for the running couplings will have very small $L,N,R$ dependent radii of convergence. But they will be related by a map permitting to express their values at scale $h$ in terms of the values at the preceding scales $h-1,\ldots,0$. We shall show that the relation is expressed by an analytic function, {\it the beta functional}, of the preceding couplings with a radius of convergence which is uniform in $N,R,L$. Thus if {\it by some other means}, see \S7, one can be sure that the beta functional generates a sequence $\undr_h$, $h=0,-1,\ldots$, of running couplings which stay small uniformly in the index $h$, then one will have shown the possibility of a resummation of the perturbation series for the full effective potential kernels, which is uniform in $R,N,L$ and a theory of the ground state will have been constructed (up to the {\it technicalities} analyzed in \S6). In the next section we begin the discussion on the beta functional and its analyticity properties. \vskip2.truecm \vglue1.truecm {\it\S4 The effective potential in the infrared region. Failure of normal scaling.} \vglue1.truecm\numsec=4\numfor=1 In this section we shall begin the analysis of the infrared problem, that is of the possibility of giving a meaning to the integration in \equ(3.8) of the infrared fluctuations of the field, associated with the propagator: % $$g_{i.r.}(x) \equiv g^{(\le 0)}(x)=\int {dk_0d\kk\over (2\p )^2}{e^{-ik x} \over -ik_0+e(\kk)} e^{-[k_0^2+e(\kk)^2]p_0^{-2}}\Eq(4.1)$$ Note that the Fourier transform of $g^{(\le 0)}(x)$ has a linear divergence on the {\it Fermi surface} $k_0=0, \kk=\pm p_F$, which cannot be treated by a naive multiscale decomposition as the one used for the u.v. problem, because of the presence of the built-in scale $p_F$. It is possible, however, to rewrite the problem in terms of {\it quasi particle fields} in the way presented in [BG], that we briefly summarize here. We write the {\it particle field} $\ps{\s (\le0)}{x}$ of covariance $g^{(\le 0)}(x)$ as a sum of independent {\it quasi particle fields}: % $$\ps{\s (\le0)}{x}\equiv \sum_{\o =\pm 1}e^{i\s p_F\o\xx} \ps{\s (\le0)}{\o ,x}\Eq(4.2)$$ % and, as usual, the fields $\ps{\s (\le0)}{\o ,x}$, essentially describing the fluctuations around the two points of the Fermi surface, are decomposed as sums of independent fields in the following way: % $$\ps{\s (\le0)}{\o ,x}=\sum_{h=-\infty}^0\ps{\s (h)}{\o ,x} \Eq(4.3)$$ % where $\ps{\s (h)}{\o ,x}$ has covariance: % $$g^{(h)}_{\o}(x) =e^{i p_F\o\xx}\int_{\g^{-2h}}^{\g^{-2h+2}} d\a \int {dk\over (2\p )^2} e^{-ik\cdot x} (ik_0+e(\kk)) e^{-\a p_0^{-2} [k_0^2+e(\kk)^2]} \c(\o\g^{-h}\kk) \Eq(4.4)$$ Here $\c(t)=\p^{-1/2}\int_{-\i}^t ds\exp(-s^2)$ is a regularization of the step function. In App. 1 we show (see also [BG], App. A) that, for any integer $m\ge 0$: $$|\dpr^m g^{(h)}_{\o}(x)| \le C_m \g^{h(1+m)} e^{-\k\g^h| x|}\Eq(4.5)$$ for some suitable constants $C_m$ and $\k$, independent of $h$. In the following we shall use also the definitions: $$\ps{\s(\le h)}{\o,x} = \sum_{k=-\i}^h \ps{\s(k)}{\o,x} \quad,\quad \ps{\s(\le h)}{x} = \sum_{\o=\pm 1} e^{i\s p_F\o\xx} \ps{\s(\le h)}{\o,x} \Eq(4.6)$$ In order to evaluate $V_{eff}(\f)$, by \equ(3.8) and \equ(3.9), we should study the functional integral % $$\int P(d\psi^{\le 0}) e^{- \bar V^{(0)}(\psi^{\le 0}+\f)} \Eq(4.7)$$ % However, the analysis of this integral is more delicate in comparison to the analogous ultraviolet problem, because of the anomalous scaling. Therefore we split the problem into the simpler problem of defining the {\it running couplings} and into that of evaluating the effective potential. The first problem already emerges from the study of the integral \equ(4.7) for $\f=0$, \ie from the study of the normalization constant in \equ(3.8); this analysis will be performed in this section and in the following one. The second problem will be faced up in \S6 indirectly, through the analysis of the Schwinger functions, which are the physically relevant quantities. Setting $\psi=\psi^{(\le 0)}$ to simplify the notation, we represent the potential $\bar V^{(0)}(\psi)$, see \equ(3.17), in terms of quasi particle fields and we obtain: % $$\eqalign{ & \bar V^{(0)}(\psi) = \l \int dx\,dy \sum_{\o_1\ldots\o_4} e^{ip_F[(\o_1-\o_2)\xx+(\o_3-\o_4)\yy]} \ps{+}{\o_1x}\ps{-}{\o_2x} v(x-y)\ps{+}{\o_3y}\ps{-}{\o_4y} \,+ \cr &\quad + \n \int dx \sum_{\o_1,\o_2}^{ }e^{ip_F(\o_1-\o_2)\xx} \,\ps{+}{\o_1x}\ps{-}{\o_2x} \,+ \cr &\quad + \a \int dx\sum_{\o_1,\o_2}^{ }e^{ip_F(\o_1-\o_2)\xx} \, \ps{+}{\o_1x}i\b\o_2{\cal D}^-_{\o_2}\ps{-}{\o_2x} \,+ \cr &\quad + \sum_{n=1}^{\infty}\sum_{n_1,n_2\atop n_1+n_2=2n} \sum_{\o_1\ldots \o_n\atop \o_1'\ldots\o_n'} \int dx_1\ldots dx_{2n} e^{ip_F \sum_{i=1}^n (\o_i \xx_i-\o'_i \xx_{n+i})} \,\cdot \cr &\quad \cdot \ps{+}{\o_1x_1}\ldots\ps{+}{\o_nx_n}\ps{-}{\o_1'x_{n+1}} \ldots\ps{-}{\o_{n-n_2}'x_{2n-n_2}} \,\cdot \cr &\quad \cdot i\o_{n-n_2+1}'{\cal D}^-_{\o_{n-n_2+1}'}\ps{-}{\o_{n-n_2+1}'x_{2n-n_2+1}} \ldots i\o_{n}'{\cal D}^-_{\o_{n}'}\ps{-}{\o_nx_{2n}} \bar W_{n_1n_2}(z,x_1\ldots x_{2n}) \cr } \Eq(4.8)$$ % where $\b\equiv p_F/m$, the {\it covariant derivative} ${\cal D}^-_{\o}$ is a differential operator acting only on the space coordinate, defined by: $${\cal D}^-_{\o}=\partial_{\xx}+{i\o\partial_{\xx}^2\over 2p_F}\Eq(4.9)$$ and the contribution of the third line in \equ(3.17) has been included in the last term (see discussion related to \equ(4.38) and \equ(4.39) below). The ${\cal D}^-_\o$ operator satisfies the following identity, which will play an important role in the following: % $$\int dx \ps{+(\le h)}{x} e(i\dpr_\xx) \ps{-(\le h)}{x} = \sum_{\o_1,\o_2} \ig dx \,e^{ip_F(\o_1-\o_2)\xx} \, \ps{+(\le h)}{\o_1x}i \b \o_2{\cal D}^-_{\o_2}\ps{-(\le h)}{\o_2x} \Eq(4.10)$$ It is now very natural to define the {\it effective potential on scale $\g^{-h}$}, for $h<0$, as in \equ(3.25), through the expression: $$e^{-\bar{V}^{(h)}(\psi^{(\le h)})} = {1\over \NN} \int P(d\ps{(h+1)}{})\ldots \int P(d\ps{(0)}{}) e^{-\overline V^{(0)}(\ps{(\le 0)}{})} \Eq(4.11)$$ We shall see in the following that {\it this is is not the correct definition}, because of the anomalous scaling properties of the model. However we proceed for the moment with this definition in order to show where and why the problem arises. As explained in Ref. [BG], we can isolate the relevant part of the effective potential by introducing a localization operator $\LL$ which acts linearly on the monomials in the fields of the form $\prod_i \ps{\s_i}{\omega _i x_i}$ and is zero on all monomials of degree $\ge 6$. Its action on the monomials of degree $2$ and $4$ is generated by linearity from: % $$\eqalignno{ \LL(\psi ^{+}_{\o_{1}x_{1}}\psi^{+}_{\o _{2}x_{2}} \psi ^{-}_{\o_{3}x_{3}}\psi^{-}_{\o_{4}x_{4}}) &= {1\over 2} [\psi^{+}_{\o_{1}x_{1}}\psi^{+}_{\o_{2}x_{1}}\psi^{-}_{\o_{3}x_{1}} \psi^{-}_{\o_{4}x_{1}} + \psi^{+}_{\o_{1}x_{2}}\psi^{+}_{\o_{2}x_{2}}\psi^{-}_{\o_{3}x_{2}} \psi^{-}_{\o_{4}x_{2}}] \cr \LL(\psi ^{+}_{\o_{1}x_{1}}\psi^{-}_{\o_{2}x_{2}}) &= \psi^{+}_{\o_{1}x_{1}}\psi^{-}_{\o_{2}x_{1}}+ (x_{2}-x_{1}) \psi^{+}_{\o_{1}x_{1}}{\cal D}_{\o_{2}}\psi^{-}_{\o_{2}x_{1}} &\eq(4.12) \cr}$$ % where (see \equ(4.9)): $${\cal D}_{\o}\equiv (\partial_{t} ,{\cal D}^-_{\o}) \Eq(4.13)$$ We used in the second line of \equ(4.12) the covariant derivative \equ(4.9) instead of the normal space derivative, which could perhaps look more natural, for a reason which will be explained later (see remark following \equ(4.30) below); in any event our choice \equ(4.12) differs from the other one only by an irrelevant term. If $\RR =1-\LL $, we have also: % $$\eqalignno{ &\qquad \RR (\psi ^{+}_{\o_{1}x_{1}}\psi ^{+}_{\o _{2}x_{2}} \psi ^{-}_{\o _{3}x_{3}}\psi ^{-}_{\o_{4}x_{4}}) = &\eq(4.14)\cr &= {1\over 2}[\psi^{+}_{\o_{1}x_{1}}D_{21{\o_2}}^{+}\psi^{-}_{\o_{3}x_{3}} \psi^{-}_{\o_{4}x_{4}} + \psi^{+}_{\o_{1}x_{1}}\psi^{+}_{\o_{2}x_{1}} D_{31{\o_3}}^{-}\psi^{-}_{\o_{4}x_{4}} + \psi^{+}_{\o_{1}x_{1}} \psi^{+}_{\o_{2}x_{1}}\psi^{-}_{\o_{3}x_{1}}D_{41{\o_4}}^{-}]+ \cr & + {1\over 2} [D_{12{\o_2}}^{+}\psi^{+}_{\o_{2}x_{2}}\psi^{-}_{\o_{3}x_{3}} \psi^{-}_{\o_{4}x_{4}} + \psi^{+}_{\o_{1}x_{2}}\psi^{+}_{\o_{2}x_{2}} D_{32{\o_3}}^{-}\psi^{-}_{\o_{4}x_{4}} + \psi^{+}_{\o_{1}x_{2}} \psi{+}_{\o_{2}x_{2}}\psi^{-}_{\o_{3}x_{2}}D_{42{\o_4}}^{-}]\cr}$$ % where: $$\eqalign{ D^{\s}_{ji\o_{j}} &= \ps{\s}{\o_j x_j} - \ps{\s}{\o_j x_i} = (x_{j}-x_{i})\int_0^1 dr \, \dpr \psi^{\s}_{\o_{j}x_{ji}(r)} \cr x_{ji}(r) &= x_{i}+r (x_j-x_i) \;, ~~~ \partial \equiv (\partial_{t},\partial_{\xx})\cr } \Eq(4.15)$$ and for the quadratic term in the fields we have: $$\eqalign{ \RR (\psi^{+}_{\o_{1}x_{1}}\psi^{-}_{\o_{2}x_{2}}) &= - {i\o_{2}\over 2p_F}(\xx_{2}-\xx_{1})\psi^{+}_{\o_{1}x_{1}} \partial^{2}_{\xx}\psi^{-}_{\o_{2}x_{1}} + \cr &+ (x_{2}-x_{1})^{2} \psi^{+}_{\o_{1}x_{1}}\int_0^1 dr \int_0^r ds \dpr^2 \ps{-}{\o_2x_{21}(s)}\cr} \Eq(4.16)$$ % where: $$(x_{2}-x_{1})^{2} \int_0^1 dr \int_0^r ds \partial^{2}\psi^{-}_{\o_2 x_{21}(s)}= \psi^{-}_{\o_{2}x_{2}}-\psi^{-}_{\o_{2}x_{1}}- (x_{2}-x_{1})\partial \psi^{-}_{\o_{2}x_{1}} \Eq(4.17)$$ We plan to evaluate iteratively the integrals in the r.h.s. of \equ(4.11), by rewriting at each step $\bar V^{(h)}$ in the form $\LL \bar V^{(h)} + \RR \bar V^{(h)}$. This implies that we have to consider the action of $\LL$ also on other monomials of second and fourth order, besides those appearing in \equ(4.8). We shall give now the complete list of the monomials that one has to take into account, for which the action of $\LL$ does not give zero, together with the result of the application of $\LL$ and $\RR$, deduced from \equ(4.12) by linearity. In the case of the fourth order monomials there is only one more term on which $\LL$ is not trivial, in principle; it is the one of the form $\ps{+}{x_1\o_1} \dpr \ps{+}{x_2\o_2}\ps{-}{x_3\o_3} \ps{-}{x_4\o_4}$. This term can only appear if $x_1$ is an interpolated point, see \equ(4.15), so that we really need the following equation: $$\eqalign{ & \LL \ps+{x_1\oo_1} D^+_{{x_2x_{2'}\oo_2}}\ps-{x_3\oo_3}\ps-{x_4\oo_4} =\cr & \quad = {1\over 2} [\ps+{{x_2}\oo_1}\ps+{{x_2}\oo_2} \ps-{{x_2}\oo_3}\ps-{{x_2}\oo_4} - \ps+{{x_{2'}\oo_1}}\ps+{{x_{2'}\oo_2}}\ps-{{x_{2'}\oo_3}} \ps-{{x_{2'}\oo_4}} ]\cr}\Eq(4.18)$$ By the anticommutation properties of the field, the r.h.s. can be different from zero only if $\o_1=-\o_2, \o_3=-\o_4$. However, in this case, the integration on the $x$-variables cancels it, because the monomial in the l.h.s. appears multiplied by a translation invariant function of the $x$-variables; furthermore the oscillating factor $e^{ip_F (\o_1\xx_1 +\o_2\xx_2 -\o_3\xx_3 -\o_4\xx_4)}$ is also translation invariant, if $\o_1=-\o_2, \o_3=-\o_4$. Hence, for our purposes: $$\LL \ps{+}{x_1\o_1} \dpr \ps{+}{x_2\o_2}\ps{-}{x_3\o_3} \ps{-}{x_4\o_4}=0 \Eq(4.19)$$ and we do not have to consider any other localization operation on the fourth order monomials, besides that of \equ(4.12). In the case of the second order monomials, we have to consider the following localization operations: $$\eqalign{ \LL (\psi^{+}_{\o_{1}x_{1}}{\cal D}_{\o_{2}}\psi^{-}_{\o_{2}x_{2}}) &= \psi^{+}_{\o_{1}x_{1}}{\cal D}_{\o_{2}}\psi^{-}_{\o_{2}x_{1}} \cr \LL (\psi^{+}_{\o_{1}x_{1}}\partial \psi^{-}_{\o_{2}x_{2'}}) &= \psi^{+}_{\o_{1}x_{1}}{\cal D}_{\o_{2}}\psi^{-}_{\o_{2}x_{1}} \cr \LL (\partial \psi^{+}_{\o_{1}x_{1'}}\psi^{-}_{\o_{2}x_{2}}) &= \partial (\psi^{+}_{\o_{1}x_{1'}}\psi^{-}_{\o_{2}x_{1'}})- \psi^{+}_{\o_{1}x_{1'}}{\cal D}_{\o_{2}}\psi^{-}_{\o_{2}x_{1'}}+ \cr &+(x_{2}-x_{1'})\partial (\psi^{+}_{\o_{1}x_{1'}} {\cal D}_{\o_{2}}\psi^{-}_{\o_{2}x_{1'}}) \cr \LL (\partial \psi^{+}_{\o_{1}x_{1'}} {\cal D}_{\o_{2}}\psi^{-}_{\o_{2}x_{2}}) &= \partial (\psi^{+}_{\o_{1}x_{1'}} {\cal D}_{\o_{2}}\psi^{-}_{\o_{2}x_{1'}})\cr \LL (\partial \psi^{+}_{\o_{1}x_{1'}} \partial \psi^{-}_{\o_{2}x_{2'}}) &= \partial (\psi^{+}_{\o_{1}x_{1'}} {\cal D}_{\o_{2}}\psi^{-}_{\o_{2}x_{1'}})\cr} \Eq(4.20)$$ % and the corresponding $\RR$ operations: % $$\eqalignno{ \RR (\psi^{+}_{\o_{1}x_{1}}{\cal D}_{\o_{2}}\psi^{-}_{\o_{2}x_{2}}) &= (x_{2}-x_{1})\psi^{+}_{\o_{1}x_{1}} \int_0^1 dr {\cal D}_{\o_{2}}\partial \psi^{-}_{\o_{2}x_{21(r )}} & \eq(4.21) \cr \RR(\psi^{+}_{\o_{1}x_{1}}\partial \psi^{-}_{\o_{2}x_{2'}}) &= - {i\o_{2}\over 2p_{F}}\psi^{+}_{\o_{1}x_{1}} \partial_{\xx}^{2}\psi^{-}_{\o_{2}x_{1}} + (x_{2'}-x_{1})\psi^{+}_{\o_{1}}(x_{1})\int_0^1 dr \partial \partial \psi^{-}_{\o_{2}x_{2'1(r )}} \cr \RR (\partial \psi^{+}_{\o_{1}x_{1'}}\psi^{-}_{\o_{2}x_{2}}) &= (x_{2}-x_{1'})\partial \psi^{+}_{\o_{1}x_{1'}} \int_0^1 dr \partial \psi^{-}_{\o_{2}x_{21'(r )}}- {i\o_{2}\over 2p_{F}}\psi^{+}_{\o_{1}x_{1'}} \partial^{2}_{\xx}\psi^{-}_{\o_{2}x_{1'}}- \cr &-(x_{2}-x_{1'})\partial \psi^{+}_{\o_{1}x_{1'}} {\cal D}_{\o_{2}}\psi^{-}_{\o_{2}x_{1'}}- (x_{2}-x_{1'})\psi^{+}_{\o_{1}x_{1'}}\partial {\cal D}_{\o_{2}} \psi^{-}_{\o_{2}x_{1'}} \cr \RR (\partial \psi^{+}_{\o_{1}x_{1'}} {\cal D}_{\o_{2}}\psi^{-}_{\o_{2}x_{2}}) &= \partial \psi^{+}_{\o_{1}x_{1'}} {\cal D}_{\o_{2}}\psi^{-}_{\o_{2}x_{2}}- \partial (\psi^{+}_{\o_{1}x_{1'}} {\cal D}_{\o_{2}}\psi^{-}_{\o_{2}x_{1'}})\cr \RR (\partial \psi^{+}_{\o_{1}x_{1'}} \partial \psi^{-}_{\o_{2}x_{2'}}) &= \partial \psi^{+}_{\o_{1}x_{1'}} \partial \psi^{-}_{\o_{2}x_{2'}}- \partial (\psi^{+}_{\o_{1}x_{1'}} {\cal D}_{\o_{2}}\psi^{-}_{\o_{2}x_{1'}}) \cr} $$ % where the symbol $x_{j'}$ is used to stress that $x_{j'}$ is a point on the segment connecting $x_{j}$ and some other point. In the r.h.s. of the last three of equations \equ(4.20), appear some new local terms with respect to the second relation in the r.h.s. of \equ(4.12). However, the field $\partial \psi^{+}_{\o_{1}x_{1'}}$ in the l.h.s. of \equ(4.20) can appear only through a field $D^+_{12}$ by interpolation, see \equ(4.15). Hence one has really to consider the following localization operations: % $$\eqalign{ \LL (D^+_{12\o_1}\ps-{x_3\o_2}) &= \ps+{x_1\o_1}\ps-{x_1\o_2} - \ps+{x_2\o_1}\ps-{x_2\o_2} +\cr &+ (x_3-x_1)\ps+{x_1\o_1}{\cal D}_{\o_2}\ps-{x_1\o_2} -(x_3-x_2) \ps+{x_2\o_1}{\cal D}_{\o_2}\ps-{x_2\o_2} \cr \LL (D^+_{12\o_1}{\cal D}_{\o_2}\ps-{x_3\o_2}) &= \ps+{x_1\o_1}{\cal D}_{\o_2}\ps-{x_1\o_2} - \ps+{x_2\o_1}{\cal D}_{\o_2} \ps-{x_2\o_2} \cr \LL (D^+_{12\o_1}D^-_{34\o_2}) &= (x_3-x_4) (\ps+{x_1\o_1}{\cal D}_{\o_2}\ps-{x_1\o_2} - \ps+{x_2\o_1} {\cal D}_{\o_2} \ps-{x_2\o_2}) \cr} \Eq(4.22)$$ % And we can conclude that, as a result of the localization operation on the effective potential, we get, for each scale, the following local monomials: % $$\ps{+}{x+}\ps{+}{x-}\ps{-}{x+}\ps{-}{x-},~~~~~ \ps{+}{x\o}\ps{-}{x\o'},~~~~~ \ps{+}{x\o}i\o'\b{\cal D}^-_{\o'}\ps{-}{x\o'}, ~~~~~\ps{+}{x\o}\partial_{t}\ps{-}{x\o'}\Eq(4.23)$$ % multiplied by some constants, the {\it running coupling constants} of the model, that we shall indicate, respectively, with $\l_h$, $\g^h \n_h$, $\a_h$, $\z_h$. At first sight, the running coupling constants depend on the $\o $ variables; however, we shall see that they are actually $\o $-independent. The fourth order local part must have the form: % $$\sum_{\o }\int dx\, \l_{h}(\o_{1},\o_{2},\o_{3},\o_{4}) \, e^{ip_{F}(\o_{1}+\o_{2}-\o_{3}-\o_{4})\xx} \psi^{+(\le h)}_{\o_{1}x}\psi^{+(\le h)}_{\o_{2}x} \psi^{-(\le h)}_{\o_{3}x}\psi^{-(\le h)}_{\o_{4}x}\Eq(4.24)$$ % and recalling the anticommutation properties of fermions, we can write: $$\l_{h}(\o_{1},\o_{2},\o_{3},\o_{4})=-{\l_{h} \over 4} \o_1\o_3 \d_{\o_1,-\o_2} \d_{\o_3,-\o_4}\Eq(4.25)$$ % Hence we can rewrite the quartic relevant part in the simpler way: % $$\l_{h}\int dx\, \psi^{+(\le h)}_{+1x}\psi^{+(\le h)}_{-1x} \psi^{-(\le h)}_{-1x}\psi^{-(\le h)}_{+1x}\Eq(4.26)$$ Let us now investigate the $\o$-dependence of the running coupling constants associated with the quadratic terms in the effective potential on scale $\g^{-h}$. By the linearity of $\LL$, we can calculate the local part in a different way. First we can do all the integrations in \equ(4.11) without introducing the quasi particle field representation; then we represent the effective potential in terms of the quasi particle fields and finally we apply the localization operator. After the first step, the quadratic part of the effective potential on scale $\g^{-h}$, expressed in terms of particle fields, looks as follows: % $${\bar V}^{[2](h)}=\int dxdy\ v_{h}(x-y)\psi^{+}(x)\psi^{-}(y)+ \int dxdy\ w_{h}(x-y)\psi^{+}(x)e(i\dpr_\yy) \psi^{-}(y)\Eq(4.27)$$ % where $v_{h}$ and $w_{h}$ are rotation invariant kernels (this means, in one dimension, that they are even functions in the spatial coordinate); such property follows from the fact that the free propagator of the theory and the interaction are indeed rotation invariant. We represent now ${\bar V}^{[2](h)}$ in terms of quasi particle fields: $$\eqalign{ {\bar V}^{[2](h)} &= \sum_{\o ,\o'}\; \left[ \int dxdy\ v_{h}(x-y)e^{ip_{F}(\o \xx-\o'\yy)} \psi^{+}_{\o x} \psi^{-}_{\o'y}+ \right.\cr &+ \left. \int dxdy\ w_{h}(x-y)e^{ip_{F}(\o \xx-\o'\yy)}\psi^{+}_{\o x} i\b \o '{\cal D}^-_{\o '}\psi^{-}_{\o 'y} \right] \cr }\Eq(4.28)$$ Hence the second order local part has the form: $$\eqalignno{ \LL {\bar V}^{[2](h)} &= \g^h \n_h \sum_{\o \o '}\int dxe^{ip_{F}(\o -\o') \xx} \psi^{+}_{\o x}\psi^{-}_{\o 'x}+ &\eq(4.29) \cr &+ \a_{h}\sum_{\o \o '}\int dxe^{ip_{F}(\o -\o')\xx}\psi^{+}_{\o x} i{\b }{\o '}{\cal D}^-_{\o '}\psi^{-}_{\o 'x}+ \z_{h}\sum_{\o \o '}\int dxe^{ip_{F}(\o -\o')\xx} \psi^{+}_{\o x}\partial_{t}\psi^{-}_{\o 'x}\cr }$$ where, if $\zz $ is the spatial part of the two dimensional space-time vector $z$ and $z_{0}$ is its time component: $$\eqalign{ \g^h \n_h &= \int dzv_{h}(z)e^{ip_{F}\o '\zz}\cr \a_{h} &= \int dze^{ip_{F}\o '\zz }[w_{h}(z)+{i\over \b} \o '\zz v_{h}(z)]\cr \z_{h} &= \int dze^{ip_{F}\o '\zz }(-z_{0})v_{h}(z)\cr }\Eq(4.30)$$ The latter definitions immediately imply that $\n_h$, $\a_h$ and $\z_h$ are independent of the $\o $'s, as a consequence of the rotation invariance of the theory. The previous observation has another consequence, which will play an important role in the following analysis. The structure of \equ(4.22) is, in fact, not suitable for the dimensional bounds that we want to discuss: the r.h.s. of \equ(4.22) is written as a sum of terms which do not vanish when $x_1=x_2$, i.e. we loose track of the fact that the l.h.s. of \equ(4.22) vanishes for $x_1=x_2$, a property which is manifest in the l.h.s. through the field $D^+_{12\o_1}$; this is disappointing because the property of vanishing of the l.h.s. must be used to regularize the vertex where the field $D^+_{12\o_1}$ appeared at a previous scale, along the iterative construction of $\bar V^{(h)}$. As a consequence we cannot have good bounds for the contributions to $\n_h, \a_h, \z_h$, coming from the individual terms in the r.h.s. of \equ(4.22). However, if $\o_1=\o_2$, it is easy to see that the contributions arising from the second and the third of \equ(4.22) cancel out, because of the translation invariance of the theory, by an argument similar to that used in the remark following \equ(4.18) and leading to the ``effective validity'' of \equ(4.19) (see also [BG], Sect. 11). In the first of \equ(4.22) the translation invariance implies that, if $\o_1=\o_2=\o$, the r.h.s. can be replaced by $(x_1-x_2)\ps{+}{x_1\o}{\cal D}_\o\ps{-}{x_1\o}$, and in this way the needed $(x_1-x_2)$ factor is explicitly exhibited. To summarize, if $\o_1=\o_2=\o$, we can replace \equ(4.22) with: $$\LL (D^+_{12\o}\ps-{x_3\o}) = (x_1-x_2)\ps{+}{x_1\o}{\cal D}_\o\ps{-}{x_1\o} \;,\quad \LL (D^+_{12\o}{\cal D}\ps-{x_3\o}) = \LL (D^+_{12\o}D^-_{34\o}) = 0 \Eq(4.31)$$ The previous properties are not valid anymore, if $\o_1\not=\o_2$; hence there would be a serious problem, if we had to bound the contributions to the effective potential associated with the local terms in the r.h.s. of \equ(4.22) for all $\o_1,\o_2$. But this is not the case, since we know {\it a priori} that $\n_h, \a_h, \z_h$ are {\it independent} of $\o_1,\o_2$ and we are not interested in the single contributions building the running coupling constants expansions, but only in their sums. Hence we can choose to compute the running coupling constants via their expansions valid for $\o_1=\o_2$, which does not give any trouble, as we shall see. Before starting the inductive evaluation of \equ(4.11), we write: $$\bar V^{(0)} (\psi^{\le (0)})= \LL \bar V^{(0)}(\psi^{\le (0)})+ \RR \bar V^{(0)}(\psi^{\le (0)})\Eq(4.32)$$ % It is easy to see that: % $$\eqalign{ \LL \bar V^{(0)}(\psi^{\le (0)}) &= \l_{0}\int dx \, \psi^{+(\le 0)}_{+1x}\psi^{+(\le 0)}_{-1x} \psi^{-(\le 0)}_{-1x}\psi^{-(\le 0)}_{+1x} + \cr &+ \n_{0}\int dx\sum_{\o_{1}\o_{2}} e^{ip_{F}(\o_{1}-\o_{2})\xx}\psi^{+(\le 0)}_{\o_{1}x} \psi^{-(\le 0)}_{\o_{2}x}+ \cr &+ \a_{0}\int dx\sum_{\o_{1}\o_{2}}e^{ip_{F}(\o_{1}-\o_{2})\xx} \psi^{+(\le 0)}_{\o_{1}x} i\b {\cal D}^-_{\o_{2}}\psi^{-(\le 0)}_{\o_{2}x}+ \cr &+ \z_{0}\int dx\sum_{\o_{1}\o_{2}}e^{ip_{F}(\o_{1}-\o_{2})\xx} \psi^{+(\le 0)}_{\o_{1}x}\partial_{t}\psi^{-(\le 0)}_{\o_{2}x} }\Eq(4.33)$$ for suitably chosen $\l_0,\n_0,\a_0,\z_0$, and: % $$\RR \bar V^{(0)}(\psi^{\le (0)}) = \sum_{n=1}^\i \sum_{\r\in I_n} \int d\undx \bar W_\r (z,\undx) M_\r(\psi^{\le (0)}) \Eq(4.34)$$ % Here $I_n$ is the finite set of different monomials of the form: % $$ M_\r(\psi) = (\prod_{i=1}^n \F^+_i) (\prod_{j=1}^n \F^-_j) \Eq(4.35)$$ where $\F^+_i$ has to be chosen between the fields (see \equ(4.21)): % $$e^{ip_F\o\xx} \ps{+}{\o x} \quad,\quad e^{ip_F\o\xx_2} \int_0^1 dr \dpr\ps{+}{\o x_{21}(r)} \Eq(4.36)$$ % and $\F^-_j$ has to be chosen between the fields: % $$\eqalign{ &e^{-ip_F\o\xx} \ps{-}{\o x}\;,\; e^{-ip_F\o\xx} \DD_\o \ps{-}{\o x}\;,\; ({-i\o\over 2p_F}) e^{-ip_F\o\xx_2} \dpr_\xx^2 \ps{-}{\o x_1} \;,\; e^{-ip_F\o\xx_2} \int_0^1 dr \dpr\ps{-}{\o x_{21}(r)} \cr & e^{-ip_F\o\xx_2} \int_0^1 dr \DD_\o\dpr \ps{-}{\o x_{21}(r)}\;,\; e^{-ip_F\o\xx_2} \int_0^1 dr \int_0^r ds \dpr^2\ps{-}{\o x_{21}(s)}\cr} \Eq(4.37)$$ % Moreover, in \equ(4.34) $\undx$ represents the set of points appearing as labels of the fields in the monomial $M_\r$. {\bf Remark}: the running couplings $\l_0$, $\n_0$, $\a_0$ and $\z_0$ are in fact convergent series of the {\it bare couplings} $z=(\l,\n,\a)$, uniformly in the u.v. cutoff $N$. This follows from \equ(3.18) for the contributions coming from $\bar W_{n_1n_2}$, with $n_1+n_2=2$ or $4$, but there is, at first sight, a problem for the contributions to $\n_0$, $\a_0$ and $\z_0$, coming from $\tilde W_2$. However we can use here \equ(3.20), which implies, for example, that the contribution of $\tilde W_2$ to $\a_0$ is: % $$-2m\int dx e^{ip_F\xx}\tilde W_2(x) = 2m\int dx [1- e^{ip_F\xx}] \tilde W_2(x)\Eq(4.38)$$ % which can be bounded by: % $$2mp_F\int dx |\xx| |\tilde W_2(x)|\Eq(4.39)$$ % a finite bound uniformly in $N$ by \equ(3.19). It is also important to stress that, by \equ(3.18) and \equ(3.19), the kernels of \equ(4.34) are convergent series of $z$, which satisfy for $|z|$ small enough the bound: % $$\int d\undx |\bar W_\r(z,\undx)| e^{{\k\over 2}d^{(0)}(\undx)} \le |\L|(C|z|)^{\max \{1,n-1\}} \Eq(4.40)$$ % and the power on the r.h.s. can be really $1$ only in the case of the term coming from the action of $\RR$ on the first term of \equ(4.8). The first order in the bare constants gives: % $$\eqalign{ \l_{0} &= 2\l \int d\xx \bar v(\xx)[1-\cos(2p_{F}\xx)] \cr \n_{0} &= \n + 2\l \int d\xx \bar v(\xx) [e^{ip_F\xx}R(0,\xx)-R(0)] \cr \a_{0} &= \a + 2\l {i\over\b} \int d\xx \bar v(\xx)R(0,\xx)\xx e^{ip_F\xx} \ ,\qquad \z_{0}=0 \cr} \Eq(4.41)$$ We can now start the inductive evaluation of \equ(4.11), by applying at each step the localization operator to the effective potential. We will obtain for $\LL V^{(h)}$ a formula like \equ(4.33), with $\l_{0},~ \n_{0},~ \a_{0},~ \z_{0}$ replaced by $(\l_{h}, \g^h \n_h, \a_{h}, \z_{h})$; and $(\l_{h}, \n_h, \a_{h}, \z_{h})\equiv r_h$ will be called the {\it running coupling constants of frequency $h$}. The $r_h$ can be expressed as a series of the running coupling constants of frequencies $k \ge h+1$, i.e. $r_{h+1}\dots r_{0}$. We could show that this series, called the {\it beta functional}, is convergent if all the running coupling constants $r_{h+1},\dots, r_0$ stay bounded within a certain radius of convergence, and we could show as well that the irrelevant part of the effective potential can be written as a convergent series of $r_{h+1},\dots, r_0$ (for a general discussion on the beta-functional see for example [G]). Of course, in order to use this result, we would also have to prove that the running constants {\it really do stay bounded}, at least if the bare constants are small enough. However, if we try to pursue this program, we immediately find a difficulty. In fact, if we calculate the beta functional at second order, we find ([BG], [G]): % $$\eqalign{ \l_{h-1} &=\l_{h} \cr \a_{h-1} &=\a_{h}+\b_2 \l^{2}_{h} + O(\g^h) \cr \z_{h-1} &=\z_{h}+\b_2 \l^{2}_{h} + O(\g^h) \cr} \Eq(4.42)$$ % with $\b_2\ne0$. The latter equations imply that, at the second order, $\l_{h}$ neither does increase nor does decrease; so we need the third order to decide what happens to $\l_{h}$. However, even if we suppose that the third order for $\l_{h}$, once calculated, will imply that $\l_{h}$ goes to zero when $h\to -\i$, the best that we can hope to find for its behaviour is clearly a rate $\sqrt{1/| h|}$. Looking at second order equations for $\a_{h}$ and $\z_{h}$, this implies that $\a_{h}$ and $\z_{h}$ go to infinity at least as $\sum_{h}1/| h|$, \ie we get out of the established domain of convergence of the beta functional in a finite number of steps. >From the mathematical point of view this is a big trouble, because it makes impossible to construct a perturbation theory for the model; from the physical point of view this means, as it is well known, that the expectation of the number of particles with fixed momentum, in the one dimensional Fermi gas, has a singularity, at the Fermi momenta $\pm p_{F}$, of a different kind with respect to the free case, where it is simply discontinuous. Hence we need to introduce a different type of scaling, allowing us to study the nature of the singularity on the Fermi surface via a consistent perturbation theory. \vskip2truecm ENDBODY \vglue1.truecm {\it\S5 The effective potential in the infrared region. Running couplings and anomalous scaling. The ground state energy.} \vglue1.truecm\numsec=5\numfor=1 A new and more general scaling approach is based on a representation of the field $\psi^{(\le0)}$ alternative to the one described by \equ(4.2)-\equ(4.4). In fact there are many ways to represent the grassmanian integration $P(d\psi^{(\le0)})$ with $\psi^{(\le0)}=\sum_{h=-\io}^0\psi^{(h)}$, each parameterized by an {\it arbitrary} sequence $Z_0=1,Z_{-1},Z_{-2},\ldots$ of non zero numbers. Denote $P_{Z_h}(d\psi)$ the grassmanian integration with propagator ${1\over Z_h}g^{(\le h)}$ and $\tilde P_{Z_h}(d\psi)$ the integration with propagator ${1\over Z_h}\tilde g^{(h)}$ where $\tilde g^{(0)}= g^{(0)}$ and $\tilde g^{(h)}$ will be fixed below. The $\tilde g^{(-1)}$ will be fixed, given the sequence $Z_h$, starting from the following obvious identities: % $$\eqalign{ P_{Z_0}(d\psi^{(\le0)})=&\tilde P_{Z_0}(d\psi^{(0)})\,P_{Z_0}(d\psi^{\le(-1)})=\cr =&\tilde P_{Z_0}(d\psi^{(0)})\,\Big[P_{Z_0}(d\psi^{\le(-1)}) e^{-(Z_{-1}-Z_0)(\psi^{(\le-1)+},T\,\psi^{(\le-1)-})-t'_{-1}|\L|}\Big] \cr &e^{+(Z_{-1}-Z_0)(\psi^{(\le-1)+},T\,\psi^{(\le-1)-})+t'_{-1}|\L|} \cr}\Eq(5.1)$$ % where $T$ is the differential operator $\dpr_t+e(i\dpr_\xx)$ and $t'_{-1}$ is a normalization constant such that the term in square brackets is a normalized grassmanian integration with propagator: % $$[Z_0 (g^{(\le-1)})^{-1}+(Z_{-1}-Z_0) T ]^{-1}\Eq(5.2)$$ % and, according to \S4: % $$g^{(\le h)}(k)={C_h(k)^{-1}\over -ik_0+e(\kk)},\qquad C_h(k)=e^{\g^{-2h}(k_0^2+e(\kk)^2)p_0^{-2}}=e^{\g^{-2h}\b(k)}\Eq(5.3)$$ % with $\b(k)$ being defined here. Therefore the normalization constant is: % $$ t'_{-1}=\ig{d^2 k\over (2\p)^2} \log(1+ {Z_{-1}-Z_0\over Z_0} e^{-\g^{2}(k_0^2+e(\kk)^2)p_0^{-2}})\Eq(5.4)$$ % and, finally, from \equ(5.2) we define $\tilde g^{(-1)}$ as: % $${[Z_0 C_{-1}(k)+zZ_0]^{-1} \over -ik_0+e(k_1)} = {[Z_{-1} C_{-2}(k)]^{-1} \over -ik_0+e(k_1)} + {1\over Z_{-1}} \tilde g^{(-1)}(k) \Eq(5.5)$$ % where, if $z=(Z_{-1}-Z_0)/Z_0$: % $$\eqalign{ \tilde g^{(-1)}(k) &= g^{(-1)}(k) + r^{(-1)}(k),\qquad g^{(-1)}(k)={e^{-\g^2\b(k)}-e^{-\g^4\b(k)}\over-i k_0+ e(k_1)}\cr r^{(-1)}(k) &= { e^{-\g^2\b(k)}(1-e^{-\g^2\b(k)})\over -ik_0+e(k_1)} {z\over 1+z e^{-\g^2\b(k)}}\cr}\Eq(5.6)$$ % Hence \equ(5.1) becomes: % $$\eqalign{ P(d\psi^{(\le0)})=&\tilde P_{Z_0}(d\psi^{(0)})\,P_{Z_0}(d\psi^{(\le-1)})=\cr =&\tilde P_{Z_0}(d\psi^{(0)})\,\tilde P_{Z_{-1}}(d\psi^{(-1)})\,P_{Z_{-1}}(d\psi^{(\le-2)})\cdot\cr &\cdot e^{(Z_{-1}-Z_0)(\psi^{(\le-1)},T\,\psi^{(\le-1)})+t'_{-1}|\L|} \cr}\Eq(5.7)$$ % By iteration we define $z_h=(Z_h-Z_{h+1})/Z_{h+1}$ and $\tilde g^{(h)}$ as: % $${[Z_{h+1} C_h(k)+z_hZ_{h+1}]^{-1} \over -ik_0+e(k_1)} = {[Z_h C_{h-1}(k)]^{-1} \over -ik_0+e(k_1)} + {1\over Z_h} \tilde g^{(h)}(k) \Eq(5.8)$$ % so that we must take: % $$\eqalign{ \tilde g^{(h)}(k) &= g^{(h)}(k)+r^{(h)}(k) \cr r^{(h)}(k) &= { e^{-\g^{-2h}\b(k)}(1 - e^{-\g^{-2h}\b(k)})\over -ik_0+e(k_1)} {z_h\over 1+z_h e^{-\g^{-2h}\b(k)}}\cr}\Eq(5.9)$$ % arriving at the representation, valid for all $k\le-1$: % $$\eqalign{ P_{Z_0}(d\psi^{(\le0)})=&\Big(\prod_{h=-\io}^0\tilde P_{Z_h}(d\psi^{(h)})\Big)\cdot\cr&\cdot \Big(\prod_{h=-\io}^{-1} e^{(Z_h-Z_{h+1})(\psi^{(\le h)},T\,\psi^{(\le h)})+t'_h|\L|}\Big)=\cr =&\Big(\prod_{h=k+1}^0\tilde P_{Z_h}(d\psi^{(h)})\Big)\cdot\cr&\cdot \Big(\prod_{h=k+1}^{-1} e^{(Z_h-Z_{h+1})(\psi^{(\le h)},T\,\psi^{(\le h)})+t'_h|\L|}\Big)\,P_{Z_{k+1}}(d\psi^{(\le k)})\cr}\Eq(5.10)$$ % with $\psi^{(\le p)}=\sum_{h=-\io}^p\psi^{(h)}$. We recover the decomposition of \S4 by setting $Z_h\equiv 1$. The freedom in the choice of the sequence $Z_h$ can be used to cancel terms proportional to $(\psi^{(\le h)},T\,\psi^{(\le h)})$ arising in the calculation of the effective potential. We define the {\it anomalous effective potentials} $V^{(h)}$ via: % $$\eqalign{ &e^{-V^{(h)}(\sqrt{Z_h}\psi^{(\le h)})}= \ig\prod_{h'=h+1}^0\tilde P_{Z_{h'}}(d\psi^{(h')})\cdot\cr&\cdot e^{-V^{(0)}(\sqrt{Z_0}\psi^{(\le0)})+ \sum_{h'=h}^{-1} [(Z_{h'}-Z_{h'+1})(\psi^{(\le h')},\,T\,\psi^{(\le h')})+ t'_{h'}|\L|]} \cr}\Eq(5.11)$$ % where $V^{(0)}(\psi^{(\le0)})\equiv \bar V^{(0)}(\psi^{(\le0)})$; so that: % $$\eqalign{ &\ig P(d\psi^{(\le0)})\,e^{V^{(0)}(\psi^{(\le0))}}=\cr &=\ig P_{Z_{h+1}}(d\psi^{(\le h)}) e^{-(Z_h-Z_{h+1})(\psi^{(\le h)},\,T\,\psi^{(\le h)})- t'_{h}|\L|}e^{- V^{(h)}(\sqrt{Z_h}\psi^{(\le h)})}=\cr &=\ig\tilde P_{Z_h}(d\psi^{(h)})\,P_{Z_h}(d\psi^{(2} V^{(-1)[2n]} (\sqrt{Z_{0}\over Z_{-1}}\psi)\,+ (t_{-1}+t'_{-1})|\L|\cr} \Eq(5.19)$$ % where the constant $l$ in front of the quartic relevant term is of course the {\it old} $\l_{-1}$ (because $Z_0=1$). Hence we have only four relevant terms, including the vacuum terms $(t_{-1}+t'_{-1})$ in $V^{(-1)}(\psi)$, and therefore only four running coupling constants which, by \equ(5.19), are given by the equations: % $$\g^{-2}\th_{-1}=(t_{-1}+t'_{-1}),\qquad\g^{-1}\n_{-1}={Z_{0}\over Z_{-1}}n,\qquad \d_{-1}={Z_{0}\over Z_{-1}}(a-z),\qquad \l_{-1}={Z_{0}^{2}\over Z_{-1}^{2}}l \Eq(5.20)$$ % with $z,n,l,a$ and $t_{-1},t'_{-1}$ convergent series of the bare constants.} \vskip.3truecm We repeat step by step for all single scale integrations the procedure followed in going from \equ(5.13) to \equ(5.18). We define: % $$e^{-\tilde V^{(h)}(\sqrt{Z_{h+1}}\psi^{(\le h)})}=\ig P_{Z_{h+1}}(d\psi^{(h+1)})e^{-V^{(h+1)}[\sqrt{Z_{h+1}}(\psi^{(h+1)}+ \psi^{(\le h)})]}\Eq(5.21)$$ % where $\psi^{(h)}$ is the field of propagator $\tilde g^{(h)}/Z_h$ defined above. And we write also the analogous of equations \equ(5.14) and \equ(5.15) (we shall call $n_h$, $a_h$, $z_h$, $l_h$ the coefficients of the local terms) and we define the ({\it anomalous}) effective potential of frequency $h$, with running coupling constants $\l_h$, $\n_h$, $\d_h$, as in \equ(5.16), that is: % $$\eqalign{ &\int P_{Z_{h+1}}( d\psi^{(\le h)} ) e^{ -\tilde V^{(h)} ( \sqrt{Z_{h+1}} \psi^{(\le h)} ) } =\cr &=\ig \tilde P_{Z_h}(d\psi^{(h)})\,P_{Z_{h}}(d\psi^{(From \equ(5.41) we get a recurrence relation for $V^{(k)}(\t,P_{v_{0}},\undx_{v_0})$: % $$\eqalignno{ &V^{(k)}(\t ,P_{v_{0}},\undx_{v_0})= ({Z_{k+1}\over Z_{k}})^{{1\over 2}| P_{v_{0}}| } (\x_{v_0}-\x'_{v_0})^{\bar z_{v_0}} \sum_{P_{v_{0}^{1}},\ldots ,P_{v_{0}^{s_{v_{0}}}}} \prod^{s_{v_{0}}}_{i=1}V^{({k+1})}(\t^{i},P_{v_{0}^{i}},\undx_{v_0^i}) \cdot \cr &\quad \cdot {1\over s_{v_{0}}!}\ET_{{k+1}} [{\tilde \psi}^{({k+1})}(P_{v_{0}^{1}}\backslash Q_{v_{0}^{1}}), \dots {\tilde \psi}^{({k+1})} (P_{v_{0}^{s_{v_{0}}}}\backslash Q_{v_{0}^{s_{v_{0}}}})]&\eq(5.42)\cr}$$ By iterating \equ(5.42) and using \equ(5.35), we can write the following closed expression: % $$\eqalign{ & V^{(k)}(\t ,P_{v_{0}}, \undx_{v_{0}}) = \sum_{\{ P_{v}\} } \prod_\vnotep ({Z_{h_v}\over Z_{h_v-1}})^{{1\over 2}| P_{v}| } (\x_v-\x'_v)^{\bar z_v}\,\cdot \cr &\cdot\, {1\over s_{v}!}\ET_{h_{v}} [{\tilde \psi}^{(h_{v})}(P_{v^{1}}\backslash Q_{v^{1}}), \dots ,{\tilde \psi}^{(h_{v})}(P_{v^{s_{v}}}\backslash Q_{v^{s_{v}}})]\,\cdot\,\prod_{i\in S_\l}\l_{h_i} \prod_{i\in S_\n} \g^{h_i}\n_{h_i}\prod_{i\in S_\d}\d_{h_i} \cr }\Eq(5.43)$$ % where $S_\a$ denotes the set of endpoints of type $\a$ (recall that we are supposing there is no endpoint of type $M_\r$) and $h_i$ is the frequency of the non trivial vertex which precedes the endpoint $i$. The symbol $\sum_{\{P_{v}\}}$ denotes the sum over all the compatible choices of the subsets $P_{v}$ in all the non trivial vertices of the tree, except $v_0$; such subsets are constrained by the same inclusion relations of the ultraviolet case. Hence the following constraints must hold: % $$ Q_v\subset P_v,\qquad P_v=\bigcup Q_{v_i}\Eq(5.44)$$ % As in the u.v. case, we now define the kernels: % $$W^{(k)}(\t,P_{v_0},\undx^{(P_{v_0})}) = \int d( \undx\backslash\undx^{(P_{v_0})} ) V^{(k)}(\t,P_{v_0},\undx) \Eq(5.45)$$ % so that: % $$V^{(k)}(\t, Z_k^{1\over 2} \psi^{(\le k)})= \sum_{P_{v_0}} \int d\undx^{(P_{v_0})} W^{(k)}(\t,P_{v_0},\undx^{(P_{v_0})}) Z_k^{{1\over 2}|P_{v_0}|} \tilde{\psi}^{(\le k)} (P_{v_0})\Eq(5.46)$$ % Here $\undx^{(P_{v_0})}$ is the set of points on which the monomial $\tilde{\psi}^{(\le k)} (P_{v_0})$ depends (recall that there can be more than one point for each field). In particular $\undx^{(P_{v_0})}$ is a single point (or an empty set), if the tree contributes to a local (or vacuum) term, and in that case $W^{(k)}$ is a constant, (by translation invariance), whose value is used to calculate the running coupling constants of frequency $k$. {\it Let us now suppose that we know all the constants} $\l_h,\n_h,\d_h,Z_h,\th_h$, with $h>k$. In order to get from \equ(5.46) the values of the kernels, we must first calculate $Z_k$. It is easy to see, by using \equ(5.23) and \equ(5.29), that we can write: % $${Z_k\over Z_{k+1}} = 1+z_k = 1 + \sum_{n=2}^\i \sum_{\t\in\TT_n} \tilde W^{(k)}_{2,t} (\t) \Eq(5.47)$$ % where $\tilde W^{(k)}_{2,t} (\t)$ is obtained by applying the $\LL$ operator to the monomials with two external lines associated with the tree and then summing the coefficients of $\ps{+(\le k)}{\o x}\dpr_t \ps{-(\le k)}{\o x}$, divided by $Z_{k+1}$. We can now calculate the new coupling constants and we get, $\forall k\le -1$: % $$\eqalignno{ \l_k &= ({Z_{k+1}\over Z_k})^2 [ \l_{k+1} + \sum_{n=2}^\i \sum_{\t\in\TT_n,r_{v_0}=L_1} W^{(k)}_4 (\t) ] \cr \d_k &= ({Z_{k+1}\over Z_k}) [ \d_{k+1} + \sum_{n=2}^\i \sum_{\t\in\TT_n,r_{v_0}=L_3} W^{(k)}_{2,\d} (\t) ] & \eq(5.48) \cr \n_k &= ({Z_{k+1}\over Z_k}) [\g \n_{k+1} + \g^{-k}\sum_{n=2}^\i \sum_{\t\in\TT_n,r_{v_0}=L_2} W^{(k)}_{2,\n} (\t)] \cr \th_k &= [\g^2 \th_{k+1} + \g^{-2k}\sum_{n=2}^\i \sum_{\t\in\TT_n,r_{v_0}=L_0} W^{(k)}_{0}(\t)] \cr}$$ % where the constants $W^{(k)}_\a$ are defined in an obvious way through \equ(5.45), taking into account the remark about the independence of $\o$ of the r.c.c., allowing us to restrict to consider only the terms with two external lines having the same $\o$ label. Furthermore $\l_0$, $\n_0$ and $\d_0=\a_0$ are defined as in \S4 (see \equ(4.41) for their first order values in the bare constants). Let us define: % $$r_h=(\l_h,\d_h,\n_h),\qquad \e_k = \max_{h\ge k} |r_h| \Eq(5.49)$$ % where the $r_h$ can take also complex values; then we can formulate the main result of this section: \vskip.3truecm {\bf Theorem 2}: {\it There exists a constant $\bar\e>0$, such that, if: % $$\e_{k+1} \le \bar\e \Eq(5.50)$$ % and, for some $c_2>0$: % $$\sup_{k0$: % $$\int d \undx^{(P_{v_0})} \sum_{\t\in\TT_n}| W^{(k)}(\t,P_{v_0},\undx^{(P_{v_0})})| e^{\bar \k \g^k d(P_{v_0})}\le \g^{-kD(P_{v_0}) }(C\e_k)^n|\L| \Eq(5.52)$$ % where $d(P_{v_0})$ is the length of the shortest tree graph connecting the set of points $\undx^{(P_{v_0})}$ and $D(P_{v_0})$ is the ``scaling dimension'' of the monomial $\tilde{\psi}^{(\le k)} (P_{v_0})$, defined by: % $$D(P_{v_0}) = -2+ \sum_{f\in P_{v_0}} ({1\over 2} + m_f) \Eq(5.53)$$ % with $m_f$ being the order of the derivative operator applied to the field of label $f$. Our proof will also imply that, see \equ(5.47): % $$ \sum_{\t} |\tilde W_{2,t}^{(k)}(\t)| \le (C\e_k)^n \Eq(5.54)$$} \vskip.3truecm {\bf Remarks}: {\bf 1)} it is easy to see that \equ(5.52) and \equ(5.54) imply that the series in the r.h.s. of \equ(5.47) and \equ(5.48) are convergent, uniformly in $k$, if \equ(5.50) holds. Hence the condition \equ(5.51) is satisfied for any $k$, for a suitable $c_2$, if $\bar\e$ is small enough and \equ(5.50) stays valid. {\it However it is not obvious at all that it is possible to choose $\bar \e$ so that the condition \equ(5.50) is satisfied for all $k$}. In order to get this result, one has to choose in a suitable way the constant $\n$ of \equ(2.29) and one has to compare the beta functional with that of the exactly soluble Luttinger model, as suggested in [BG], [BGM], see \S1. The problem will be discussed in detail in \S7 below (and solved). {\bf 2)} It is important to keep in mind that \equ(5.45) and the bound \equ(5.58) below allow us to get a version of the bound \equ(5.52) without the integration, \ie in the form: % $$\sum_{\t\in\TT_n}|W^{(k)}(\t,P_{v_0},\V x^{(P_{v_0})})|\le \sum_{h=k}^0 (C\e_h)^n \g^{-hD(P_{v_0})+2h(|x^{(P_{v_0})}|-1)} e^{-\bar\k\g^h d(P_{v_0})} \Eq(5.55)$$ % which is valid if the points belonging to $x^{(P_{v_0})}$ are pairwise at distance greater than $p_0^{-1}$, say (in order to avoid the trivial ultraviolet divergences due to the irrelevant terms present on scale $1$, see \equ(4.34)). Hence the {\it dimensionless potential}, \ie the kernels obtained from those of $V^{(k)}$ by multiplying them by $\g^{kD(P_{v_0})-2k(|x^{(P_{v_0}}|-1)}$ and by replacing their $\xx$'s arguments by $\g^{-k}\V x$ (we do not need to apply also a wave function renormalization, because the $Z_h$ factors were already extracted from the definition of the kernels, see \equ(5.46)), verify: % $$\eqalign{ W^{(k)}_{dimless}(\V x^{(P_{v_0})})\equiv & \g^{kD(P_{v_0})-2k(|x^{(P_{v_0})}|-1)} \sum_\t W^{(k)}(\t,P_{v_0},\g^{-k}x^{(P_{v_0})})\cr |W^{(k)}_{dimless}(\V x^{(P_{v_0})})|<& e^{-\bar\k d(\V x^{(P_{v_0})})} (C\e_k)^n\cr}\Eq(5.56)$$ % and the bound can be improved by replacing $\e_k$ with $\e_k'=\sum_{h\ge k}\g^{-\th(h-k)}|r_h|$ for some $\th>0$ (so that if $r_k\tende{k\to-\io}0$ the dimensionless potential tends to zero: a situation not arising in our problem but which can arise in asymptotically free theories). {\bf 3) } The discussion of \S7 will imply that the dimensionless potentials have a well defined limit as $k\to-\io$, which can be interpreted as an exact fixed point of the renormalization group transformations that we consider, if regarded as a transformation of the dimensionless potentials. However the Schwinger functions are related to the non rescaled potential, see \equ(2.33). The latter also has a limit as $k\to-\io$, but this cannot be seen directly from the discussion in this section, because of the divergence of $Z_h$, which will be also proved in \S7. Hence we cannot use \equ(2.33) to study the Schwinger functions; in the next section we shall solve this problem by developing a more refined tree expansion, based on the application of the method of this section directly to the Schwinger functions. The structure of the effective potential {\it on all scales} found in this section will play an essential role, especially trough the bound \equ(5.68), in getting the ``right'' bounds on the asymptotic behaviour of the Schwinger functions. It is important to remark that, also if the wave function renormalization constants were finite, we could not hope to use directly \equ(2.33) to estimate the asymptotic behaviour of the Schwinger functions. We could only obtain a convergent expansion for their values at fixed distances. {\bf 4)} The following general statement, ultimately relying on \equ(5.52) and the latter improvement \equ(5.68), can be also be derived from the estimates of \S5 and \S6: the dimensionless effective potential $V^{(-\io)}_{dimless}$ governs the corrections to the free asymptotic behaviour at large distances of the Schwinger functions. By ``free'' we mean here that the Schwinger functions can be evaluated from the pair Schwinger functions via the Wick rule, to leading order in the arguments distance. While the dimensional effective potential $V_{eff}$ (formally equal to $\lim_{h\to-\io} V^{(h)}(\sqrt{Z_h}\cdot)$ ) describes the correlations on all scales. Hence the vanishing of $V^{(-\io)}_{dimless}$ has the physical meaning of {\it trivial}, \ie free, asymptotic behaviour. The $V^{(-\io)}_{dimless}$ is quite inedependent from the initial potential, it is {\it universal}; while $V_{eff}$ is, of course, explicitly dependent on the initial potential. \vskip.3truecm The proof of \equ(5.52) is based on the following estimates of the truncated and simple expectations, which are very similar to those used in the u.v. case, and which are proved in appendix 2: % $$\eqalignno{ &{1\over s!} |\ET_h [ {\tilde \psi}^{(h)} (P_{1}), \dots ,{\tilde \psi}^{(h)} (P_{s}) ]|\le\cr &\le C^{\sum_i |P_i|} \g^{ {h\over 2} \sum_i \sum_{j=0}^2 (2j+1) |P_i^j| } \cdot\, {1\over s!} \sum_T \int d{\underline r}^{(P)} e^{-\k \g^h d^{*}_T (P_{1},\dots ,P_{s})} & \eq(5.57)\cr}$$ % where: \item{1)} $P^{j}$ denotes the subset of $P$ related to the fields containing a derivative operator of order $j$. \item{2)} ${\underline r}^{(P)}$ is the set of interpolation parameters, appearing in the definition of some of the fields in $P$, see \equ(4.36) and \equ(4.37); \item{3)} $T$ is an {\it anchored tree graph} between the clusters of space vertices (depending on ${\underline r}^{(P)}$) from which the fields labeled by $P_{1},\dots ,P_{s}$ emerge; this means that $T$ is a set of lines connecting pairs of points in different clusters, and $T$ becomes a tree graph if one identifies all the points in the same cluster; $d^{*}_T (P_{1},\dots ,P_{s})$ is the sum of the lenghts of the lines in $T$. Hence, after some algebra, we can bound \equ(5.46) as % $$\eqalignno{& |V^{(k)}(\t,P_{v_{0}}, \undx_{v_{0}})| \le \sum_{\{ P_{v}\} } \prod_\vnotep \left|{Z_{h_v}\over Z_{h_v-1}}\right|^{{1\over 2}| P_{v}| } C^{\sum_i |P_{v^i}| -| P_{v}| } \,\cdot& \eq(5.58) \cr &\cdot\,J(\t,P_{v_{0}}, \undx_{v_{0}}) \,\cdot \g^{{h_v\over 2} \sum_{j=0}^2 (2j+1) \sum_{i}(|P_{v^i}^j| - |Q_{v^i}^j|)}\, ( \prod_{i\in S_\l}|\l_{h_i}|) ( \prod_{i\in S_\n}|\g^{h_i} \n_{h_i}|) (\prod_{i\in S_\d}|\d_{h_i}|) \cr }$$ % where $Q^j_{v}$ is defined analogously to $P^j_{v}$ and: % $$J(\t ,P_{v_{0}}, \undx_{v_{0}}) = \prod_\vnotep (\x_v-\x'_{v})^{\bar z_v} {1\over s_v!} \sum_{T_v} \int d{\underline r}^{(P_v)} e^{-\k \g^{h_v} d^{*}_{T_v} (P_{v^1}\backslash Q_{v^1},\dots ,P_{v^{s_v}} \backslash Q_{v^{s_v}})} \Eq(5.59) $$ In Appendix 3 we prove also that: % $$\eqalign{ & \int d \undx_{v_0} J(\t,P_{v_0},\undx_{v_0}) e^{\bar \k \g^k d(P_{v_0})} \le \cr & \qquad \le \; |\L| \prod_\vnotep C^{\sum_{i}(| P_{v^{i}}| -| Q_{v^{i}}|) } \g^{-2h_{v}(s_{v}-1) -h_v\bar z_v} \cr} \Eq(5.60)$$ if: $$ \bar \k < {1\over 2}\k(1-\g^{-1}) \Eq(5.61)$$ % Note that in the bound \equ(5.60) we took into account also the sum over the $\o $'s, giving at worst an extra factor $2^{4n}$. The bounds \equ(5.58) and \equ(5.60) imply that: % $$\eqalignno{ & {1\over |\L|} \int d \undx_{v_0} e^{\bar \k \g^k d(P_{v_0})} |V^{(k)}(\t,P_{v_{0}}, \undx_{v_{0}})| \le \g^{-k D(P_{v_0}) } \sum_{\{P_{v}\}} \prod_\vnotep \cr & \quad \left|{Z_{h_v}\over Z_{h_v-1}}\right|^{{1\over 2}| P_{v}| } \g^{ -D(P_v) } C^{\sum_{i}| P_{v^{i}}| -| P_{v}| } (\prod_{i\in S_\l}|\l_{h_i}|) ( \prod_{i\in S_\n}|\n_{h_i}|) (\prod_{i\in S_\d}|\d_{h_i}|) & \eq(5.62)\cr\cr }$$ Remarking that, if $v\not= v_0$, $D(P_v)>0$ (the $\RR$ operation was defined so to obtain this result, see [BG]); furthermore, we have: % $$D(P_v) \ge {1\over 6}| P_{v}| \Eq(5.63)$$ % Then, by using also \equ(5.51), we find, if ${1\over 2}c_2\bar\e^2<1/6-1/8$: % $$\eqalign{ & \g^{k D(P_{v_0}) }\ig d \undx_{v_0} e^{\bar \k \g^k d(P_{v_0})} |V^{(k)}(\t ,P_{v_{0}}, \undx_{v_{0}})| \le \cr &\qquad \le (C\e_k)^n \sum_{\{P_{v}\}} \prod_{v \ge v_{0}\atop v \hbox{\seven \ not e.p.}} \g^{ -{1\over 8} | P_{v}| }\cr } \Eq(5.64)$$ % and this shows that the leading terms in the estimate are given by the contributions from the trees without trivial vertices (note that such trees are "concentrated" near the infrared scales in the sense that all their frequencies are between $k+1$ and $k+n$: this happened also in the u.v. case of \S3 but for a somewhat different mechanism). We can now proceed as in the ultraviolet case and show that actually: % $$\sum_{\t\in \TT_n} \sum_{\{P_{v}\}} \prod_\vnotep \g^{-{1\over 8}| P_{v}|}\le C^{n}\Eq(5.65)$$ % ending the proof of the bound \equ(5.52) and of the above thorem. {\bf Remarks:} {\bf 1)} The bounds \equ(5.64) can be easily converted into bounds on the functional derivatives: % $${\d\over\d\psi^+_x}V^{(h)}(\sqrt{Z_h}\psi)=\sqrt{Z_h} V^{(h)'}_{x+}(\sqrt{Z_h}\psi)\Eq(5.66)$$ where, by \equ(5.32), \equ(5.33) and \equ(5.46): % $$\eqalignno{ &\qquad\qquad V^{(h)'}_{x+}(\sqrt{Z_h}\psi)=&\eq(5.67) \cr & = \sum_{n=1}^\i \sum_{\t\in\TT_n} \sum_{P_{v_0}} \sum_{f\in P_{v_0}^+} \int d\undx^{(P_{v_0})} W^{(h)}(\t,P_{v_0},\undx^{(P_{v_0})}) Z_h^{{1\over 2}(|P_{v_0}|-1)} \partial ^{m_f}\d(x_f-x) \tilde{\psi}^{(\le h)} (P_{v_0}\backslash f) \cr}$$ % $ P_{v_0}^+$ being the set of field labels associated with fields of type $\psi^+$ or $\dpr\psi^+$ in $\tilde{\psi}^{(\le h)} (P_{v_0})$. The functional derivative will be used in next section to study the Schwinger functions and we should get in trouble with the representation \equ(5.67), unless there are no terms with $m_f>0$ in the r.h.s.\ . Of course the definitions used so far do not imply such property; however we could easily change them so that the field of label $f$ selected in \equ(5.67) always appears (in $\tilde{\psi}^{(\le h)} (P_{v_0})$) exactly in the form $\psi^+_x$, without any derivative acting on it. This is achieved by considering the path $\CS$ on the tree joining the root to the top vertex $v_f$, whose graph element contains the selected field of label $f$, and {\it undoing} all the $\RR$ operations, acting in the vertices of $\CS$ and involving subgraphs with four external lines. Then we recombine the various terms by using a new localization operation consisting in choosing as localization point always $x$ (\ie we do not use the localization prescription of \S4 in which the two $\psi^+$ fields in a four external lines subgraph are treated symmetrically: this means eliminating the factor $1/2$ in \equ(4.12) and keeping only one of the two addends in the first line of \equ(4.12), and precisely the first if $x_1=x$ or the second if $x_2=x$). The new prescription does not affect the running coupling constants, by symmetry reasons, as the two terms in \equ(4.12) produce the same contributions to the running couplings. It will be useful in the following section a bound of the kernels of \equ(5.67) analogous to \equ(5.52). We consider the contribution to $\d\,V^{(h)}/\d\psi^+_{x}$ coming from a monomial containing $|P_{v_0}^{j+}|$ fields of type $\dpr^j\psi$, $j=0,1,2$, and look for the part of degree $g$ in the running coupling constants. We immediately get the bound: % $$\eqalign{ \sqrt{Z_h} \int d\undx^{(P_{v_0})} & \sum_{\t\in\TT_g}\vert W^{(h)}(\t,P_{v_0},\undx^{(P_{v_0})})\vert \d(x_f-x) e^{\bar\k\g^h d(P_{v_0})}\le \cr & \le (C\e)^g\sqrt{Z_h}\g^{-{1\over2}h}\g^{2h} \g^{ -{h\over2} \sum_j (2j+1) |P_{v_0}^{j+}| } \cr} \Eq(5.68)$$ Note the factor $\sqrt{Z_h} \g^{-h/2}$ to be associated with the field selected by the functional derivative. {\bf 2)} Note that the above arguments {do not hold} for the functional derivatives with respect to $\psi^-_x$. The reason is simply that fields $\dpr^m\psi^-_x$, $m>0$, arise also in the $\RR$ and $\LL^*$ operations on second order monomials. Of course, however, the role of $\psi^+$ and $\psi^-$ is symmetric. This means that the same bounds hold for the functional derivatives of $V^{(h)}$ with respect to $\psi^-_y$! A way to check explicitly the latter (obvious) statement would be to do once more the whole theory so far developed, by exchanging the role of $\psi^+$ and $\psi^-$. Hence one would start by writing the kinetic part with the laplacian operating on the $\psi^+$ field and so on, and in particular the localization operations would have to be defined by localizing over the points corresponding to the $\psi^-$ fields. \vskip2.truecm \vglue1.truecm {\it\S6 The two point Schwinger function.} \vglue1.truecm\numsec=6\numfor=1 As discussed in \S2, in order to study the Schwinger functions (our results are summarized in the theorem at the end of this section), one has to calculate $V_{eff}(\f)$, which is related to the effective potential $\bar V^{(0)}$ by the relation (recall that $Z_0=1$): $$e^{-V_{eff}(\f) } = {1\over \NN} \int P_{Z_{0}}(d\psi^{(\le 0)}) e^{-\bar V^{(0)}[ \sqrt{Z_{0}}(\psi^{(\le 0)}+ \f)] }\Eq(6.1)$$ By using \equ(4.7) and the formal change of variables $\psi+\f \to \psi$ (to be correctly interpreted as in \S2), one can easily check that: $$e^{-V_{eff}(\f) } = {1\over \NN} \int P_{Z_{0}}(d\psi^{(\le 0)}) e^{-\bar V^{(0)}( \sqrt{Z_{0}} \psi ) -(\f^+,C_0 g^{-1} \f^-) + (\psi^+,C_0 g^{-1} \f^-) + (\f^+,C_0 g^{-1} \psi^-) }\Eq(6.2)$$ where $\psi=\psi^{(\le 0)}$, $C_0$ is the convolution operator defined by \equ(5.3) and $g^{-1}$ is the differential operator $\partial_{t}+ e(i\dpr_\xx) $. By using \equ(2.31), we find: $$q(\f) = (\f^+,(1-C_0) g \f^-) + q^{(\le 0)}( C_0\f) \Eq(6.3)$$ with the functional $q^{(\le 0)}(\f)$ defined by: $$e^{ q^{(\le 0)}(\f) } = {1\over \NN} \int P_{Z_{0}}(d\psi^{(\le 0)}) e^{-\bar V^{(0)}( \sqrt{Z_{0}} \psi ) + (\psi^+, \f^-) + (\f^+, \psi^-) }\Eq(6.4)$$ Equation \equ(6.3) implies a simple relation between the two point Schwinger function $S(x-y)$ and $S^{(\le 0)}(x-y) = \d^2 q^{(\le 0)}(\f) / \d\f^+_x \d\f^-_y|_{\f=0}$, that is, in terms of Fourier transforms: $$\hat S(k) = {1- e^{[k_0^2 + e(k_1)^2]p_0^{-2}} \over -ik_0+ e(k_1) } +S^{(\le 0)}(k) \; e^{2[k_0^2 + e(k_1)^2]p_0^{-2}} \Eq(6.5)$$ % which means that, if we are interested in the infrared behaviour of the theory (\ie $k_0$, $e(\kk)$ small), it is sufficient to study $q^{(\le 0)}(\f)$, as we shall do in the following. We could study directly $q(\f)$, by the same technique discussed below, obtaining in this way information also on the ultraviolet behaviour of the two point Schwinger function; we would find results in agreement with the discussion of \S3. In order to study $S^{(\le 0)}(\f)$, we shall use a tree expansion similar to that used in \S5, by suitably taking into account the new terms, linear in the external field $\f$, which are added in \equ(6.4) to the effective potential $\bar V^{(0)}$. The expansion is generated inductively, as in \S5, by integrating step by step the fields of decreasing frequency index. We shall suppose again, for simplicity, that only the local terms are present in $\VB{0}$. The first step will be the integration of the field of frequency index $h=0$ in \equ(6.4); we obtain the identity: % $$S^{(\le 0)} = \gt{0} + \gt{0} * K_2^{(-1)} * \gt{0} + S^{(\le -1)} \Eq(6.6)$$ % where $*$ denotes convolution, $K_2^{(-1)}$ is $Z_0$ times the kernel of $\VT{-1}_2 (\psi)$ (\ie it is the kernel of $\VT{-1}_2 (\sqrt{Z_0}\psi)$ as a functional of $\psi$) and $S^{(\le -1)}(x-y) = \d^2 q^{(\le -1)}(\f) / \d\f^+_x \d\f^-_y |_{\f=0}$, with $q^{(\le -1)}(\f)$ defined by: % $$e^{ q^{(\le -1)}(\f) } = {1\over \NN'} \int P_{Z_{-1}}(d\psi) e^{-V^{(-1)}( \sqrt{Z_{-1}} \psi ) - \WB{-1}(\f,\psi) }\Eq(6.7)$$ % Here $P_{Z_{-1}}(d\psi)$ is an abbreviation for $P_{Z_{-1}}( d\psi^{(\le -2)} ) P_{Z_{-1}}( d \pb^{(-1)} )$, see \equ(5.18), and: $$\eqalign{ &\WB{-1}(\f,\psi) = (\psi^+, Q_0 \f^-) + (\f^+, Q_0 \psi^-) + \cr &\quad + [\f^+ * G_{-1} *\sqrt{Z_0} \tilde{V}^{(-1)'}_{x+} ] (\sqrt{Z_0}\psi) + [\sqrt{Z_0}\tilde{V}^{(-1)'}_{y-}* G_{-1}* \f^-] (\sqrt{Z_0}\psi) + \cr &\quad + [\f^+* G_{-1}*{Z_0}\tilde{V}^{(-1)''}_{\ge 2}* G_{-1}* \f^-] (\sqrt{Z_{0}}\psi) + \WB{-1}_R(\f,\psi)\cr }\Eq(6.8)$$ % where: % $$Q_0 = 1 \quad\quad G_{-1} = \gt{0} * Q_0 \Eq(6.9)$$ % and we used for $\tilde{V}^{(-1)'}_{x\pm}$ a definition analogous to \equ(5.66) and $\tilde{V}^{(-1)''}_{\ge 2}$ represents the terms of the second functional derivative of $\tilde{V}^{(-1)}$ with two or more external legs; moreover $\WB{-1}_R(\f,\psi)$ represents the terms which do not contribute to $S^{(\le -1)}(x-y)$ (because either they are of order $\f^3$ or they contain a factor $(\f^+)^2$ or $(\f^-)^2$). In order to use the bounds on the functional derivative, that we found at the end of \S5, we have to write \equ(6.8) in terms of $V^{(-1)}$ instead of $\tilde{V}^{(-1)}$. Therefore we localize $\tilde{V}^{(-1)}$ and then we extract the local terms proportional to $[\dpr_t+e(i\dpr_\xx)]$. The terms proportional to $[\dpr_t+e(i\dpr_\xx)]$ can be conveniently added to the terms \hfill\break $(\psi^+, Q_0 \f^-)$ and $(\f^+, Q_0 \psi^-)$, so obtaining the following representation of \equ(6.8): $$\eqalignno{ &\WB{-1}(\f,\psi) = (\psi^+, Q_{-1}\f^-) + (\f^+, Q_{-1}\psi^-) + \cr &\quad +[\f^+ * G_{-1} *\sqrt{Z_{-1}} {V}^{(-1)'}_{x+} ] (\sqrt{Z_{-1}}\psi) + [\sqrt{Z_{-1}}{V}^{(-1)'}_{y-}* G_{-1}* \f^-] (\sqrt{Z_{-1}}\psi) + \cr &\quad + [\f^+* G_{-1}*{Z_0}\tilde{V}^{(-1)''}_{\ge 2}* G_{-1}* \f^-] (\sqrt{Z_{0}}\psi) + \WB{-1}_R(\f,\psi) & \eq(6.10)\cr }$$ % where $$\eqalign{ Q_{-1}=&Q_0 -z_{-1} Z_0 [\dpr_t+e(i\dpr_\xx)] G_{-1} =Q_0-z_{-1}w_0*Q_0 \cr w_0 &= [\dpr_t+e(i\dpr_\xx)] \tilde g^{(0)}(k) \cr} \Eq(6.11)$$ Note that no localization operation is performed on $\tilde{V}^{(-1)''}$. The construction can be iterated and, at each step, we get new contributions to the two point Schwinger function, as in \equ(6.6). We build in this way an expansion for $S^{(\le 0)}(x-y)$ of the following type: $$S^{(\le 0)}(x-y) = \sum_{h=-\i}^0 \sum_{k=-\i}^{h-1} \sum_{n=0}^\i \sum_{\t\in\TT_n^{h,k}} S_{h,k,\t} (x-y) \Eq(6.12) $$ % where the family of labeled trees $\TT_n^{h,k}$ can be described as in \S5, with the following modifications (see Fig. 9). \insertplot{300pt}{150pt}{fig92}{fig92} 1) There are $n+2$ endpoints, $n\ge 0$, and two of them, denoted $v_x$ and $v_y$ in the figure, represent the following functions: % $$\eqalign{ &\ig dx\,\f^+_x \left[ Q_h*\psi_x^{(\le h)-}+ G_h*{\d\over{\d\psi_x^+}}\Big(V^{(h)}(\sqrt{Z_h}\psi) \Big)\right] \cr &\ig dy\, \left[ \psi_y^{(\le h)+} * Q_h + {\d\over{\d\psi_y^-}}\Big(V^{(h)}(\sqrt{Z_h}\psi) \Big) * G_h \right] \f^-_y\cr} \Eq(6.13)$$ % where the following recursive relations for the convolution operators $Q_h,G_h$, hold: % $$\eqalignno{ Q_{h-1} =& Q_h - z_{h-1} Z_h [\dpr_t+e(i\dpr_\xx)] \left[ G_h + \gt{h} * Q_h \right]= \cr =&Q_h- z_{h-1}\sum_{j=h}^0{Z_h\over Z_j} w_j*Q_j \quad,\quad Q_0=1 & \eq(6.14)\cr G_{h-1} =& G_h + \gt{h} * Q_h = \sum_{j=h}^0{1\over Z_j}\tilde g^{(j)}*Q_j \quad,\quad G_0=0\cr w_h =& [\dpr_t+e(i\dpr_\xx)]\,\tilde g^{(h)}(k)\cr}$$ 2) The two special endpoints of item 1) belong to the vertical line with frequency index $h+1$ and are attached at the {\it same} tree vertex $v_{xy}$ bearing a frequency label $h$. This implies that $h$ is the scale at which the lines $\f_x^+$ and $\f_y^-$ become connected by graph lines. 3) There are no external lines in the root of the tree. 4) There are no $\RR$ labels associated with the tree vertices $v$ belonging to the line $\CS$ joining the root to $v_{xy}$. \vskip.5truecm In the quasi particle representation (which is used for the bounds) the {\it renormalized propagator} $G_h(x)$ can be written as % $$G_h(x) = \sum_\o G_{h,\o}(x) e^{-ip_F\o\xx},\quad G_{h,\o}(x) = \sum_{j=h+1}^0 {1\over Z_j} \tilde g^{(j)}_{Q,\o}(x) \Eq(6.15)$$ % where $\tilde g^{(j)}_{Q,\o}(x)$ has a definition similar to that of $\tilde g^{(j)}_\o(x)$, see \equ(5.9) and \equ(4.4), that is: $$\tilde g^{(h)}_{Q,\o}(x) = e^{i p_F\o\xx} \int {dk\over (2\p )^2} e^{-ik\cdot x} \tilde g^{(h)}(k) Q_h(k) \c(\o\g^{-h}\kk)\Eq(6.16)$$ In Appendix 1 we show that $\tilde g^{(h)}_{Q,\o}(x)$ satisfies a bound like \equ(5.27): % $$|\tilde{g}^{(h)}_{Q,\o}(x)| \le C \g^h e^{-\k'\g^h|x|} \Eq(6.17)$$ % for any $\k'<\k$, provided the $z_h$ verify $|z_h| \le C\e^2$ for all $h$, with $\e$ small enough ({\it i.e.}, by the bounds of \S5, provided the running couplings $r_h$ verify $|r_h| < \e$ for $\e$ small enough). Hence, for the purpose of establishing bounds in $x$ space, we could replace $Q_h$ by $1$. It is possible to prove that a similar property is valid in $k$ space, but we shall not give the details. Eq. \equ(6.17) easily implies that, for any $\bar \k <\k$: % $$\int dx e^{\bar \k \g^h |x|} |G_{h,\o}(x)| \le C'{\g^{-h} \over Z_h} \Eq(6.18)$$ We want to show that: $$\sum_{k=-\i}^{h-1} \sum_{\t\in\TT_n^{h,k}} |S_{h,k,\t} (x-y)| \le (C \e)^n {\g^{h} \over Z_h} e^{-\bar\k \g^h|x-y|} \Eq(6.19) $$ We shall treat explicitly only the bound of the contributions to the values of the tree in Fig. 9, coming from the second terms in \equ(6.13) only; the other three possibilities can be (more easily) treated along the same lines. For such contributions we can write: $$S_{h,k,\t} (x-y) = \sum_{\o\o'} \int dx_0 dy_0 G_{h,\o}(x-x_0) \bar S_{h,k,\t} (x_0-y_0) G_{h,\o'}(y_o-y) \Eq(6.20)$$ By \equ(6.18), it is sufficient to show that: $$\sum_{k=-\i}^{h-1} \sum_{\t\in\TT_n^{h,k}} |\bar S_{h,k,\t} (x_0-y_0)| \le (C \e)^n Z_h \g^{3h} e^{-\bar\k \g^h|x_0-y_0|} \Eq(6.21) $$ Note that, if we are interested in the contribution of order $m$ in the running coupling constants, we have to pick out of \equ(6.13) the terms of order $n_\pm=0,1,\ldots$ and consider trees with $n=m-n_+-n_-$. The remarks at the end of \S5 and the bound \equ(5.68) play a key role. In order to prove \equ(6.21) we have to consider the contributions to the functional derivatives $\d \,V^{(h)}/\d\psi^+_{x_0}$ and $\d V^{(h)}/\d\psi^-_{y_0}$ coming from the monomials containing $|P_{v_{xy}}^{j\pm}|$ fields of $\dpr^j \psi$-type and of degree $n_\pm$ in the running coupling constants. In this way we expand $\bar S_{h,k,\t} (x_0-y_0)$ in a sum of term, that we can bound proceeding as in \S5. We obtain: % $$\eqalign{ &[ (C\e)^{n_+} \sqrt{Z_h} \g^{-{1\over2}h} \g^{2h} \g^{-{h\over2} \sum_j (2j+1) |P_{v_{x}}^{j+}|} ] \cdot [ (C\e)^{n_-} \sqrt{Z_h} \g^{-{1\over2}h} \g^{2h} \g^{-{h\over2} \sum_j (2j+1) |P_{v_{y}}^{j-}|} ] \cdot \cr & (C\e)^n \; \big[\g^{2h}\big] \; \prod_{v \, \hbox{\sette e.p.}} \g^{h_v\d_2(v)} \cdot \prod_\vnotep \Big({Z_{h_v}\over Z_{h_v-1}}\Big)^{{1\over 2}|P_v|} \g^{ h_v \big( {1\over2} \sum_{i,j} ( |P_{v^i}^j| - |Q_{v^i}^j| ) (2j+1) - 2(s_v-1) \big) } \cr} \Eq(6.22)$$ % where the first two factors arise from the bound \equ(5.68) and $\d_2(v)=1$ if the end point of the tree represents a chemical potential running coupling $\n_h\g^h$. The {\it extra} factor $[\g^{2h}]$ in square brackets is there because there is no integration on $y_0$ but we count $s_v-1$ space integrations for all vertices, while there are really only $s_{v_{xy}}-2$ for $v=v_{xy}$ (recall that $h_{v_{xy}}=h$ in our notations). >From \equ(6.22), after a power counting computation, we get: % $$\eqalign{ &(C\e)^{n+n_+ + n_-}\,Z_h\,\g^{3h}e^{-\bar\k\g^h|x_0-y_0|} \cdot\cr &\prod_{\vnotep \atop v\not\in\CS} \g^{-D(P_v)-\h|P_v|} \prod_{v\in\CS} \g^{- \sum_j |P_v^j| ({2j+1\over2}+\h) } \cr} \Eq(6.23)$$ % where we have written $\g^{-2\h}$, with % $$2\h = \liminf_{h\to -\i} {\log Z_h\over |h|} \Eq(6.24)$$ % instead of the correct value $Z_{h_v} Z_{h_{v}-1}^{-1}$ (asymptotic to it), to simplify the notations, and $D(P_v)$ is defined in \S5, \equ(5.53). The bound \equ(6.23) implies \equ(6.21) by the same arguments used in \S5. The conclusion is that: \vskip0.5truecm % \\{\bf Theorem 3: \it The pair Schwinger function can be written in the form: $$\sum_{h=-\infty}^0 {1\over Z_h} (g^{(h)} +\e \bar g^{(h)} ) \Eq(6.25)$$ % where $\e$ is {\it supposed} to be small enough and to be a {\it bound on the running couplings on all scales}; and % $$|\bar g^{(h)}(x-y)| \le B \g^h e^{-\bar \k|x-y|} \Eq(6.26)$$ % $B>0$ being a suitable constant, independent on $\e$. Furthermore, under the same conditions, $S(x-y)$ is analytic in the running couplings with a domain independent on $x-y$.} % \vskip0.5truecm An immediate corollary of the Theorem is that the pair Schwinger function decays, for $|x-y|\to\i$, as $|x-y|^{-1-2\h}$, with $\h$ defined by \equ(6.24), if the sequence in the r.h.s. is convergent, as the analysis of the following section implies. Furthermore, by explicit calculation it is easy to prove that $\h=c\l_{-\io}^2 + O(\e^3)$, with $c>0$. In \S7 we shall also prove that the running couplings are analytic functions of $\l_0$ near $0$; hence, since the analysis of \S3 implies that $\l_0$ is analytic in $\l$, we have, using \equ(4.41): % $$\h = c [2\l(\hat v(0)-\hat v(2p_F))]^2 + O(\e^3) \Eq(6.27)$$ \vskip2.truecm \vglue1.truecm {\it\S7 The vanishing of the beta function and completion of the theory of spinless Fermi systems.} \vglue1.truecm\numsec=7\numfor=1 It remains to prove that there is a small $\e$ such that $|\l_h|,|\d_h|,|\n_h|<\e$, $\forall\, h$, if the initial coupling constant $\l$ is small enough and the parameters $\a$ and $\n$ (see \equ(1.3)) are suitably chosen. This was conjectured in [BG] on the basis of heuristic arguments; some further heuristic arguments for the proof of such statement were presented also in [BGM]. Here we want to reduce the proof to some technical lemma, which we think are easy consequences of the analysis of \S5, of the results of [GS], and of some properties of the exact solution of the Luttinger model (see [ML] and [BGM]). A more detailed proof will probably be published elsewhere as it would make this paper too long, but we think that it is not really necessary as all the steps are clearly outlined below referring to the estimates of the previous sections, and no further information is needed. A direct proof of the boundedness of the running coupling constants might also be possible, by using the symmetry properties of the propagators (see [S],[DM] for a heuristic discussion), but we met serious obstacles in trying to do it, although we succeeded in proving the key property \equ(7.6) below (\ie the {\it vanishing of the beta function} in the scaling limit), to fourth order and to see several cancellations to all orders. Let us call $\m_h=(\l_h,\d_h)$; by eliminating the $Z_h$ constants from the r.h.s of equations \equ(5.48) through \equ(5.47) and using the theorem and the remark following them in \S5, it is possible to prove that we can rewrite the beta functional as: % $$\eqalign{ \m_{h-1}=&\ \m_h+B^{h,\m} (\m_{h},\n_h,\m_{h+1},\n_{h+1},\ldots,\m_0,\n_0)\cr \n_{h-1}=&\g\,\n_h+B^{h,\n} (\m_{h},\n_h,\m_{h+1},\n_{h+1},\ldots,\m_0,\n_0)\cr} \Eq(7.1)$$ % where the $B^h$ are analytic in $\m_{h'},\n_{h'}$, $h'\ge h$, if $|\m_{h'}|,|\n_{h'}|< \e$, for a suitable small $\e$. The property $\g>1$ can be used to show that the above relation is equivalent to: % $$\eqalign{ \m_{h-1}=&\ \m_h+\BB_\m^h(\m_h,\ldots,\m_0;\n_h)\cr \n_{h-1}=&\g\,\n_h+\BB_\n^h(\m_h,\ldots,\m_0;\n_h)\cr}\Eq(7.2)$$ % with $\BB^h$ analytic for $|\m_{h'}|<\e$, $h'\ge h$, and $|\n_h|<\e$. See Appendix 4 for the proof. By direct calculation one checks also that: % $$\BB^h_\n(\m_h,\ldots,\m_0;\n_h)=\n_h\l_h^2\BB^{'h}(\m_h,\ldots,\m_0;\n_h)+ \g^h\BB^{''h}(\m_h,\ldots,\m_0;\n_h)\Eq(7.3)$$ % with $|\BB^{'h}|\le C,\,|\BB^{''h}|\le C\e^2$ for a suitable $C$ and for $\e$ small enough, see [BGM]. The relations \equ(7.2),\equ(7.3), given any infinite sequence $\m_h$ with $|\m_h|<\e$, imply that there is a unique $\n_0$ such that $|\n_h|<\e$ and $\n_h\to0$ as $h\to-\io$, and $\n_0$ is analytic in the running constants $\m_h$ for $|\m_h|<\e$; moreover the convergence to $0$ will be at the rate $\n_h=O(\g^h)$(see [BG]). This is a version of the existence of an unstable manifold theorem. Furthermore, since the analysis of \S3 implies that $\n_0$ is an analytic function of $\l,\a,\n$, this value of $\n_0$ is obtained, given $\a$ and $\l$, by a unique choice of $\n$. In [BG] it was shown that $\d_{h-1} = \d_h + O(\l_h^2\d_h)$; hence, if $\l_h$ stays bounded away from zero, as $h\to -\io$, one can apply the previous arguments to show that also $\d_0$ can be chosen so that $\d_h\to 0$, as $h\to -\io$; this choice would fix also the value of $\a$. However, the following analysis shows that this choice is not necessary to control to flow of $\m_h$, while of course the choice of $\n_0$ is essential. \vskip.3truecm {\bf Remark:\ } the previous considerations imply that we can consider the running couplings as functions of $\m\equiv (\l,\a)$. If we also take into account the results of \S3, we can claim that there is a small $\e_0$, such that, {\it if for all $\m$ with $|\m|\le \e_0$} (so that $\m_0$ is well defined as an analytic function of $\m$ and $|\m_0|\le\e$) {\it it happens that $|\m_{h'}|\le \e$ for $h'\ge h$}, then $\m_{h'}$, $h'\ge h-1$, is holomorphic in $\m$, for $|\m|\le\e_0$. \vskip.3truecm We want to show that the running couplings stay really bounded (and analytic in $\m$) for all $h\le 0$, if $|\m| \le \bar\e \le \e_0$. In order to do that, we shall need the following function: % $$\lim_{h\to-\io}\BB^h_{\m,\LL}(\bar\m,\bar\m,\ldots,\bar\m;0)\= \BB_\LL(\bar\m)\Eq(7.4)$$ % where $\BB^h_{\m,\LL}$ is the beta function of the Luttinger model, defined in a way entirely analogous to the above $\BB^h_\m$. {\it Such a definition is rather delicate in the part concerning the ultraviolet cut off} (\ie in the part corresponding to the contents of \S3) but it has been discussed in detail in [GS]. The part concerning the infrared cut off requires an analysis identical to the one just carried out (this was pointed out in [BG], [BGM]); such analysis and the fact that $\n_h=O(\g^h)$ also imply that: % $$\BB^h_i(\m_h,\m_{h+1},\ldots,\m_0;\n_h) = \BB^h_{i,\LL}(\m_h,\m_{h+1},\ldots,\m_0;0) + \g^h R^h_i(\m_h,\m_{h+1},\ldots,\m_0;\n_h) \Eq(7.5)$$ % where $i=\m,\n$ and $R^h$ has the same structure and satisfies the same bounds as $\BB^h$. This essentially follows from the observation that the single scale propagator $g_\o^{(h)}(x)$ (see \equ(4.4)) differs from the analogous Luttinger model propagator (obtained by linearizing $e(\kk)$ around $\kk=\o p_F$) by terms of order $\g^h$, and exponentially decaying in $\g^h|x|$ (see [BG]). Furthermore the function $\BB_{\n,\LL}^h(\m_h,\ldots,\m_0;0)$ {\it vanishes because of the special symmetries of the Luttinger model}, see [BGM], \ie in such case the unstable manifold is the plane $\n=0$. The main point of our analysis will be the proof that, in the Luttinger model (with $\n=0$, see above), the running couplings stay bounded for all $h\le 0$, if $\m$ is small enough. From this property we shall deduce the strong property: % $$\BB_\LL(\bar\m)=0, \qquad {\rm for\ all\ small\ } \bar\m \Eq(7.6)$$ % The latter equality will, in turn, be used to prove that the running couplings are bounded also in our model. We start by remarking that the Luttinger model is exactly soluble, even if the particles are constrained in a finite space box of size $L$, with periodic boundary conditions, [ML]. Furthermore the analysis of the previous sections and the results of [GS] could be applied to the model in finite volume without any uniformity problem, and we would get bounds uniform in $L$. By some refinement of our techniques, we can also prove a ``continuous $L$--dependence'' of the running couplings in the following sense. Let $\m_h^{(L)}$ be the running couplings for the model in finite volume, while $\m_h$ still denotes the infinite volume running couplings and let $\e$ be the radius of convergence of the beta function, independent of $L$; and we define $L_h\equiv \g^{-h} p_0^{-1}$. \vskip.3truecm {\bf Lemma 1.\ } {\it If $\m_{h'}$ is defined and $|\m_{h'}|\le \etd \le \e/2$, for $h'\ge h$, then there exists $n_0>0$ such that also $\m_{h'}^{(L_{h-n})}$ is defined for $h'\ge h$ and for any positive integer $n\ge n_0$; furthermore: % $$|\m_{h'}^{(L_{h-n})} - \m_{h'}| \le b_0\etd^2 e^{-\k n} \quad,\quad h'\ge h \Eq(7.7)$$ % for some positive constants $b_0$ and $\k$.} \vskip.3truecm It is very easy to prove this statement at any order of perturbation theory (in the running couplings), by using the exponential decay of the single scale covariances (which makes very slightly dependent on $L$, for $L$ large, the integrals involved in the definition of the beta function) and the remark that $g^{(h,L)}_\o - g^{(h)}_\o$ is of order $1/L$. It is also easy to see that the completion of the proof rests on a ``good'' bound of the difference between the finite and infinite volume expectations of a generic monomial $\tilde\psi^{(h)}(P)$. In appendix 2 we show that this ``good'' bound can be indeed obtained in a simple way. Another key remark is that the finite volume acts as an infrared cut off, so that the running couplings $\m_h^{(L)}$ ``stop'' flowing after the scale corresponding to $L$ has been reached. This property can be formalized in the following lemma. \vskip.3truecm {\bf Lemma 2.\ } {\it There exists $\e_1\le \e$, such that, for any fixed $h$, if $\m_{h'}^{(L_h)}$ is defined and $|\m_{h'}^{(L_h)}|\le \etd\le \e_1$ for $h'\ge h$, then $\m_{h'}^{(L_h)}$ is defined also for all $h'0 \Eq(7.11)$$ for some constant $b_2$ and % $$\tilde\m = ({\l\over 2} \hat v(0), \a+ {\l\over 4\p} \hat v(0))\Eq(7.12)$$ } \vskip.3truecm Let us now suppose that, given $\etd\le \e/2$, there exists $h_0>-\i$, such that: % $$|\m_h| \le {\etd\over 2} < |\m_{h_0}| < \etd \quad,\quad h>h_0 \Eq(7.13)$$ Note that if $|\m_{h'}| \le\etd\le\e$, $h'>h$, then the bounds of \S5 imply that: % $$|\m_{h'}^{(L)} - \m_{h'+1}^{(L)}| \le b\etd^2, \qquad {\rm for\ all}\ h'\ge h \Eq(7.14)$$ % for some positive $b$, independent of $L$. We start with a small $\m$, say $|\m| \le \bar\e \le {1\over 4}\etd$ and remark that $\m_{h'}$ stays close to the finite volume running couplings $\m_{h'}^{(L_{h_0}-n)}$ for $h'\ge h_0$: $|\m_{h'}-\m_{h'}^{(L_{h_0}-n)}|\le b_0\tilde\e^2 e^{-\k n}\le {1\over 8}\etd$ (see lemma 1), having fixed once and for all $n$ to be such that the second inequality holds. But we know that $\m_{h_0}^{(L_{h_0-n})}$ is close to $\m_{h_0-n}^{(L_{h_0-n})}$ by $2b\tilde\e^2 n$ (by \equ(7.14)) (the factor $2$ takes into account the small growth of $\m_{h'}^{(L_{h_0-n})}$ for $h'> h_0$); the latter is close to $\m_{-\io}^{L_{h_0-n}}$ by $b_1\tilde\e^2$, (by lemma 2); and the latter is close to $\tilde\m$ by $b_2\bar\e^2$ by lemma 3. Hence $\m_{h_0}$ is close to $\tilde\m$ by $b_2\bar\e^2+b_1\tilde\e^2+2b\tilde\e^2 n+{1\over 8}\tilde\e$. It is now sufficient to choose $\tilde\e$ small enough to conclude that: % $$|\m_{h_0}|\le{1\over 2}\tilde\e\Eq(7.15)$$ % in contradiction with \equ(7.13). \vskip0.3truecm {\bf Remark:} The above formal proof has a simple meaning. If, starting with $|\m|<\bar\e \le \etd/4$, it is nevertheless $|\m_{h_0}|>{1\over2}\tilde\e=2\bar\e$, this means that the running couplings can start arbitrarily small and reach size $O(1)$ (actually $O(\tilde\e)$, as in this argument $\tilde\e$ has to be regarded fixed) in finitely many steps. However the value that they reach is (lemma 1) close to the value that they would reach in the theory with cut off at scale $L_{h_0}$ (lemma 2). But {\it by the exact solution}, we know that such value is still of $O(\bar\e)$, hence it cannot be of size $O(1)$ (\ie $>\tilde\e/2$), and this is a contradiction. \vskip0.3truecm The previous considerations can be summarized in the following theorem. \vskip.3truecm {\bf Theorem 4:\ } {\it In the infinite volume Luttinger model, for any $h\le 0$, the running coupling $\m_h$ is a well defined analytic function of $\m$, if $|\m|\le \bar\e$, for a suitable $\bar\e$, and: % $$|\m_h-\tilde\m| \le C|\m|^2 \Eq(7.16)$$ } \vskip.3truecm We are now ready to prove \equ(7.6). We can write: % $$\BB^h_{\m,\LL}(\m_h,\m_{h+1},\ldots,\m_0;0) = \BB^h_{\m,\LL}(\m_h,\m_h,\ldots,\m_h;0) + \sum_{k=h+1}^0 D^{h,k}(\m_h,\m_{h+1},\ldots,\m_0) \Eq(7.17)$$ % where % $$\eqalign{ &D^{h,k}(\m_h,\m_{h+1},\ldots,\m_0) = \cr &\qquad = \BB^h_{\m,\LL}(\m_h,\ldots,\m_h,\m_k,\m_{k+1},\ldots,\m_0;0) - \BB^h_{\m,\LL}(\m_h,\ldots,\m_h,\m_h,\m_{k+1},\ldots,\m_0;0) \cr}\Eq(7.18)$$ >From the analysis of \S5, it is not difficult to deduce that: % $$\BB^h_{\m,\LL}(\m_h,\m_h,\ldots,\m_h;0) = \BB_\LL(\m_h) + O(\g^h) \Eq(7.19)$$ % if, of course, $|\m_h|\le \tilde\e$ for all $h\le0$. The function $\BB_\LL(\bar\m)$ is holomorphic near $\bar\m=0$ ($|\bar\m|\le\e$). Let us suppose that \equ(7.6) is not true; hence there exists $r\ge 2$ such that: % $$\BB_\LL(\m_h) = b_r \m_h^r + O(\m_h^{r+1}) \quad,\quad b_r \not=0 \Eq(7.20)$$ % and in fact, by explicit calculation, one verifies that $r\ge 3$, see for instance [BG] or [BGM] for this (well known) fact. We want to show that this is in contradiction with theorem 4 above and the structure of the beta function. In fact, by theorem 4, if $|\m| \le \bar\e$: % $$\m_h = \tilde\m + \sum_{n=2}^r c_n^{(h)} \m^n + O(\m^{r+1}) \Eq(7.21)$$ % and for each fixed $n$ the sequence, labeled by $h\le0$, $\{c_n^{(h)}\}_{h\le 0}$ is a bounded sequence. Hence, if we insert the power expansions \equ(7.21) in the first equation of \equ(7.2) and in equation \equ(7.17) and use \equ(7.19), \equ(7.20), we can write: % $$\sum_{n=2}^r c_n^{(h-1)} \m^n = \sum_{n=2}^r c_n^{(h)} \m^n + b_r\m^r + \sum_{k=h+1}^0 \sum_{n=3}^r d_n^{h,k} \m^n + O(\g^h) \Eq(7.22)$$ % where $\sum_{n=3}^r d_n^{h,k} \m^n$ represents the Taylor expansion of $D^{h,k}$ up to order $r$. The coefficients $d^{h,k}_n$ can be bounded by recalling the analysis of \S5. We see that for all complex $\m$'s, $|\m|<\bar\e$, it is $D^{h,k}=(\m_h-\m_k)\lis D^{h,k}$ because $D^{h,k}$ is at least of first order in $\m_h-\m_k$; it is also of third order in $\m$, because $\m_h-\m_k$ is of order $\m^2$. So that for some constant $b_3$ it is $|D^{h,k}|\le\bar\e^3 b_3\g^{-{1\over2}(k-h)}$, where the exponential decay in $k-h$ is due to the tree estimates of \S5 and this can be used to get bounds on the coefficients $\bar d_n^{h,k}$ of the Taylor expansion of $\lis D^{h,k}$ in $\m$ via the Cauchy's theorem. It also follows that the coefficients $d^{h,k}_n$ depend only on $\d_m\equiv c^{(h)}_m-c^{k)}_m$ with $2\le m\le n-1$ and are $\sum_{m=2}^{n-1}\d_m\lis d^{h,k}_{n-m}$, so that: % $$|d_n^{h,k}| \le d_n \g^{-{k-h\over 2}} \sup_{2\le m \le n-1} |c_m^{(h)} - c_m^{(k)}| \Eq(7.23)$$ % with $d_n$ than can be taken $\bar\e^3 b_3\bar\e^{-n}n$. Hence, if we define $d_2=0$, by \equ(7.22) and \equ(7.23), if $2\le n\le r-1$ it is: % $$|c_n^{(h-1)} - c_n^{(h)}| \le d_n \sum_{k=h+1}^0 \g^{-{k-h\over 2}} \sup_{2\le m \le n-1} |c_m^{(h)} - c_m^{(k)}| + O(\g^h) \Eq(7.24)$$ % which easily implies that, if $n\le r-1$, $c_n \equiv \lim_{h\to-\io} c_n^{(h)}$ does exist and: % $$|c_n^{(h)} - c_n| \le \bar b \g^{\theta h} \Eq(7.25)$$ % for some constant $\bar b$, depending on $r$, and $0 < \theta < 1/2$. In fact, \equ(7.25) is trivial for $n=2$; for $n>2$ it can be proved by induction, noting that $|c_n^{(h)} - c_n^{(k)}|$ does not appear in the r.h.s. of \equ(7.24). Finally, we have: % $$c_r^{(h-1)} = c_r^{(h)} + b_r + \sum_{k=h+1}^0 d_r^{h,k} + O(\g^h) \Eq(7.26)$$ % which would imply that $\{c_r^{(h)}\}_{h\le 0}$ is a diverging sequence, in contradiction with the remark following \equ(7.21), if the hypothesis \equ(7.20) were verified; this easily follows by noting that, by \equ(7.23) and \equ(7.25), $\sum_{k=h+1}^0 d_r^{h,k}$ is small of order $\g^{\theta h}$. Hence \equ(7.6) is proved. \vskip0.3pt {\bf Remark:} The idea behind the above argument is simply the following. The recursion relation is {\it essentially local} or {\it with short memory}: \ie \equ(7.2) is essentially a memoryless dynamical system because \equ(7.23) shows that the memory, \ie the number of scales $h'$ above $h$ at which one must know $\m_{h'}$ in order to compute $\m_{h-1}$ is essentially finite (by the exponential decay factor in \equ(7.23)). On the other hand a dynamical system without memory of the form $\m_{h-1}=\m_h+B(\m_h)$ with $B$ analytic and vanishing at least to second order cannot have trajectories bounded by a constant $\tilde\e$ for all small enough initial data unless $B\equiv0$. \vskip0.3pt We can now come back to our model; from now on $\m_h$ will again denote the corresponding running couplings. We note that, by \equ(7.5), \equ(7.6), \equ(7.18) and \equ(7.19): % $$\BB^h_\m(\m_h,\m_{h+1},\ldots,\m_0;\n_h) = \sum_{k=h+1}^0 D^{h,k}(\m_h,\m_{h+1},\ldots,\m_0) + O(\g^h) \Eq(7.27)$$ % Furthermore, the analysis of \S5 implies that, if $|\m_k|\le\etd\le\e$, $k\ge h$, and $\etd$ is small enough: % $$|D^{h',k}(\m_{h'},\m_{h'+1},\ldots,\m_0)| \le b \etd \g^{-{k-h'\over 2}} |\m_k -\m_{h'}|, \quad h'\ge h \Eq(7.28)$$ % which implies that, for all $h'\ge h$: % $$|\m_{h'-1}-\m_{h'}| \le b\etd \sum_{k=h'+1}^0 \g^{-{k-h'\over 2}} |\m_k-\m_{h'}| + O(\g^{h'}) \Eq(7.29)$$ % By induction on $h'$, it is easy to prove that $|\m_{h'-1}-\m_{h'}| \le \tilde b \g^{\theta h'}$, for any positive $\theta$ smaller than $1/2$ and a suitably chosen $\tilde b$, independent of $h' \ge h$. Hence it follows that, if $|\m|\le\bar\e$, with $\bar\e$ small enough, the sequence $\m_h$, $h\le 0$, is well defined and: % $$\m_{-\io} = \lim_{h\to-\io} \m_h \Eq(7.30)$$ % does exist as an analytic function of $\m$, for $|\m|\le\bar\e$, if $\n$ is suitably chosen (as an analytic function of $\m$). Furthermore we can choose $\a$ (as a holomorphic function of $\l$ near $\l= 0$), so that $\d_{-\i}=0$, if we want to impose that the Fourier transform of the pair Schwinger function behaves as $[k_0^2+e(\kk)^2]^{2\h} [-ik_0+e(\kk)]^{-1}$ near the Fermi surface (see [BG], [BGM]). Hence our theory of the one dimensional spinless Fermi systems is complete, and it can be summarized in the theorem of \S1. \vskip2truecm \pagina \vglue1.truecm {\it Appendix 1: Bounds on the free propagators.} \vglue1.truecm\numsec=1\numfor=1 In this Appendix we want to prove the bounds \equ(4.5), \equ(5.27) and \equ(6.17) on the single scale quasi particle covariances. We consider first $g_\o^{(h)}(x)$; if $x=(t,\xx)$, we have: % $$g_\o^{(h)}(t,\xx) = g_{+1}^{(h)}(t,\o\xx) \Eqa(A1.1)$$ % hence it is sufficient to consider the case $\o=+1$. We write: % $$g_{+1}^{(h)}(x) = \g^h \int_1^{\g^2} d\a \ \bar g_h(\a,\x) \Eqa(A1.2)$$ % where $\x=\g^h x$ and: % $$\bar g_h(\a,\x) = \int {dk\over (2\p )^2} e^{-ik\cdot \x -\a b(k)} (ik_0+\b\bar e(\kk)) \c(\kk + \g^{-h}p_F) \Eqa(A1.3)$$ % where $b(k)=(k_0^2+\b^2 \bar e(\kk)^2)p_0^{-1}$, % $$\bar e(\kk) = \kk (1+\kk{\g^h\over 2p_F}) \Eqa(A1.4)$$ The $k_0$ integration can be explicitly performed and we get, if $\x=(\x_0,\xxi)$: % $$\bar g_h(\a,\x) = {e^{-{p_0^2\x_0^2\over 4\a}} \over 4\p^{3/2}\sqrt{\a}} \int d\kk e^{-i\kk \xxi -\a\b^2 p_0^{-2} \kk^2 (1+\e \kk)^2 } [{\x_0\over 2\a} + \b \kk (1+\e \kk )] \c(\kk + {1\over 2\e}) \Eqa(A1.5)$$ % where $\e=\g^h/(2p_F)$. The integrand in the r.h.s. of \equ(A1.5) is an analytic function of $\kk $ in all the complex plane; hence we can shift the integration path in the imaginary direction, by putting $\kk = p +i q$, with $q$ a fixed real number, having the same sign of $\xxi $. It is now very easy to show, by using the fact that $\a\ge 1$, that: % $$|\bar g_h(\a,\x)| \le c(q) e^{-|q| |\x|} \Eqa(A1.6)$$ where $c(q)$ is a suitable constant, independent of $h$. The estimate \equ(A1.6) and equation \equ(A1.2) immediately imply the bound \equ(4.5), for $m=0$. The bound on the derivatives of the covariance is obtained by a straightforward extension of the previous arguments. Let us now come to the bound \equ(5.27); by \equ(5.9), we have to prove that a bound like \equ(A1.6) is valid for the function: % $$\bar r_h(\a,\x) = \int {dk\over (2\p )^2} e^{-ik\cdot \x -\a b(k)} (ik_0+\b\bar e(\kk )) {z_h\over 1+z_h e^{-b(k)}} \c(\kk + \g^{-h}p_F) \Eqa(A1.7)$$ % for $1\le\a\le2$. There are two differences with respect to the previous case. The first one is that we can not explicitly perform the $k_0$ integration; we can solve this problem by shifting also the $k_0$ integration path. The second difference is that the integrand is not analytic in all the complex plane, as a function of $k_0$ and $\kk $, because of the factor ${z_h\over 1+z_h e^{-b(k)}}$, which has an infinite number of poles. However, if $z_h$ is sufficiently small, for example $|z_h|\le 1/2$, it is easy to see that we can find a strip around the real axis in both variables, so that the integrand is bounded and fast decreasing at infinity. Hence we can prove a bound like \equ(A1.6), for $|q|$ small enough, say $|q|=\k$. \vskip.3truecm Finally, we shall prove the bound \equ(6.17). The recursive relation defining $Q_h$, in the first line of \equ(6.14), can be easily solved; the solution can be graphically represented as a sum of chains of single scale propagators, separated by operators $z_jZ_{j+1} [\dpr_t + e(i\dpr_\xx]$. If we insert the solution in \equ(6.16), we get the following representation of $\tilde g^{(h)}_{Q,\o}$ (see Fig. 10): % $$\eqalign{ & {1\over Z_h} \tilde g^{(h)}_{Q,\o_0}(x-x_0) = {1\over Z_h} \tilde g^{(h)}_{\o_0}(x-x_0) + \sum_{p=1}^{|h|} \sum_{h_0=h0 \cr |q_{j}| &= q_{j}^{t}+ q_{j}^{\xx}\cr } \Eqa(A2.2)$$ % with $q_j^t$, $q_j^\xx$ non negative integers. We shall also denote: % $$|q| = |q_{1}| + \dots + |q_{2m}| \Eqa(A2.3)$$ % the total number of derivative operations present in the monomial $\pt^{(h)}(P)$. Note that, when $h\le 0$, the field variables depend also on the quasiparticle $\o$-indices, but we have omitted them for the moment, to simplify the notation. We will prove the following estimate: % $$|\E_{h}[\pt^{(h)}(P)] | \le C^{|P|}~\g^{{a(h)\over 2}|P|}~\g^{h|q|} \Eqa(A2.4)$$ % where $a(h)=h$, if $h\le 0$, and $a(h)={h\over 2}$, if $h> 0$. The bound \equ(A2.4) immediately implies \equ(3.43) and \equ(5.57) in the case of the simple expectation ($s=1$). By the definition of simple expectation we can write: % $$\E_{h}[ \pt^{(h)}(P)] \le \sum_\CC (-1)^{\p} \prod_{(i,j)\in\CC} \E_{h}[ \partial^{|q_{m+i}|}\psi_{x_{m+i}}^{-(h)} \partial^{|q_{j}|}\psi_{x_j}^{+(h)}] \Eqa(A2.5)$$ % where the sum is over all the {\it couplings}, that is over all the possible ways to join each $\psi^-$ variable with a $\psi^+$ variable, and $(-1)^\p$ is the parity of the permutations which bring next to each other the joined variables, with the $\psi^-$ variable on the left. It is an easy task to show that \equ(A2.5) may be rewritten as a determinant, up to a sign: % $$\E_{h}[ \pt^{(h)}(P)] = \pm \det g^{(h)} \Eqa(A2.6)$$ % where $g^{(h)}$ is the $m\times m$ matrix with elements (see \equ(3.22) and \equ(5.22)): % $$g^{(h)}_{ij}= \cases{ \partial^{|q_{m+i}|}\partial^{|q_{j}|} C_{h}(x_{m+i} - x_{j}) &if $h>0$\cr \partial^{|q_{m+i}|}\partial^{|q_{j}|} \d_{\o_i \o_j} \tilde g^{h}_{\o_i} (x_{m+i} - x_{j}) & if $h\le 0$\cr} \Eqa(A2.7)$$ In order to show \equ(A2.4) we need {\it a good bound} of the determinant in \equ(A2.6); we shall use the well known {\it Gramm-Hadamard inequality}. Let $M$ be a square matrix, with elements $M_{\a\b}$, and suppose that $M_{\a\b}$ can be written as: % $$M_{\a\b} = ( A_{\a} , B_{\b} ) \Eqa(A2.8)$$ % where $A_{\a}$ and $B_{\b}$ are vectors in a Hilbert space with scalar product $(\cdot,\cdot)$. Then the following inequality is satisfied: % $$|\det M|\le \prod_{\a}\Vert A_{\a}\Vert~ \Vert B_{\a}\Vert \Eqa(A2.9)$$ % where $\Vert~\cdot~\Vert$ is the norm induced by the scalar product. Hence \equ(A2.4) will be proved, if we show that, both in the ultraviolet and in the infrared case, the matrix $g^{(h)}$ can be written as in \equ(A2.8), with: % $$\Vert A_{i}\Vert\le C\g^{a(h)/2+h|q_{m+i}|}, \Vert B_{j}\Vert\le C\g^{a(h)/2+h|q_j|}\Eqa(A2.10)$$ Let us define: % $$g^{(h)}(x) = \cases{ C_h(x) & if $h>0$ \cr \tilde g^{(h)}_\o(x) & if $h\le 0$ \cr} \Eqa(A2.11)$$ % and note that the Fourier transform $g^{(h)}(x)$ satisfies, for any $n\ge 0$, the following bound: % $$\int |k|^n |{\hat g}^{(h)}(k)| d^2k \le C_n \g^{a(h)+n}\Eqa(A2.12)$$ % This immediately follows, for $h>0$, from the definition \equ(3.21), that is: % $$C_h(x) = f(\g^h x_0) h(\xx) e^{x_0p_F^2\over 2m} \left( {m\over 2\p x_0} \right)^{d/2} e^{-{m\xx^2\over 2x_0}} \Eqa(A2.13)$$ % and from the remark that the functions $f(x_0)$ and $h(\xx)$ were chosen as smooth functions. For $h\le 0$, \equ(A2.12) follows very easily from the expression for the Fourier transform of $\tilde g_\o^{(h)}$, given in Appendix 1. Let us now observe that we can write: % $$\eqalign{ &\partial^{|q_{m+i}|}\partial^{|q_{j}|} g^{(h)}(x_{m+i} - x_{j})= \int {dk \over (2\p)^2} e^{-ik(x_{m+i} - x_{j})} (-ik)^{(|q_{m+i}|+|q_j|)} {\hat g}^{(h)}(k) = \cr &\quad =\int dz \int {dk \over (2\p)^2} e^{-ik(x_{m+i} - z)} (-ik)^{|q_{m+i}|} {\hat A}_h(k) \int {dk \over (2\p)^2} e^{-ik'(x_{j} - z)} (-ik')^{|q_j|} {\hat B}_h(k') \cr} \Eqa(A2.14)$$ % where $(-ik)^{|q|} \equiv (-ik_0)^{q^t} (-i\kk)^{q^\xx}$ and: % $$\eqalign{ {\hat A}_h(k) &= (\vert {\hat g}^{(h)}(k)\vert ^2)^{3/4} {\hat g}^{(h)*}(k)^{-1}\cr {\hat B}_h(k) &= (\vert {\hat g}^{(h)}(k)\vert ^2)^{1/4} \cr} \Eqa(A2.15)$$ % Hence, if $h>0$, we define: % $$\eqalign{ A_i^{(h)}(z) &= \int {dk \over (2\p)^2} e^{ik(x_{m+i} - z)} (+ik)^{|q_{m+i}|} {\hat A}^*_h(k) \cr B_j^{(h)}(z) &= \int {dk \over (2\p)^2} e^{-ik(x_{j} - z)} (-ik)^{|q_j|} {\hat B}_h(k) \cr} \Eqa(A2.16)$$ % so that, by \equ(A2.12), $A^{(h)}_i$ and $B^{(h)}_j$ are ${ L}_2$ functions, satisfying the relations \equ(A2.8) and \equ(A2.10) with respect to the ${L}_2$ scalar product. If $h\le 0$, we have to take in account also the $\o$ dependence. This is easily done by considering, in the tensor product of ${ L}_2({\bf R}^2)$ and ${\bf C}^2$: % $$\eqalign{ A_i &= A_{i}^{(h)}(z)\otimes S_{\o_i}\cr B_{j} &= B_{j}^{(h)}(z)\otimes S_{\o_j}\cr} \Eqa(A2.17)$$ % where $S_{\o}\in {\bf C}^{2}$ id defined by: % $$S_{\o} = \cases{ \left( _{0}^{1}\right) & if $\o= +1$ \cr \left( _{1}^{0}\right) & if $\o= -1$ \cr} \Eqa(A2.18)$$ so that: % $$(S_{\o_{i}}, S_{\o_{j}})= \d_{\o_{i}\o_{j}},~~~~~~ \Vert S_{\o_{i}}\Vert = \Vert S_{\o_{j}}\Vert = 1 \Eqa(A2.19)$$ This concludes our discussion for the simple expectations. The bounds on the truncated expectations are obtained by using for them the well known expansion in terms of interpolating parameters (see, for example [B]), as in Ref. [Le]. It turns out that the sum of the connected graphs can be written in the following way: % $$\eqalign{ &\ET_h (\widetilde{\psi}(P_1), \dots ,\widetilde{\psi}(P_k)) =\cr &\quad =\int\prod_{j=1}^k\prod_{i=1}^{p_j}d\eta_{j,i} \prod_{i=1}^{q_j}d\bar{\eta}_{j,i}\ \sum_{\tilde T} \prod_{(j,j')\in {\tilde T}}(V_{j,j'}+V_{j',j})\int dP_{\tilde T}(s)\ e^{-V(s)} \cr}\Eqa(A2.20)$$ % where: 1) $\eta_{j,i}$ and $\bar{\eta}_{j,i}$ are Grassmanian variables, each associated with the $i$-th field of the $j$-th monomial (cluster) of fields appearing in \equ(A2.20). The fields on scale $h$ will be denoted from now on with $\psi^{\s_{j,i}}_{x_{j,i}}$ and $g^{(h)}(x_{j',i'}-x_{j,i})$ will denote the corresponding covariance; $p_j$ ($q_j$) are the number of $\psi^+\,(\psi^-)$ fields in the $j$-th cluster and $\sum_j p_j= \sum_j q_j=n$. We are assuming for sake of simplicity that no derivative fields are present. 2) $\sum_{\tilde T}$ is the sum over all the tree graphs between the clusters thought as points. 3) $V_{j,j'}=\sum_{i'=1}^{q_{j'}}\sum_{i=1}^{q_j} \bar\eta_{j',i'} g^{(h)}(x_{j',i'}-x_{j,i}) {\eta}_{j,i}$ and $V(s)=\sum_{j=1}^kV_{j,j}+\sum_{j\neq j'}S_{jj'}V_{j,j'}$ 4) $S_{jj'}$ is a product of interpolating parameters $s_n$, $n=1,\ldots ,k-1$, valued in $[0,1]$, and the clusters can be ordered in such a way that $S_{jj'}=\prod_{n=j}^{j'-1}s_n\quad(j'>j)$. 5) $ dP_{\tilde T}(s)$ is a normalized measure, $\int dP_{\tilde T}(s)=1$, which depends on the interpolating parameters $s_n$ and on $\tilde T$. It is easy to extract from \equ(A2.20) the exponential factor appearing in \equ(3.43) and \equ(5.57). Let us in fact develop, for a fixed tree graph ${\tilde T}$, the product $\prod_{(j,j')}(V_{j,j'}+V_{j',j})$; we get: $$\eqalign{ \prod_{(j,j')}(V_{j,j'}+V_{j',j}) &= \sum_{i_{1},\cdots ,i_{k-1}} \sum_{i_{1}',\cdots ,i_{k-1}'} {\bar \h}_{j_{1}',i_{1}'} \h_{j_{1},i_{1}} \cdots {\bar \h}_{j_{k-1}',i_{k-1}'} \h_{j_{k-1},i_{k-1}} \,\cdot \cr &\cdot\, g^{(h)}(x_{j_{1}',i_{1}'}-x_{j_{1},i_{1}}) \cdots g^{(h)}(x_{j_{k-1}',i_{k-1}'}-x_{j_{k-1},i_{k-1}}) \cr} \Eqa(A2.21)$$ Recalling the definitions of \S3 and \S5 of {\it anchored tree graph}, it is now obvious that once $\tilde T$ and the sets $i_{1},\cdots ,i_{k-1}$, $i_{1}',\cdots ,i_{k-1}'$ are fixed, an {\it anchored tree graph} $T$ is also uniquely chosen. We remind that $T$ is a set of $k-1$ difference vectors $x_{j_{1}',i_{1}'}-x_{j_{1},i_{1}}, \cdots ,x_{j_{k-1}',i_{k-1}'}-x_{j_{k-1},i_{k-1}}$ which realize the connection between the $k$ clusters of fields $\psi(P_{1}),\cdots ,\psi(P_{k})$, see remarks after \equ(3.43) and \equ(5.57). Thus we can rewrite: % $$\sum_{\tilde T}\sum_{i_{1},\cdots ,i_{k-1}} \sum_{i_{1}',\cdots ,i_{k-1}'} =\sum_{T} \Eqa(A2.22)$$ % Now, using the bounds \equ(3.22) and \equ(5.27), we can also write: % $$|g^{(h)}(x_{j_{1}',i_{1}'}-x_{j_{1},i_{1}}) \cdots g^{(h)}(x_{j_{k-1}',i_{k-1}'}-x_{j_{k-1},i_{k-1}})|\le C^{k-1}\g^{a(h)(k-1)} e^{-\k d^{h}_{T}(P_{1},\cdots ,P_{s})}\eqa(A2.23)$$ % where $d^{h}_{T}(P_{1},\cdots ,P_{s})$ is defined as in \equ(3.43) or as in \equ(5.57). Hence we can bound \equ(A2.20) as: % $$\eqalign{ |\ET_h (\widetilde{\psi}(P_1), \dots ,\widetilde{\psi}(P_k))| & \le C^{k-1}\g^{a(h)(k-1)} \sum_{T} e^{-\k d^{h}_{T}(P_{1},\cdots ,P_{s})} \,\cdot \cr & \cdot\, |\int\prod_{j=1}^k\prod_{i=1}^{p_j}d\eta_{j,i} \prod_{i=1}^{q_j}d\bar{\eta}_{j,i}\eta^{T}{\bar \eta}^{T} \int dP_T(s)\ e^{-V(s)}| \cr}\Eqa(A2.24)$$ % where $\eta^{T}{\bar \eta}^{T}= \h_{j_{1},i_{1}}{\bar \h}_{j_{1}',i_{1}'}\cdots \h_{j_{k-1},i_{k-1}}{\bar \h}_{j_{k-1}',i_{k-1}'}$. It is now a standard task to prove, using the properties of the Grassmanian variables, that, for a fixed {\it anchored tree graph} $T$, the integration over the the variables $\eta_{j,i}$, ${\bar \eta}_{j,i}$ gives: % $$\int\prod_{j=1}^k\prod_{i=1}^{p_j}d\eta_{j,i} \prod_{i=1}^{q_j}d\bar{\eta}_{j,i}\eta^{T}{\bar \eta}^{T} e^{-V(s)}=det~G^{T}(s)\Eqa(A2.25)$$ % where $G^{T}(s)$ is a $(n-k+1)\times (n-k+1)$ matrix whose elements are $G^{T}_{jij'i'} =S_{jj'}g^{(h)}(x_{j',i'}-x_{j,i})$ with $x_{j',i'}-x_{j,i}$ not belonging to the anchored tree graph $T$. Such determinant can be bounded again using Gramm-Hadamard inequality. In, $G^{T}_{jij'i'}$ can be rewritten as a scalar product of two vectors, as in \equ(A2.8), performing the tensor product between the $A_{ji}$ and $B_{i'j'}$ defined as in \equ(A2.16) (taking care of the indices) and the vector $e_j$ defined as follows [Le]. Let $v_i\in{\bf C}^k$ be the unit vector $(v_i)_j=\d_{ij}$; then the $e_j$ are defined inductively by: % $$e_1=v_1\quad\quad e_j=s_{j-1}e_{j-1}+(1-s_{j-1}^2)^{1/2}v_j \quad j=2,\ldots,k-1 \Eqa(A2.26)$$ % which implies that: % $$\Vert e_j\Vert =1\;,\qquad (e_i,e_j)= s_is_{i+1}\ldots s_{j-1}=S_{jj'} \Eqa(A2.27)$$ % where $(\cdot,\cdot)$ denotes the usual scalar product in ${\bf C}^k$. Hence, writing $G^{T}_{jij'i'}=(e_{j}\otimes A_{ji},e_{j'}\otimes B_{j'i'})$ and performing the same steps as before we obtain the following bound: $$|det G^{T}(s)|\le \g^{{a(h)\over 2}\sum_{j=1}^{k}|P_{j}|-a(h)(k-1)} ~C^{{1\over 2}\sum_{j=1}^{k}|P_{j}|-(k-1)}\Eqa(A2.28)$$ Inserting now \equ(A2.28) in \equ(A2.24) and taking into account item 5) above, we obtain the bounds \equ(3.43) and \equ(5.57) for the case that no derivative is acting on the fields. The generalization to the case in which also derivative fields are allowed is trivial and we left it to the reader. \vskip.3truecm Finally we want to show the result claimed in lemma 1 of section 7; i.e. we want to compare the finite and infinite volume expectations of a generic monomial ${\tilde \psi}^{(h)}(P)$. We can obviously define in the finite volume the vectors $A_{i}^{h,L}, B_{j}^{(h,L)}$ such that: % $$(A_{i}^{(h,L)},B_{j}^{(h,L)})=g_{ij}^{(h,L)} \Eqa(A2.29)$$ % This is done in a way totally analogous to \equ(A2.16), with the integral replaced by a sum. The result that now we want to prove is therefore the following. $$|det~(A_{i}^{(h)},B_{j}^{(h)})- det~(A_{i}^{(h,L)},B_{j}^{(h,L)})|\le {C^{|P|}\over L} \g^{{a(h)\over 2}|P|}\g^{h|q|}\Eqa(A2.30)$$ This is easily achieved using the obvious property: $$\parallel A_{i}^{(h)}-A_{i}^{(h,L)}\parallel\le {C\over L} \g^{{a(h)\over 2}+h|q_{m+i}|},~~~~ \parallel B_{j}^{(h)}-B_{j}^{(h,L)}\parallel\le {C\over L} \g^{{a(h)\over 2}+h|q_{j}|} \Eqa(A2.31)$$ and the well known relation: $$det~(M+M')=\sum_{m}det_{m}(M)~det_{m^c}(M')+det~M+det~M'\Eqa(A2.32)$$ where $\sum_{m}$ is the sum over the (non void) minors of the matrix $M+M'$, and $det_{m}(M)$ ($det_{m^c}(M')$) is the determinant of the minor $m$ (of the complementary $m^c$ of $m$) of the corresponding matrix. In fact, let us write: $$det~(A_{i}^{(h)},B_{j}^{(h)})= det~(A_{i}^{(h)}+A_{i}^{(h,L)}-A_{i}^{(h,L)},B_{j}^{(h,L)})=$$ $$=det~\left((A_{i}^{(h)}-A_{i}^{(h,L)},B_{j}^{(h,L)})+ (A_{i}^{(h,L)},B_{j}^{(h)})\right)\Eqa(A2.33)$$ Using now \equ(A2.32) we have: $$det~(A_{i}^{(h)},B_{j}^{(h)})= det~(A_{i}^{(h,L)},B_{j}^{(h)})+ \sum_{m}det_{m}(A_{i}^{(h)}-A_{i}^{(h,L)},B_{j}^{(h)})~ det_{m^c}(A_{i}^{(h,L)},B_{j}^{(h)})+$$ $$~~~~~~~~~~~~~~+det(A_{i}^{(h)}-A_{i}^{(h,L)},B_{j}^{(h)})~\Eqa(A2.34)$$ Using \equ(A2.31) and the Gramm-Hadamard inequality it is now obvious that the second and third addend in the r.h.s. of \equ(A2.34) can be bounded by ${C_{1}^{|P|}\over L}\g^{{a(h)\over 2}|P|}\g^{h|q|}$ (note that the total number of non void minors is $4^{|P|/2}-2$). Repeating the same argument for $B^{(h)}_{j}$ we obtain \equ(A2.30) \vskip2truecm \vglue1.truecm {\it Appendix 3: The bound \equ(5.60).} \vglue1.truecm\numsec=3\numfor=1 In this section we want to prove the bound \equ(5.60). We can write: % $$J(\t ,\{ h_v\},P_{v_{0}}, \undx_{v_{0}}) = \sum_T \left( \prod_\vnotep {1\over s_v!} \int d{\underline r}^{(P_v)} \right) J_T(\undx,\undr) \Eqa(A3.1)$$ % where % $$J_T(\undx,\undr)= \left( \prod_{l\in T} e^{-\k \g^{h_l}|\x_l-\h_l|} \right) \left( \prod_v (\x_v-\x'_{v})^{\bar z_v} \right) \Eqa(A3.2)$$ % Here $\undr$ is the set of all interpolation parameters and $T$ is a set of lines obtained by choosing one of the anchored trees $T_v$ in each non trivial vertex. Moreover, if $l\in T$, we denote $h_l$ the corresponding frequency index and $\x_l,\h_l$ its endpoints; $h_l$ is the frequency of the contraction between the two field variables, emerging from the space vertices $\x_l$ and $\h_l$, which gave rise to the factor $e^{-\k \g^{h_l}|\x_l-\h_l|}$ (see App. 2). Note that $\x_l$ and $\h_l$ \item{a)} either coincide with one of the integration variables $\undx$, and in this case we shall say that they are {\it simple space vertices}; \item{b)} or are convex combinations of the integration variables trough the interpolation parameters, and we shall say that they are {\it interpolated space vertices}. Note also that $T$ is not in general a tree, if some space vertex is an interpolated one. However, we can uniquely associate to $T$ a tree $\tilde T$ connecting the set $\undx$ of the integration variables, by substituting $\x_l$ and $\h_l$ with the space vertices $x_l$ and $y_l$ (which can coincide with them), from which the corresponding field variables emerge before the application of the $\RR$ operations (see \S5, item 5 before \equ(5.31)). There is of course a one to one correspondence between the lines of $T$ and $\tilde T$. Given a non trivial vertex $v\in \t$, we shall denote $\tilde S_v$ the subset of $\tilde T$, connecting the points in $\undx_v$ (recall that $\undx_v$ is the set of integration variables associated to the vertex $v$) and $S_v$ the corresponding subset of $T$; of course: $$S_v = \bigcup_{ \hbox{\seven n.t. } \bar v \ge v} T_{\bar v} \Eqa(A3.3)$$ Finally, we shall say that a line in $T$ is a {\it simple line} if it connects two simple space vertices, an {\it interpolated line} if one of its endpoints is an interpolated space vertex; note that, if the line $l\in T$ is a simple line, then it is also true that $l\in \tilde T$. The main point of this appendix is the proof that $$|J_T(\undx,\undr)| \le \left( \prod_{l\in \tilde T} e^{-\bar\k \g^{h_l} |x_l-y_l|} \right) (\prod_v C \g^{-h_v \bar z_v}) \Eqa(A3.4) $$ where $C$ is a suitable constant and $$\bar\k < {\k\over 2} (1-{1\over\g}) \Eqa(A3.5)$$ As in \S5, we shall suppose, for simplicity, that only local terms are associated to the endpoints of $\t$. We first bound the factors $(\x_v-\x'_{v})^{\bar z_v}$; recall that $\bar z_v$ is a positive integer $\le 2$ and that $(\x_v-\x'_{v})^{\bar z_v}$ denotes a tensor of rank two, if $\bar z_v=2$. We can write: $$\x_v=\sum_{i=1}^r \l_i x_i \quad,\quad \x'_v=\sum_{j=1}^s \m_j y_j \quad, \qquad x_i,y_j \in \undx_v \Eqa(A3.6)$$ where $\l_i$ and $\m_j$ are interpolation parameters, hence they are positive and $\sum_{i=1}^r \l_i = \sum_{j=1}^s \m_j = 1$. We have, for any $\e >0$: $$|\x_v-\x'_{v}| \le \sup_{i,j} |x_i-y_j| \le C_\e \g^{-h_v} e^{{\e\over 2} \g^{h_v} \sum_{l\in \tilde S_v} |x_l-y_l|} \Eqa(A3.7)$$ where $C_\e$ is a suitable constant and $\tilde S_v$ is defined before \equ(A3.3). Since $\bar z_v\le 2$, \equ(A3.7) implies that: $$|\x_v-\x'_{v}|^{\bar z_v} \le C_\e^2 \g^{-h_v \bar z_v} e^{ \e \g^{h_v} \sum_{l\in \tilde S_v} |x_l-y_l|} \Eqa(A3.8)$$ We observe now that, given any line $l\in \tilde T$, we can associate to it all the factors $e^{\e\g^{h_l}|x_l-y_l|}$ coming from the r.h.s of \equ(A3.8), for each non trivial vertex containing that line; the product of these factors can be bounded by $e^{2\e|x_l-y_l| \sum_{h\le h_l}\g^h}$ (the factor $2$ in the exponent comes from the observation that, in each line of $\t$ connecting two non trivial vertices, at most two trivial vertices can carry a factor $|\x_v-\x'_{v}|^{\bar z_v}$ with $\bar z_v>0$). Hence we have: % $$\prod_v |\x_v-\x'_v|^{\bar z_v} \le (\prod_v C_\e^2 \g^{-h_v \bar z_v} ) \left( \prod_{l\in \tilde T} e^{{2\e\over 1-1/\g} \g^{h_l}|x_l-y_l|}\right) \Eqa(A3.9)$$ In order to complete the proof of \equ(A3.4), we have to bound the first factor in the r.h.s. of \equ(A3.2). Let us define $\k'$ so that $$\k \g^h = 2\k' \sum_{p=0}^\i \g^{h-p} \Eqa(A3.10)$$ that is $$\k'={\k\over 2} (1-{1\over \g}) \Eqa(A3.11)$$ Hence we have, for any $l\in T$: $$e^{-\k\g^{h_l}|\x_l-\h_l|} = \prod_{h\le h_l} [ e^{-\k'\g^h |\x_l-\h_l| }]^2 \Eqa(A3.12)$$ If the line $l\in T$ is a simple line, we associate to it a factor $e^{-\k'\g^{h_l}|\x_l-\h_l|}$, taken from the r.h.s. of \equ(A3.12), whose remaining part will be used as explained below. Note that all the lines associated to the higher non trivial vertices in $\t$, different from the endpoints, are simple lines. Let us now suppose that the line $l\in T$ is an interpolated line, but $y_l$ is a simple space vertex. We can write $\x_l=\sum_{i=1}^r \l_i x_i$, with $x_i$ simple space vertices associated to some non trivial vertex $v$ of $\t$, having frequency index $h_v>h_l$; the set $\{x_1,\ldots,x_r\}$ has to contain the special space vertex $x_l$ (see remark before \equ(A3.4)). We have: $$\eqalign{ &|x_l-y_l| \le |x_l-\x_l| + |\x_l-y_l| \le \sum_{i=1}^r \l_i |x_i-x_l| + |\x_l-y_l| \le \cr &\quad \le |\bar x-x_l| + |\x_l-y_l| \cr} \Eqa(A3.13)$$ where $\bar x$ is defined so that $|\bar x-x_l| = \sup_i |x_i-x_l|$. In the tree $\tilde S_v$ we can find a unique path $\CC$ connecting $\bar x$ to $x_l$. We shall distinguish two cases. \smallskip \\a) $\CC$ is made by lines belonging also to $T$. In this case, for any $\lb \in \CC$, since $h_{\bar l} > h_l$, we can extract from the r.h.s of \equ(A3.12) a factor $e^{-\k'\g^{h_l}|x_{\bar l}-y_{\bar l}|}$; then we associate to $l$ all these factors, together with the factor $e^{-\k'\g^{h_l}|\x_l-y_l|}$ coming again from \equ(A3.12), applied to the line $l$ itself. Hence, by using \equ(A3.13) and the trivial inequality $$|\bar x-x_l| \le \sum_{\lb\in \CC} |x_{\bar l} - y_{\bar l}| \Eqa(A3.14)$$ we can bound the overall factor associated to the line $l$ by $e^{-\k'\g^{h_l}|x_l-y_l|}$, as in the case of the simple lines. \smallskip \\b) At least one line of $\CC$ does not belong to $T$. In this case, the inequality \equ(A3.14) is not useful; however, if we can associate to $\CC$ a subset $T_l$ of $S_v$, such that $$|\bar x-x_l| \le \sum_{\lb\in T_l} |\x_{\bar l} - \h_{\bar l}| \Eqa(A3.15)$$ the argument of item a) can be immediately generalized. We shall now prove that this is in fact possible. Let $\bar v$ be the higher non trivial vertex containing $\CC$ and let $v_1,\ldots,v_s$, $2\le s\le s_{\bar v}$, be the non trivial vertices or endpoints following $\bar v$ in $\t$, which {\it are intersected} by $\CC$, that is such that at least one line of $\CC$ has an endpoint belonging to $x_{v_i}$, for any $i=1,\ldots,s$; at least one of these vertices has to be different from an endpoint of $\t$, otherwise we would be in the situation of item a), since all the lines associated to the higher non trivial vertices of $\t$ are simple lines. The $v_i$ are ordered so that, if we fix a positive direction in the path $\CC$, going from $\bar x$ to $x_l$, $v_i$ is crossed by $\CC$ before $v_j$, if $i0$: $${1\over |\L|} \int d\undx_{v_0} \prod_{ \hbox{\seven n.t. } v \ge v_{0}\atop v \hbox{\seven \ not e.p.}} \left[ {1\over s_v!} \sum_{\tilde T_v} \prod_{l\in \tilde T_v} e^{-\e \g^{h_l} |x_l-y_l|} \right] \le \prod_v C^{\sum_i^{s_v} N_{v^i}} \g^{-2h_v(s_v-1)}\Eqa(A3.18) $$ where $\tilde T_v$ is the anchored tree corresponding to $T_v$, $v^1,\ldots, v^{s_v}$ are the non trivial vertices immediately following $v$, and $N_{v^i} = |P_{v^i}|-|Q_{v^i}|$ is the number of the external lines in $v^i$. If we fix in an arbitrary way a point in $x_{v_0}$, we can bound the other integrations in the l.h.s. of \equ(A3.18) as usual, starting from the endpoints of $\tilde T$, and we get: $$\prod_{ \hbox{\seven n.t. } v \ge v_{0}\atop v \hbox{\seven \ not e.p.}} (C\g^{-2h_v})^{s_v-1} \;{1\over s_v!} \;|\tilde T_v| \Eqa(A3.19) $$ where $|\tilde T_v|$ is the number of possible choice for $\tilde T_v$, which can be bounded in the standard way, by observing that the number of anchored trees with $d_i$ lines branching from the vertex $v^i$ can be bounded by: $${(s_v-2)! \over (d_1-1)! \cdots (d_{s_v}-1)! } N_{v^1}^{d_1} \cdots N_{v^{s_v}}^{d_{s_v}} \Eqa(A3.20) $$ The bound \equ(A3.18) easily follows from \equ(A3.19) and \equ(A3.20). \vskip2truecm \vglue1.truecm {\it Appendix 4: Simplified beta functional.} \vglue1.truecm\numsec=4\numfor=1 To show that the ratios $Z_h/Z_{h'}$ can be eliminated we remark that they can be computed recursively, from \equ(5.47), \equ(5.48) provided \equ(5.51) holds. On the other hand, if we suppose that $|r_h|<\e$ for all $|h|k$. Hence the ratios $Z_{h+1}/Z_h$, regarded as recursively defined functions of $r_{h+1},\ldots,r_0$, are holomorphic in the domain $|r_j|<\e$, $j>h$. It follows that the r.h.s. of \equ(5.48), as a function of $r_h$, $h>k$, is holomorphic in the domain $|r_h|<\e$. In this appendix we prove the equations \equ(7.2). Let us consider the second of \equ(7.1) for $h=-1$: % $$\n_{-1}=\g\,\n_0+B^{0,\n}(\m_0,\n_0)\Eqa(A4.1)$$ % with $B^{0,\n}$ holomorphic in $\m_0,\n_0$ for $|\m_0|,|\n_0|<\e$, and: % $$\sup_{|\m_0|,|\n_0|<\e}|B^{0,\n}(\m_0,\n_0)|< b\e^2\Eqa(A4.2)$$ % for some $b>0$. The image of the disk $|\n_0|<\e$ under the map $\n_0\to\g\n_0+ B^{0,\n}(\m_0,\n_0)$ will contain the disk of radius $r=\g\e-b\e^2$, which is larger than $\e$, if $\e\le\bar\e=(\g-1)/b$, as we shall suppose from now on. Hence for all $|\n_{-1}|<\e$ there is a point $\n_0$ with $|\n_0|<\e$ such that \equ(A4.1) holds: such point is clearly unique if $\e$ is small enough. % Then \equ(A4.1) can be inverted in the form: % $$\n_0=\g^{-1}\n_{-1}+C(\n_{-1},\m_0)\Eqa(A4.3)$$ % with $C(\n_{-1},\m_0)$ holomorphic if $|\n_{-1}|,|\m_0|<\e$ and $|C(\n_{-1},\m_0)| = \g^{-1}|B^{0,\n}(\m_0,\n_0)|\le b\e^2\g^{-1}$. In fact we see that the analyticity domain in $\n_{-1}$ of $C(\n_{-1},\m_0)$ could be taken as large as $\e\g^{1-\x}$ with $\x>0$ prefixed and for $\e$ small enough (depending on $\x$). Let us consider now the equation: % $$\n_{-2}=\g\n_{-1}+ B^{-2,\n}(\m_{-1},\n_{-1}, \m_0,\g^{-1}\n_{-1}+C(\n_{-1},\m_0))\Eqa(A4.4)$$ % \equ(A4.4) has the same form as \equ(A4.1) if one sets $$B(\n_{-1},\m_{-1},\m_0)=B^{-2,\n}(\m_{-1},\n_{-1}, \m_0,\g^{-1}\n_{-1}+C(\n_{-1},\m_0))$$ and $B$ verifies the bound $b\e^2$ and $b$ can be taken to be {\it the same} $b$ as in \equ(A4.2), by the bounds of \S5; hence we can proceed inductively. By repeating the argument we arrive at: % $$\n_{h-1}=\g\n_{h}+\BB^h_\n(\m_h,\m_{h+1},\ldots,\m_0;\n_h)\Eqa(A4.5)$$ % with $\BB^h_\n$ analytic for $|\m_{h'}|<\e$, $h'\ge h$, and $|\n_h|<\e$. And, by the same substitutions, we get also: % $$\m_{h-1}=\m_h+\BB^h_\m(\m_h,\m_{h+1},\ldots,\m_0;\n_h)\Eqa(A4.6)$$ % with $\BB^h_\m$ analytic for $|\m_{h'}|<\e$, $h'\ge h$, and $|\n_{h}|<\e$. \vskip2.truecm \vglue1.truecm {\it References.} \vglue1.truecm \halign{[#]& \vtop{\hsize=14.truecm\\#}\cr B&{D. 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