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{}\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
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\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
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\ciao