text:
\section*{Figure Captions\markboth
        {FIGURECAPTIONS}{FIGURECAPTIONS}}\list
        {Figure \arabic{enumi}:\hfill}{\settowidth\labelwidth{Figure
999:}
        \leftmargin\labelwidth
        \advance\leftmargin\labelsep\usecounter{enumi}}}
\let\endfigcap\endlist \relax
\newskip\humongous \humongous=0pt plus 1000pt minus 1000pt
\def\mathsurround=0pt{\mathsurround=0pt}
\def\eqalign#1{\,\vcenter{\openup1\jot \mathsurround=0pt
        \ialign{\strut \hfil$\displaystyle{##}$&$
        \displaystyle{{}##}$\hfil\crcr#1\crcr}}\,}
\newif\ifdtup
\def\panorama{\global\dtuptrue \openup1\jot \mathsurround=0pt
        \everycr{\noalign{\ifdtup \global\dtupfalse
        \vskip-\lineskiplimit \vskip\normallineskiplimit
        \else \penalty\interdisplaylinepenalty \fi}}}
\def\eqalignno#1{\panorama \tabskip=\humongous
        \halign to\displaywidth{\hfil$\displaystyle{##}$
        \tabskip=0pt&$\displaystyle{{}##}$\hfil
        \tabskip=\humongous&\llap{$##$}\tabskip=0pt
        \crcr#1\crcr}}
\def\ap#1#2#3{Ann.\ Phys.\ (NY) #1 (19#3) #2}
\def\ar#1#2#3{Ann.\ Rev.\ Nucl.\ Part.\ Sci.\ #1 (19#3) #2}
\def\cpc#1#2#3{Computer Phys.\ Comm.\ #1 (19#3) #2}
\def\jmp#1#2#3{J.\ Math.\ Phys.\ #1 (19#3) #2}
\def\ib#1#2#3{ibid.\ #1 (19#3) #2}
\def\np#1#2#3{Nucl.\ Phys.\ B#1 (19#3) #2}
\def\pl#1#2#3{Phys.\ Lett.\ #1B (19#3) #2}
\def\pr#1#2#3{Phys.\ Rev.\ D #1 (19#3) #2}
\def\prb#1#2#3{Phys.\ Rev.\ B #1 (19#3) #2}
\def\prep#1#2#3{Phys.\ Rep.\ #1 (19#3) #2}
\def\prl#1#2#3{Phys.\ Rev.\ Lett.\ #1 (19#3) #2}
\def\rmp#1#2#3{Rev.\ Mod.\ Phys.\ #1 (19#3) #2}
\def\sj#1#2#3{Sov.\ J.\ Nucl.\ Phys.\ #1 (19#3) #2}
\def\zp#1#2#3{Zeit.\ Phys.\ C#1 (19#3) #2}

\newcounter{hran}
\def\setcounter{hran}{\value{equation}{\setcounter{hran}{\value{equation}}
\refstepcounter{hran} \setcounter{equation}{0}
\renewcommand{\theequation}{\thehran\alph{equation}}
              \begin{eqnarray}  }
\def\setcounter{hran}{\value{equation}{\setcounter{hran}{\value{equation}}
\refstepcounter{hran} \setcounter{equation}{1}
\renewcommand{\theequation}{\thehran\alph{equation}}
              \begin{eqnarray}  }
\def\bminiG#1{
          \setcounter{hran}{\value{equation}}
          \refstepcounter{hran}
          \setcounter{equation}{-1}
          \renewcommand{\theequation}{\thehran\alph{equation}}
          \refstepcounter{equation}
    \label{#1}
          \begin{eqnarray}          }
\def\end{eqnarray}\setcounter{equation}{\value{hran}{\end{eqnarray}\setcounter{equation}{\value{hran}}
\renewcommand{\theequation}{\arabic{equation}}}

\newskip\humongous \humongous=0pt plus 1000pt minus 1000pt
\def\mathsurround=0pt{\mathsurround=0pt} \def\eqalign#1{\,\vcenter{\openup1\jot
\mathsurround=0pt   \ialign{\strut \hfil$\displaystyle{##}$&$
\displaystyle{{}##}$\hfil\crcr#1\crcr}}\,} \newif\ifdtup
\def\panorama{\global\dtuptrue \openup1\jot \mathsurround=0pt
\everycr{\noalign{\ifdtup \global\dtupfalse     \vskip-\lineskiplimit
\vskip\normallineskiplimit      \else \penalty\interdisplaylinepenalty \fi}}}
\def\eqalignno#1{\panorama \tabskip=\humongous
\as\halignto\displaywidth{\hfil$\displaystyle{##}$
\tabskip=0pt&$\displaystyle{{}##}$\hfil
\tabskip=\humongous&\llap{$##$}\tabskip=0pt     \crcr#1\crcr}}

\def\bom#1{\mbox{\bf{#1}}}
\def\mbox{\small $\frac{1}{2}$}{\mbox{\small $\frac{1}{2}$}}
\def\mbox{\small $\frac{1}{3}$}{\mbox{\small $\frac{1}{3}$}}
\def\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$<$}{\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$<$}}
\def\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$>$}{\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$>$}}
\def\VEV#1{\left\langle #1\right\rangle}
\def\frac#1#2{ {{#1} \over {#2} }}
\def\hbox{\rm et al.}{\hbox{\rm et al.}}
\def\s#1{{\small #1}}

\def\refup#1{~$^{#1}$}
\def\mathrel{\mathpalette\fun <}{\mathrel{\mathpalette\fun <}}
\def\mathrel{\mathpalette\fun >}{\mathrel{\mathpalette\fun >}}
\def\fun#1#2{\lower3.6pt\vbox{\baselineskip0pt\lineskip.9pt
  \ialign{$\mathsurround=0pt#1\hfil##\hfil$\crcr#2\crcr\sim\crcr}}}
\def\hbox{\it i.e.}{ }}      \def\etc{\hbox{\it etc.}{ }{\hbox{\it i.e.}{ }}      \def\etc{\hbox{\it etc.}{ }}
\def\hbox{\it e.g.}{ }}      \def\cf{\hbox{\it cf.}{ }{\hbox{\it e.g.}{ }}      \def\cf{\hbox{\it cf.}{ }}
\def\partder#1{{\partial   \over\partial #1}}

\def\ds#1{\ooalign{$\hfil/\hfil$\crcr$#1$}}
\def\re#1{(\ref{#1})}

\def\begin{equation}{\begin{equation}}
\def\end{equation}{\end{equation}}
\def\begin{eqnarray}{\begin{eqnarray}}
\def\end{eqnarray}{\end{eqnarray}}

\def\Gamma{\Gamma}
\def \bar \Gamma{ \bar \Gamma}
\def {\cal I}{ {\cal I}}
\def \Lambda{ \Lambda}
\def \lambda{ \lambda}
\def \gamma{ \gamma}
\def\left[\frac{q^2}{\G_2(q,\l)}\right]^2{\left[\frac{q^2}{\Gamma_2(q, \lambda)}\right]^2}
\def \bar p{ \bar p}
\def\d4#1{\frac {d^4 {#1} }{(2\pi)^4}}
\def\dLID#1{ \frac { \Lambda \partial D^{-1}_{ \Lambda, \Lambda_0}(#1)}{\partial  \Lambda}}
\def\dLD#1{  \frac { \Lambda \partial      D_{ \Lambda, \Lambda_0}(#1)}{\partial  \Lambda}}
\def\L{\partial   \over\partial \L}{ \Lambda{\partial   \over\partial  \Lambda}}
\def$\L_0\to\infty\;${$ \Lambda_0\to\infty\;$}
\def$\L\to 0\;\;${$ \Lambda\to 0\;\;$}
\def[dq]_{\lambda}\;{[dq]_{\lambda}\;}
\def\begin{itemize}{\begin{itemize}}
\def\end{itemize}{\end{itemize}}
\def\begin{enumerate}{\begin{enumerate}}
\def\end{enumerate}{\end{enumerate}}
\def\abs#1{|#1|}
\def\over{\over}
\def\Delta{\Delta}
\def\raisebox{-1.5 ex}{\rlap{\tiny $\;\;p_i^2<c\l^2$}{\raisebox{-1.5 ex}{\rlap{\tiny $\;\;p_i^2<c \lambda^2$}}
\raisebox{0ex} {$\; \mbox{Max}\;\;\;\,$}}
\def\nome#1{{\label{#1}}}

\begin{document}
\begin{titlepage}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\begin{flushright}
     UPRF 92-360  \\
     December 1992
\end{flushright}
\par \vskip 10mm
\begin{center}
{\Large \bf Perturbative renormalization and infrared \\
finiteness in the Wilson renormalization group: \\
the massless scalar case\footnote{Research supported in part by MURST,
Italy}}
\end{center}
\par \vskip 2mm
\begin{center}
        {\bf M.\ Bonini, M.\ D'Attanasio and G.\ Marchesini} \\
        Dipartimento di Fisica, Universit\`a di Parma and\\
        INFN, Gruppo Collegato di Parma, Italy
        \end{center}
\par \vskip 2mm
\begin{center} {\large \bf Abstract} \end{center}
\begin{quote}
A new proof of perturbative renormalizability and infrared finiteness for a
scalar massless theory is obtained from a formulation of renormalized field
theory based on the Wilson renormalization group. The loop expansion of the
renormalized Green functions is deduced from the Polchinski equation of
renormalization group. The resulting Feynman graphs are organized in such a
way that the loop momenta are ordered. It is then possible to analyse their
ultraviolet and infrared behaviours by iterative methods. The necessary
subtractions and the corresponding counterterms are automatically
generated in the process of fixing the physical conditions for
the ``relevant'' vertices at the normalization point.
The proof of perturbative renormalizability and infrared finiteness is
simply based on dimensional arguments and does not require the usual
analysis of topological properties of Feynman graphs.
\end{quote}
\end{titlepage}

\section{Introduction}
In perturbation theory renormalization requires a cumbersome
analysis of Feynman graph divergences and the exploitation of
topological properties in graph theory \cite{Bo}.
The origin of this complication is technical and due to the
overlapping divergence problem which arises in the procedure of
generating in a systematic way all appropriate subtractions.
This difficulty seems rather artificial since renormalization
is based on simple and general properties \cite{C} such as
dimensional analysis and the breaking of scale invariance in
interacting theories.
The perturbative study of finiteness of the vertex functions in a
massless theory \cite{LNK} brings a similar surprise.
Also in this case the reason for the finiteness is simple and general,
but in perturbation theory the analysis turns out to be rather
complex \cite{PQ}.

The deep physical meaning of renormalization can be appreciated
in the formulation due to Wilson \cite{W}-\cite{Be}.
One introduces a momentum cutoff $ \Lambda$, considers the effective
Lagrangian obtained by integrating the fields with frequencies
above $ \Lambda$ and
requires that the physical measures below such a scale are independent
of $ \Lambda$.
It is then natural to explore whether it is possible to deduce from
this formulation a proof of perturbative renormalizability which
avoids the usual cumbersome analysis of overlapping divergences.
This program has been successfully carried out by Polchinski \cite{P}.
{}From his formulation it is possible to deduce the perturbative expansion
and he was able to give a proof of perturbative renormalizability
for a scalar theory.
One does not need to control the detailed momentum dependence of
the effective couplings, thus the proof is based on simple dimensional
analysis.
However also this proof is technically quite laborious and the
physical features are not transparent. Moreover, although the proof is
perturbative, Green functions and Feynman graphs are never directly involved,
thus no connection with the usual analysis of perturbative renormalization
can be made.

In this paper we use the renormalization group equation of Ref.~\cite{P}
to obtain an alternative proof of perturbative renormalizability
for a scalar field theory.
Also in this case the proof is based on simple dimensional analysis but
in all steps we have an explicit connection to Feynman graphs.
Since power counting enters also in the infrared finiteness of the Green
functions at non exceptional momenta, we use similar
methods to obtain a perturbative proof of this feature in a massless theory.

We start by observing that the effective vertices satisfying the
renormalization group equations of Ref.~\cite{P}
correspond to the cutoff-Green functions of the theory in which
the parameter $ \Lambda$ plays the role of an infrared cutoff in the
propagators.
Thus the effective vertices evaluated at $ \Lambda=0$ correspond to
the renormalized Green functions.
The physical couplings, \hbox{\it i.e.}{ }}      \def\etc{\hbox{\it etc.}{ } the couplings with non negative
dimensions such as the mass, the wave function normalization, the
four point coupling, are fixed at the physical point $ \Lambda=0$.
In this way the renormalization group equations define in a constructive
way the renormalized quantum field theory.
This fact has been discussed also in Ref.~\cite{Be}-\cite{KK}.

We prove the renormalizability of the scalar theory by showing that by
fixing the physical couplings at $ \Lambda=0$ one generates,
order by order in perturbation theory, the necessary subtractions
which make finite any Green function.
The analysis of the ultraviolet behaviour of Feynman graphs does
not require
any complicated technical step so that the role of simple dimensional
counting and the origin of the needed subtractions are clear.
As in \cite{P}, we avoid the study of overlapping divergences.
Although we still refer to Feynman graphs we do not need a detailed
knowledge of their momentum dependence.
Solving by iteration the renormalization group equation, we find that
the Green functions are given as sum of contributions of Feynman
diagrams in which the various virtual momenta are ordered.
This allows us to control the ultraviolet behaviour by iterative methods.

In the same way we study the finiteness of the Green functions for
non exceptional momenta.
The perturbative analysis here is simpler since there is no need
to control the subtractions.
Due to the fact that the loop momenta are ordered, we need to
analyse only the infrared behaviour of the softest momentum.

The paper is organized as follows.
In Section~2 we rederive the renormalization group flow of Ref.~\cite{P}.
We show that the solution at $ \Lambda=0$ is the effective action
of the theory, deduce the perturbative expansion and perform
some one and two loop calculations. In Sections~3 and 4 we describe the proof
by induction of perturbative renormalizability and infrared finiteness.
Section~5 contains some remarks and conclusions. In the Appendix we
describe in this context the beta function.

\section{Renormalization group flow}
We consider a four dimensional massless scalar field theory
with a four point coupling $g$ at the scale $\mu$, \hbox{\it i.e.}{ }}      \def\etc{\hbox{\it etc.}{ } with
the two and four point vertex functions satisfying the conditions
\begin{equation}          \nome{norm}
\Gamma_2(0)=0 \,,
\;\;\;\;\;\;\;\;
\frac {d \Gamma_2(p^2)}{dp^2}\vert_{p^2=\mu^2}=1  \,,
\;\;\;\;\;\;\;\;
\Gamma_4( \bar p_1, \bar p_2, \bar p_3, \bar p_4)=g  \, ,
\end{equation}
where $ \bar p_i$ are the momenta at the symmetric point
\begin{equation}\nome{sp}
 \bar p_i \bar p_j=\mu^2(\delta_{ij}-\frac{1}{4}).
\end{equation}
The generating functional of Euclidean Green functions is
\begin{equation}  \nome{W}
Z[j]=e^{W[j]}=\int {\cal D}\phi\;
\exp
\left\{-\mbox{\small $\frac{1}{2}$}\int \d4 p\phi(p)D^{-1}(p)\phi(-p)
\;{-S_{int}[\phi]}\;+ {\int \d4 p \phi(p)j(-p)}
\right\} \,,
\end{equation}
where $D(p)=1/p^2$ is the free propagator of the massless theory and
$ S^{int} [\phi]$ is the self interaction
\begin{equation}                 \nome{intaction}
S^{int}[\phi]=\mbox{\small $\frac{1}{2}$} \int \d4 p
\left[  \gamma^{(B)}_2 p^2 + \gamma^{(B)}_3 \right]
\phi(p)\phi(-p) +
\frac { \gamma^{(B)}_4}{4!} \int d^4x \Phi^4(x) \, ,
\end{equation}
with $\phi(p)$ the momentum representation of the field $\Phi(x)$.
Feynman graphs  generated by \re{W} and \re{intaction} have ultraviolet
divergences, which can be regularized by
assuming that the fields have frequencies bounded by
an ultraviolet cutoff $ \Lambda_0$ ($p^2< \Lambda_0^2$) or, equivalently, that the
propagator $D(p)$ vanishes for $p^2 >  \Lambda_0^2$.
The parameters $ \gamma^{(B)}_2$ and $ \gamma^{(B)}_4$ are dimensionless,
$ \gamma^{(B)}_3$ has the dimensions of a square mass, and they are fixed by the
physical conditions \re{norm}.

\subsection{Derivation of the evolution equation}
The evolution equation which is the basis for the present treatment is
derived as follows.
We consider the functionals $Z[j; \Lambda]$ and $W[j; \Lambda]$ obtained
from \re{W} in which we replace $D(p)$ by the cutoff propagator
\begin{equation} \nome{prop}
D(p) \Rightarrow D_{ \Lambda, \Lambda_0}(p)= \frac { K_{ \Lambda, \Lambda_0}(p)}{p^2} \,,
\end{equation}
with $K_{ \Lambda, \Lambda_0}(p)=1$ in the region
$ \Lambda^2 \, \raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$<$} \,p^2\, \raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$<$} \,  \Lambda_0^2$
and rapidly vanishing outside.
As one takes the limit $ \Lambda \to 0$, the cutoff Green functions
generated from $W[j; \Lambda]$ tend, at least perturbatively, to the
physical ones, \hbox{\it i.e.}{ }}      \def\etc{\hbox{\it etc.}{ } we have $W[j; \Lambda] =W[j]$ for $\L\to 0\;\;$.

The functional $Z[j; \Lambda]$ can be computed from the following
evolution equation
\begin{equation}\nome{evZ}
\L{\partial   \over\partial \L} Z[j; \Lambda]= - \mbox{\small $\frac{1}{2}$} \int \d4 q \dLID {q} \, (2\pi)^8
            \frac { \delta^2 Z[j; \Lambda]}
                  {\delta j(q)\delta j(-q)}\,,
\end{equation}
obtained by differentiating \re{W} with the propagator $D_{ \Lambda, \Lambda_0}(p)$.
As we will discuss in detail later, the boundary conditions
are determined by requiring that the vertex functions satisfy
the normalization conditions \re{norm} at $ \Lambda=0$, and the form
\re {intaction} at $ \Lambda= \Lambda_0$. Since we are interested in the limit
$\L_0\to\infty\;$, in these functionals we do not write explicitly the dependence
on the ultraviolet cutoff.
Notice that the inverse cutoff propagator entering in \re{evZ} is cancelled by
corresponding propagators in the external legs of the Green functions.
Although inverse cutoff propagators cancel, in this
section we shall assume a cutoff function different from zero everywhere
but rapidly vanishing outside the range $( \Lambda, \Lambda_0)$.

The present treatment is related to the Wilson renormalization
formalism. Actually, Eq.~\re{evZ} is equivalent to
the renormalization group equation of Ref. \cite{P}.
To clarify this relation recall that in the Wilson formulation
one integrates \re{W} over the fields with frequency larger than $ \Lambda$
to obtain a new action $S_{ \Lambda}[\phi_{ \Lambda},j]$ and Eq.~\re{W} becomes
$$
Z[j]=e^{W[j]}=\int {\cal D} \phi_{ \Lambda} \;
\exp
\left\{ -S_{ \Lambda}[\phi_{ \Lambda},j] \; \right\} \, ,
$$
where $\phi_{ \Lambda}(p)$ are the soft fields with frequency smaller
than $ \Lambda$.
Obviously $Z[j]$ does not depends on $ \Lambda$, and this is the basis
of Wilson renormalization group.
In particular we can take $ \Lambda$ small.
If we consider sources $j(p)$ with frequencies larger than $ \Lambda$, the
integration over $\phi_{ \Lambda}$ generates only virtual loops with
soft momenta $p^2 <  \Lambda^2$.
Since the theory is infrared finite (at least perturbatively,
see Section 4), these contributions vanish for $ \Lambda \to 0$
and in this limit we can ignore the $\phi_{ \Lambda}$ dependence in the
new action and we can write
$$
S_{ \Lambda} [\phi_{ \Lambda},j]  \simeq S_{ \Lambda} [0,j]= -W[j; \Lambda] \to -W[j] \, .
$$
The two functionals $S_{ \Lambda} [0,j]$ and $-W[j; \Lambda]$ are equal,
since to use the cutoff propagator $D_{ \Lambda, \Lambda_0}(p)$
is equivalent to integrate only over fields with frequencies larger
than $ \Lambda$.
In the framework of renormalization group, it is more typical to
consider sources with frequencies softer than $ \Lambda$ and compute
$S_{ \Lambda}[\phi_{ \Lambda},0]$ which is the action at the scale $ \Lambda$.
It is easy to show that $S_{ \Lambda}[\phi_{ \Lambda},0]=S_{ \Lambda}[0,j]$ with
$\phi_{ \Lambda}(p)=D_{ \Lambda, \Lambda_0}(p)j(p)$. This explains why the
Polchinski equation for $S_{ \Lambda}[\phi_{ \Lambda},0]$ is the same as
the evolution
equation for $W[j; \Lambda]$ obtained from \re{evZ} provided one identifies
$\phi_{ \Lambda}(p)$ with $D_{ \Lambda, \Lambda_0}(p)j(p)$.
The fact that $S_{ \Lambda}[\phi_{ \Lambda},0]$ generates cutoff Green functions with
amputated external propagators has been observed also
in Refs.~\cite{Be}-\cite{KK}.

\subsection{Evolution equation for the vertex functions}

To study the renormalizability and the infrared finiteness of this
theory it is more convenient to consider the proper vertices
$\Gamma_{2n}(p_1,\cdots,p_{2n})$ (see Fig.~1a) and their generating functional
$$
\Gamma[\psi]=
\sum_{n=1} \frac 1 {(2n)!} \int \prod_{i=1}^{2n} \d4 {p_i} \psi(p_i)
\, \Gamma_{2n}(p_1, \cdots,p_{2n})
\, (2\pi)^4\delta^4(\sum_{i=1}^{2n} p_i) \; ,
$$
which is related to $W[j]$ by the Legendre transformation
$$
\Gamma[\psi]=-W[j] + W[0]+\int \d4 p j(p)\psi(-p) \, ,
\;\;\;\;\,\;\;
 (2\pi)^4  \frac {\delta \Gamma}{\delta \psi(-p)} =j(p) \, .
$$
In the same way we introduce the cutoff effective action
$\Gamma[\psi; \Lambda]$ as Legendre transformation of $W[j; \Lambda]$. Taking into account
that
$$
\L{\partial   \over\partial \L} \Gamma[\psi; \Lambda]=-\L{\partial   \over\partial \L} W[j; \Lambda]+\L{\partial   \over\partial \L} W[0; \Lambda] \,,
$$
the evolution equation for the cutoff effective action is given by
\begin{equation} \nome{ev1}
\eqalign{
\L{\partial   \over\partial \L} \biggr\{ &\Gamma[\psi; \Lambda]-\mbox{\small $\frac{1}{2}$} \int \d4p D^{-1}_{ \Lambda, \Lambda_0}(p)\,\psi(p)\psi(-p)
\biggr\}
\cr&
=\mbox{\small $\frac{1}{2}$} \int \d4q \dLID {q} \, (2\pi)^8
\frac { \delta^2 W[j; \Lambda]}{\delta j(q)\delta j(-q)}
+\L{\partial   \over\partial \L} W[0; \Lambda]\;.
}
\end{equation}
The functional in the integrand is the inverse of
$\delta^2\Gamma[\psi]/\delta\psi(q)\delta\psi(q')$
\begin{equation}\nome{inv}
\delta^4(q+q')=(2\pi)^{8}
\int d^4 {q''}
\frac{\delta^2 \Gamma[\psi]}{\delta\psi(q)\delta\psi(q'')}
\frac{\delta^2 W[j]}{\delta j(-q'')\delta j(q')}\;,
\end{equation}
and is obtained as follows.
First we isolate the two point function contribution in the
two functionals
\begin{equation}\nome{inv2}
\eqalign{
(2\pi)^8\frac{\delta^2\Gamma[\psi]}{\delta\psi(q)\delta\psi(q')}
=&
(2\pi)^4\delta^4(q+q') \Gamma_2(q; \Lambda)+
\Gamma_2^{int}[q,q',\psi]\,,
\cr(2\pi)^8\frac {\delta^2\, W[j; \Lambda]}{\delta j(q)\delta j(q')}
=&
(2\pi)^4\delta^4(q+q')\frac{1}{\Gamma_2(q; \Lambda)}+
W_2^{int}[q,q',j]\,.
}
\end{equation}
Notice that the two point function contribution in the second
equation cancels the constant term $ \Lambda\partial W[0; \Lambda]/\partial \Lambda$ in
\re{ev1}.
Then, from \re{inv} we obtain $W_2^{int}$ as functional of $\psi(p)$
$$
W_2^{int}[q,q',j]=-
\frac {1}{\Gamma_2(q; \Lambda)}
 \bar \Gamma[q,q',\psi]
\frac {1}{\Gamma_2(q'; \Lambda)} \,,
$$
where the auxiliary functional $ \bar \Gamma$ satisfies the equation
\begin{equation} \nome{bGrec}
 \bar \Gamma[q,q',\psi] = \Gamma_2^{int}[q,q',\psi]-
\int \d4 {q''} \Gamma_2^{int}[q,-q'',\psi]
\frac{1}{\Gamma_2(q''; \Lambda)}
 \bar \Gamma[q'',q',\psi]   \, .
\end{equation}
By expanding this equation we obtain the auxiliary vertices
$ \bar \Gamma_{2n+2}(q,p_1, \cdots,p_{2n},q'; \Lambda)$ (see Fig.~1b)
in terms of the proper vertices. For $n=1$ we find
$$
 \bar \Gamma_4(q,p_1,p_2,q'; \Lambda)=
 \Gamma_4(q,p_1,p_2,q'; \Lambda)\,,
$$
and in general (see Fig.~2)
\begin{equation}\nome{auxv}
\eqalign{
&
 \bar \Gamma_{2n+2} (q,p_1,\cdots,p_{2n},q'; \Lambda)=
\Gamma_{2n+2} (q,p_1,\cdots,p_{2n},q'; \Lambda)
\cr&
-\sum
\Gamma_{2k+2} (q,p_{i_1},\cdots,p_{i_{2k}},-Q; \Lambda)
\;\frac{1}{\Gamma_2(Q; \Lambda)}
 \bar \Gamma_{2n-2k+2}(Q,p_{i_{2k+1}},
\cdots,p_{i_{2n}},q'; \Lambda)
}
\end{equation}
where $Q=q+p_{i_1}+\cdots p_{i_{2k}}  $ and
the sum is over $k=1 \dots n-1$ and over the ${2n\choose 2k}$
combinations of $(i_1 \cdots i_{2n})$.

In conclusion the evolution equation for the functional $\Gamma[\psi]$ is
$$
\eqalign{
\L{\partial   \over\partial \L} \biggr\{ \Gamma[\psi; \Lambda]
&-\mbox{\small $\frac{1}{2}$} \int \d4p D^{-1}_{ \Lambda, \Lambda_0} (p)\,\psi(p)\psi(-p) \biggr\}
\cr
&= - \mbox{\small $\frac{1}{2}$} \int \d4q
\dLID {q}\biggr[ \frac {1}{\Gamma_2(q; \Lambda)} \biggr]^2\,
 \bar \Gamma[q,-q,\psi]\, .
}
$$
After isolating the interaction part of the two point function
$$
\Gamma_2(p; \Lambda)=D^{-1}_{ \Lambda, \Lambda_0}(p)+\Sigma(p; \Lambda)\,,
$$
the evolution equation for the proper vertices are
\bminiG{eveq}
\nome{eveq1}
\L{\partial   \over\partial \L}
\Sigma(q; \Lambda)
=\mbox{\small $\frac{1}{2}$} \int \d4 q
\frac{S(q; \Lambda)}{q^2}
\Gamma_{4}(q,p,-p,-q; \Lambda)\,,
\end{eqnarray}\setcounter{equation}{\value{hran}
and
\addtocounter{equation}{-1}
\setcounter{hran}{\value{equation}
\refstepcounter{equation}
\nome{eveq2}
\L{\partial   \over\partial \L}
\Gamma_{2n}(p_1,\cdots,p_{2n}; \Lambda)
=\mbox{\small $\frac{1}{2}$} \int \d4 q
\frac{S(q; \Lambda)}{q^2}
 \bar \Gamma_{2n+2}(q,p_1,\cdots,p_{2n},-q; \Lambda)\,,
\end{eqnarray}\setcounter{equation}{\value{hran}
where $S(q; \Lambda)$ is given by
\begin{equation} \nome{esse}
\frac{S(q; \Lambda)}{q^2}
\equiv
{ \frac { \Lambda \partial D_{ \Lambda, \Lambda_0}(q)}{\partial  \Lambda}}
\left[ \frac {1}{1+D_{ \Lambda, \Lambda_0}(q)\Sigma(q; \Lambda)} \right]^2\,.
\end{equation}
These equations involve in the right
hand side vertices at the infrared cutoff $ \Lambda$ with a pair of
exceptional momenta $q$ and $-q$.
For $\L\to 0\;\;$ these vertices become singular since we are dealing with a
massless theory.
In a next Section we analyse this limit and show that Eq.~\re{eveq}
allows one to obtain vertex functions $\Gamma_{2n}$ with non exceptional
momenta which are finite for $\L\to 0\;\;$, order by order in perturbation theory.

\subsection{Physical couplings and boundary conditions}
In the study of renormalization we should control the
quantities with non negative momentum dimension, \hbox{\it i.e.}{ }}      \def\etc{\hbox{\it etc.}{ } the
``relevant'' couplings. Dimensional analysis gives
$$
\Gamma_{2n} \sim  \bar \Gamma_{2n} \sim  \Lambda^{4-2n}\,.
$$
Thus the ``relevant'' couplings are
$$
 \gamma_2 ( \Lambda) = \frac {d\Sigma(p^2; \Lambda)}{dp^2}\vert_{p^2=\mu^2} \,,
\;\;\;\;\;\;\;\;
 \gamma_3 ( \Lambda) =\Sigma(0; \Lambda)\,,\;\;\;\;\;\;\;\;
 \gamma_4 ( \Lambda) =\Gamma_4( \bar p_1, \bar p_2, \bar p_3, \bar p_4; \Lambda)\,,
$$
and correspond, for $ \Lambda=0$, to the physical couplings introduced in
(\ref{norm}).
We then isolate the relevant couplings in the two and
four point vertices
\begin{equation}\nome{isol}
\eqalign{
\Sigma(p^2; \Lambda)&= p^2  \gamma_2( \Lambda) +  \gamma_3 ( \Lambda) +
\Delta_2(p^2; \Lambda) \,, \cr
\Gamma_4(p_1,p_2,p_3,p_4; \Lambda)&=  \gamma_4( \Lambda)+\Delta_4(p_1,p_2,p_3,p_4; \Lambda)\,,
}
\end{equation}
where $\Delta_2(0; \Lambda)=0$, $\partial \Delta_2(p^2; \Lambda)/\partial p^2=0$
at $p^2=\mu^2$, and $\Delta_4(\{ \bar p_i\}; \Lambda) =0 $
at the symmetric point \re{sp}.
{}From dimensional analysis we have
$$
\eqalign{
&
 \gamma_2 \sim  \gamma_4 \sim ( \Lambda)^0\;, \;\;\;\;\;\;  \gamma_3 \sim ( \Lambda)^{2}\,,
\cr&
\Delta_2\sim \Delta_4 \sim ( \Lambda)^{-2}\,.
}
$$
Notice that in $\Delta_2(p^2; \Lambda)$ four powers of momentum are
absorbed by the $p$-dependence required by the two conditions at $p^2=0$
and $p^2=\mu^2$. Similarly in $\Delta_4$ two powers of momentum are absorbed
by the $p_i$-dependence required by the condition at the symmetric
point.

To obtain the vertex functions from the evolution equation
(\ref{eveq}) we need the boundary conditions.
For the relevant couplings $ \gamma_i( \Lambda)$ they are defined at the physical
value $ \Lambda=0$ by the conditions \re{norm}
\bminiG{bound}
\nome{bound1}
 \gamma_2( \Lambda=0) =0\;,\;\;\;\;
 \gamma_3( \Lambda=0) =0\;,\;\;\;\;
 \gamma_4( \Lambda=0) =g\;.
\end{eqnarray}\setcounter{equation}{\value{hran}
The physical requirement we have to set on the remaining vertex
functions is that they are irrelevant when the ultraviolet cutoff
$ \Lambda_0$ is set to infinity.
The simplest choice is to set all these ``irrelevant'' vertices to zero
at $ \Lambda= \Lambda_0$
\addtocounter{equation}{-1}
\setcounter{hran}{\value{equation}
\refstepcounter{equation}
\nome{bound2}
\Delta_2(p^2; \Lambda_0) =0 \,,
\;\;\;\;\;
\Delta_4(p_1,p_2,p_3,p_4; \Lambda_0)=0\,,
\;\;\;\;\;
\Gamma_{2n}(p_1,\cdots,p_{2n}; \Lambda_0)=0 \,,
\;\;\;\;\; n \ge 3\,.
\end{eqnarray}\setcounter{equation}{\value{hran}
With these conditions the functional $\Gamma[\psi; \Lambda]$ has the form of the
action in (\ref{intaction}) with $ \gamma_i^{(B)}$ given by the
relevant couplings $ \gamma_i$ evaluated at $ \Lambda= \Lambda_0$.
The bare coupling constant is then
$ g^{(B)}= { \gamma_4^{(B)} }/{(1+ \gamma_2^{(B)})^2 }$.

The evolution equations (\ref{eveq}) with the boundary conditions
\re{bound}, can be converted into a set of integral
equations.
For the three relevant couplings $ \gamma_i$ the boundary conditions
(\ref{bound1}) give
\begin{equation}\nome{inte1}
\eqalign{
 \gamma_{2}( \Lambda)&=
\mbox{\small $\frac{1}{2}$} \int \d4 q \int_0^{ \Lambda}
\frac{d \lambda}{ \lambda} \frac{S(q; \lambda)}{q^2}
\frac{\partial}{\partial p^2}
\Gamma_{4}(q,p,-p,-q; \lambda)|_{p^2=\mu^2}\,,
\cr
 \gamma_{3}( \Lambda)&=
\mbox{\small $\frac{1}{2}$} \int \d4 q \int_0^{ \Lambda}
\frac{d \lambda}{ \lambda} \frac{S(q; \lambda)}{q^2}
\Gamma_{4}(q,0,0,-q; \lambda)\,,
\cr
 \gamma_{4}( \Lambda)=g &+
\mbox{\small $\frac{1}{2}$} \int \d4 q \int_0^{ \Lambda}
\frac{d \lambda}{ \lambda} \frac{S(q; \lambda)}{q^2}
 \bar \Gamma_{6}(q, \bar p_1,\ldots, \bar p_{4},-q; \lambda)\,.
}
\end{equation}
For the other vertices, the boundary conditions (\ref{bound2}) give
\begin{equation}\nome{inte2}
\eqalign{
\Gamma_{2n}(p_1\ldots p_{2n}; \Lambda)&=
-\mbox{\small $\frac{1}{2}$} \int \d4 q \int_{ \Lambda}^{ \Lambda_0}
\frac{d \lambda}{ \lambda} \frac{S(q; \lambda)}{q^2}
 \bar \Gamma_{2n+2}(q,p_1,\ldots,p_{2n},-q; \lambda)\,,
\cr
\Delta_{2}(p; \Lambda)&=
-\mbox{\small $\frac{1}{2}$} \int \d4 q \int_{ \Lambda}^{ \Lambda_0}
\frac{d \lambda}{ \lambda} \frac{S(q; \lambda)}{q^2}
\Delta \Gamma_{4}(q,p,-p,-q; \lambda)\,,
\cr
\Delta_{4}(p_1 \ldots p_4; \Lambda)&=
-\mbox{\small $\frac{1}{2}$} \int \d4 q \int_{ \Lambda}^{ \Lambda_0}
\frac{d \lambda}{ \lambda} \frac{S(q; \lambda)}{q^2}
\Delta  \bar \Gamma_{6}(q,p_1,\ldots,p_{4},-q; \lambda)\,,
}
\end{equation}
where $n>2$ in the first equation.
The subtracted vertices $\Delta \Gamma_{4}$ and $\Delta  \bar \Gamma_{6}$ are defined by
\begin{equation} \nome{inte3}
\eqalign{
\Delta \Gamma_{4}(q,p,-p,-q; \lambda)
\equiv
&
\Gamma_{4}(q,p,-p,-q; \lambda) -\Gamma_{4}(q,0,0,-q; \lambda)
\cr
&
-p^2 \frac{\partial}{\partial p'^2}
\Gamma_{4}(q,p',-p',-q; \lambda)|_{p'^2=\mu^2}\,,
\cr
\Delta  \bar \Gamma_{6}(q,p_1,\ldots,p_{4},-q; \lambda)
\equiv
&
 \bar \Gamma_{6}(q,p_1,\ldots,p_{4},-q; \lambda)
- \bar \Gamma_{6}(q, \bar p_1,\ldots,  \bar p_{4},-q; \lambda)\,.
}
\end{equation}
The subtractions in $\Delta \Gamma_{4}$ and $\Delta  \bar \Gamma_{6}$ are a consequence
of isolating in eq.~\re{isol} the relevant couplings in the two and four
vertices and of the different boundary conditions \re{bound}.
As we expect they provide the necessary subtractions to
make finite the vertex functions for $ \Lambda_0\to\infty$ at any order
in perturbation theory.

We should notice the different role of boundary conditions for the
relevant couplings at $ \Lambda=0$ and of the irrelevant vertices at
$ \Lambda= \Lambda_0 \to \infty$. For the relevant couplings this implies that
the $q$-integration is bounded above by $ \Lambda$.
This is crucial for obtaining a finite result since, as expected from
dimensional counting, the integrands grow with $q^2$.
The bare couplings, obtained by setting $ \Lambda= \Lambda_0$, are therefore
growing with $ \Lambda_0$ and give the counterterms of the Lagrangian
\re{intaction} in terms of the physical coupling $g$.
For the other vertices the $q$-integration is bounded above by the
ultraviolet cutoff $ \Lambda_0$. To show that the theory is renormalizable, one
must prove that for $q^2 \to \infty$ the vertices in the integrands in
\re{inte2} are sufficiently vanishing to allow one to take $\L_0\to\infty\;$.

For a sharp cutoff Eq.~\re{esse} can be written
\begin{equation} \nome{esse1}
\frac{S(q; \lambda)}{q^2} = - \frac {1}{ \lambda} \delta (  \lambda-\sqrt{q^2} ) s( \lambda)\,,
\;\;\;\;\;\;
s( \lambda)=\left[ \frac{1}{1+\frac{1}{ \lambda^2}\Sigma( \lambda; \lambda)} \right]^2 \,,
\end{equation}
which is independent of $ \Lambda_0$.

\subsection {Loop expansion}

By solving iteratively Eqs.~\re{inte1} and \re{inte2}
one obtains the loop expansion.
Here we compute the first terms for $ \Lambda_0 \to \infty$ so that
the cutoff propagator is simply
$$
D_{ \Lambda}(q)=K_{ \Lambda}(q)/q^2\,,
$$
where $K_{ \Lambda}(q)=1$ for $q^2\geq  \Lambda^2$ and vanishes for $q^2< \Lambda^2$.
It will be clear that the limit $ \Lambda\to 0$ can be taken only for non
exceptional momenta.
The starting point is the zero loop order in which  the only non
vanishing vertex is
$$
 \gamma_4^{(0)} ( \Lambda) =g\,,
$$
and the auxiliary vertices with $n \ge 2$ are given by (see Fig.~3)
\begin{equation} \nome{bG2n+20l}
 \bar \Gamma_{2n+2}^{(0)}(q,p_1,\cdots,p_{2n},q'; \Lambda)=
-(-g)^n \; \sum_{perm} \prod_{k=1}^{n-1}D_{ \Lambda}
 \left( q+\sum_{\ell=1}^{2k}p_{i_\ell} \right) \,,
\end{equation}
where the sum is over $(2n)!/2^n$ terms obtained from the
permutations of $(p_{i_1},\cdots, p_{i_{2n}})$ and the symmetry of
the four point coupling.

\vskip .3 true cm \noindent
1. {\it One loop vertices.}
\vskip .2 true cm \noindent
The only non vanishing contribution for the two point function is
$$
 \gamma_3^{(1)}( \Lambda) = \mbox{\small $\frac{1}{2}$} g \int\d4 q \Delta_{ \Lambda}(q)
= - \frac{1}{32\pi^2}  g  \Lambda^2\,,
$$
where
$$
\Delta_ \Lambda(q)=D_{ \Lambda}(q) - D_{0}(q)
= -\frac{1}{q^2}\theta( \Lambda^2-q^2) \,.
$$
{}From \re{bG2n+20l} one obtains
$$
 \gamma^{(1)}_4( \Lambda)\,  =\, -\frac 3 2 g^2\int\d4 q
\left\{ D_{ \Lambda}(q) D_{ \Lambda}(q+ \bar p)-
       D_{0}(q) D_{0}(q+ \bar p)
\right\}\,,
$$
where $ \bar p= \bar p_i+ \bar p_j,i \neq j$, and we used the symmetry of the
subtraction point \re{sp}.
For large $ \Lambda$ the range of integration is bounded by $q^2 \raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$<$}  \Lambda^2$
and we get
$$
 \gamma_4^{(1)}( \Lambda) \simeq \frac{3}{16\pi^2} g^2\ln( \Lambda/\mu)\,, \;\;\;\;\;
\mu \ll  \Lambda \,.
$$
For small $ \Lambda$ we have
$$
 \gamma_4^{(1)}( \Lambda) \sim  \Lambda^2\,\;\;\;\; \Lambda \ll \mu\,.
$$
The remaining part $\Delta_4$ of the four point vertex is obtained in a
similar way
$$
\eqalign{
\Delta_4^{(1)}(p_1,\cdots,p_4; \Lambda) =& -\mbox{\small $\frac{1}{2}$} g^2\int\d4 q D_{ \Lambda}(q)
\cr &
\times
\left\{ D_{ \Lambda}(q+p_1+p_2)+\cdots -3D_{ \Lambda}(q+ \bar p) \right\}\,,
}
$$
where the dots stand for the other two terms with $p_2$ replaced by
$p_3$ and $p_4$.
Due to the subtractions the integration is convergent for $q^2 \to \infty$.
For large infrared cutoff this integral vanishes as $\mu^2/ \Lambda^2$
and $P^2/ \Lambda^2$ with $P$ a combination of external momenta.
The physical value is obtained at $ \Lambda=0$ and one has
\begin{equation} \nome{onel1}
\Gamma_4^{(1)}(p_1,\cdots,p_4) = \frac{1}{32\pi^2} g^2
\left\{ \ln[\frac{(p_1+p_2)^2}{\mu^2}]+\cdots \right\}
\,.
\end{equation}
In the Appendix we use this results to obtain, in this formulation,
the one loop beta function.

For the vertices $\Gamma^{(1)}_{2n}$ with $n\ge 2$ we have
\begin{equation} \nome{onel2}
\Gamma_{2n}^{(1)}(p_1,\cdots,p_{2n}; \Lambda)=
- \frac {(-g)^n} {2n} \int\d4 q \, D_ \Lambda(q) \,
\; \sum_{perm} \prod_{k=1}^{n-1}D_{ \Lambda}
 \left( q+\sum_{\ell=1}^{2k}p_{i_\ell} \right)\,.
\end{equation}
The integral is convergent for large $q^2$.
At the physical value $ \Lambda=0$ these vertex functions become
singular for vanishing momenta.
However it is known \cite{CW} that the effective potential obtained by
summing the vertex functions for vanishing momenta is infrared finite.
We rederive here the one loop effective potential $V(\psi)$ in order to
illustrate in this framework the role of the regularization and the
physical conditions in \re{norm}.
Apart from a volume factor, $V(\psi)$ is given by $\Gamma[\psi]$ obtained
with the source $\psi(p)=(2\pi)^4\delta^4(p) \psi$.
For non vanishing $ \Lambda$ we get
$$
V(\psi)=\mbox{\small $\frac{1}{2}$}  \gamma_3^{(1)}\,\psi^2
+\frac {1}{4!}(g+ \gamma_4^{(1)})\, \psi^4 -\mbox{\small $\frac{1}{2}$} \int \d4  q \Theta(q^2- \Lambda^2)
\left\{ \sum_{n=2}^{\infty} \frac 1 n \left(\frac {-g\psi^2}{2q^2} \right)^n
- \frac {(g\psi^2)^2}{8 q^2(q+\bar p)^2} \right\}\,.
$$
The various terms diverge at $q=0$ for
$ \Lambda=0$. However, performing the sum and then taking $ \Lambda=0$, we have
$$
V(\psi)= \frac {g}{4!}\psi^4 + \mbox{\small $\frac{1}{2}$} \int \d4  q
\left\{ \ln  \left(1+\frac{g\psi^2}{2q^2}\right)
-\frac{g\psi^2}{2q^2}
+\frac{(g\psi^2)^2}{8 q^2(q+\bar p)^2}
\right\}\,.
$$
This expression does not have any infrared singularity for $q=0$
and the integral is convergent at large $q$ (see Ref.~\cite{CW}).

\vskip .3 true cm\noindent
2. {\it Two loop propagator.}
\vskip .2 true cm\noindent
By using the previous results we find
\begin{equation} \nome{two2}
\eqalign{
& \Gamma_2^{(2)}(p; \Lambda) =
\Gamma_2^{(2)}(p;0)-
\frac{g^2}{3!} \int\d4 q \Delta_ \Lambda(q) \int\d4 {q'}
\biggr\{ \Delta_ \Lambda(q')\Delta_ \Lambda(q+q'+p)
\cr&
+
\frac{3}{2}\biggr[ D^2_{ \Lambda}(q')
+2D_{0}(q')D_{ \Lambda}(q+q'+p)
-3D_{0}(q')D_{0}(q+q'- \bar p)\biggr]\biggr\} \,.
}
\end{equation}
The $q$-integration is bounded by the factor $\Delta_ \Lambda(q)$ which gives
$q^2\leq \Lambda^2$.
The $q'$-integration for the first term in the curly bracket
is also bounded by $ \Lambda$.
For the second term the $q'$-integration is convergent because of
subtractions.
The value of $\Gamma_2^{(2)}(p;0)$ is given by
$$
\Gamma_2^{(2)}(p,0)=-\frac{g^2}{3!}\int\d4 q\int\d4 {q'}
\frac{1}{q^2 q'^2}\biggr[\frac{1}{(q+q'+p)^2}-\frac{1}{(q+q')^2}
-p^2   \frac{\partial}{\partial p'^2} \frac{1}{(q+q'+p')^2}\biggr]
$$
where $p'^2=\mu^2$ and the integrations are convergent due to the
subtractions.

\section{Perturbative renormalizability}
In this section we prove that the theory is perturbative renormalizable,
namely that in \re{inte2}  we can set $\L_0\to\infty\;$.
As shown before, the loop expansion is obtained by iterating
Eqs.~ \re{inte1} and \re{inte2}.
{}From the vertices $\Gamma_{2n}^{(\ell)}$ one constructs the integrands at
the loop $\ell$ which give the next loop vertices upon $q$-integration.
The convergence of the integrals giving $\Gamma^{(\ell+1)}_{2n}$
will be ensured by dimensional counting, while the one for
$\Delta_2$ and $\Delta_4$ will require the subtractions in
$\Delta \Gamma_4$ and $\Delta  \bar \Gamma_6$ given in \re{inte3}.
The convenient way to represent the subtracted vertices $\Delta \Gamma_4$
and $\Delta  \bar \Gamma_6$ is by Taylor expansion as for
the Bogoliubov $R$ operators.
Since we are interested in the large $ \lambda$ behaviour we use the
expansion around vanishing momenta.
We need to consider only even derivatives since odd derivative
terms vanish for symmetry either directly or after integration.
The subtracted vertex $\Delta \bar \Gamma_6$ is obtained considering the expansion
\bminiG{taylor}
\nome{t1}
&  \bar \Gamma_6(q,p_1,\cdots, p_4,-q; \lambda) =  \bar \Gamma_6(q, 0,\cdots, 0,-q; \lambda) \nonumber \\
& +\int_0^1dx(1-x)\left(\sum_{i=1}^3 p_i \cdot \partial'_{i,4}\right)^2
 \bar \Gamma_6(q,p'_1,\cdots, p'_4,-q; \lambda)|_{p'_i=xp_i}
\end{eqnarray}\setcounter{equation}{\value{hran}
where $\partial'_{i,4}= {\partial}/{\partial p'_i}-
{\partial}/{\partial p'_4}$.
The first term, which is the most singular contribution, is
cancelled in the subtracted quantity $\Delta  \bar \Gamma_6$.
For $\Delta\Gamma_4$ we need to consider the expansion up to four derivatives
\addtocounter{equation}{-1}
\setcounter{hran}{\value{equation}
\refstepcounter{equation}
\nome{t3}
& \Delta\Gamma_4(q,p,-p,-q; \lambda)=
\frac 1{3!}\int_0^1dx(1-x)^3(p \cdot \partial')^4
\Gamma_4(q,p',-p',-q; \lambda)|_{p'=xp} \nonumber \\
& -\mbox{\small $\frac{1}{2}$} (p \cdot \partial')^2
\left\{ \Gamma_4(q,p',-p',-q; \lambda)|_{p'^2=\mu^2}-
       \Gamma_4(q,p',-p',-q; \lambda)|_{p'=0} \right\}
\end{eqnarray}\setcounter{equation}{\value{hran}
where $\partial'=\partial/\partial p'$.
Notice that also the second term can be expressed by the fourth derivative
of $\Gamma_4$. Similarly the integrand for $ \gamma_2$ can be expressed in terms of
the second derivative of $\Gamma_4$ with respect to the momentum components.

In order to prove that the theory is perturbatively renormalizable we
have to analyse the behaviour for large $ \lambda$ of the vertices in the
integrands and show that the integration over $ \lambda$ is convergent for
$ \lambda\to\infty$ (the convergence of the integrals for $ \lambda\to 0$ will be
discussed in the next section).
In this analysis we are not interested in the detailed dependence
of the vertices on the external momenta, except
for the fact that the integration momentum is fixed at $q^2= \lambda^2$
(see \re{esse1}).
To prove perturbative renormalizability it will be sufficient,
as in \cite{P}, to bound the large $ \lambda$ behaviour of the vertices in
which all external momenta do not exceed the cutoff.
We then introduce the following functions which depend only on $ \lambda$
\begin{equation}\nome{norma}
|f_{2n}|_{ \lambda}\equiv
\raisebox{-1.5 ex}{\rlap{\tiny $\;\;p_i^2<c\l^2$}
|f_{2n}(p_1,\cdots, p_{2n}; \lambda)|
\end{equation}
where $c$ is some numerical constant and $f(p_1,\cdots, p_n; \lambda)$ is
$\Gamma_{2n}$, $ \bar \Gamma_{2n+2}$ or one of their derivatives.
Iterating \re{inte1} and \re{inte2} in which we set $\L_0\to\infty\;$,
we obtain the following bounds.
For the relevant couplings
\bminiG{e1}
\nome{e1a}
 \gamma^{(\ell+1)}_2( \Lambda)
&\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$<$}& \int_0^{ \Lambda^2}d \lambda^2s^{(\ell-\ell')}( \lambda)
|\partial^2 \Gamma^{(\ell')}_4|_{ \lambda} \;, \\
\nome{e1b}
 \gamma^{(\ell+1)}_3( \Lambda)
&\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$<$}& \int_0^{ \Lambda^2}d \lambda^2s^{(\ell-\ell')}( \lambda)
|\Gamma^{(\ell')}_4|_{ \lambda} \;, \\
\nome{e1c}
 \gamma^{(\ell+1)}_4( \Lambda)
&\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$<$}& \int_0^{ \Lambda^2}d \lambda^2s^{(\ell-\ell')}( \lambda)
| \bar \Gamma^{(\ell')}_6|_{ \lambda} \;.
\end{eqnarray}\setcounter{equation}{\value{hran}
For the irrelevant vertices
\bminiG{e2}
\nome{e2a}
|\Gamma^{(\ell+1)}_{2n}|_{ \Lambda}
&\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$<$}& \int_{ \Lambda^2}^{\infty }d \lambda^2s^{(\ell-\ell')}( \lambda)
| \bar \Gamma^{(\ell')}_{2n+2}|_{ \lambda} \;, \\
\nome{e2b}
|\Delta_2^{(\ell+1)}|_{ \Lambda}
&\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$<$}&  \Lambda^4 \int_{ \Lambda^2}^{\infty }d \lambda^2s^{(\ell-\ell')}( \lambda)
|\partial^4 \Gamma^{(\ell')}_{4}|_{ \lambda} \;, \\
\nome{e2c}
|\Delta_4^{(\ell+1)}|_{ \Lambda}
&\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$<$}&  \Lambda^2 \int_{ \Lambda^2}^{\infty }d \lambda^2s^{(\ell-\ell')}( \lambda)
|\partial^2  \bar \Gamma^{(\ell')}_{6}|_{ \lambda} \; .
\end{eqnarray}\setcounter{equation}{\value{hran}
For the derivatives of vertices
\bminiG{e3}
\nome{e3a}
|\partial^m\Gamma_{2n}^{(\ell+1)}|_{ \Lambda}
&\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$<$}& \int_{ \Lambda^2}^{\infty} d \lambda^2 s^{(\ell-\ell')}( \lambda)
|\partial^m \bar \Gamma_{2n+2}^{(\ell')}|_{ \lambda} \; , \\
\nome{e3b}
|\partial^m\Delta_2^{(\ell+1)}|_{ \Lambda}
&\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$<$}&  \Lambda^4\,\int
_{ \Lambda^2}^{\infty} d \lambda^2 s^{(\ell-\ell')}( \lambda)
|\partial^{m+4}\Delta_4^{(\ell')}|_{ \lambda} \; , \\
\nome{e3c}
|\partial^m\Delta_4^{(\ell+1)}|_{ \Lambda}
&\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$<$}&  \Lambda^2\,\int_{ \Lambda^2}^{\infty} d \lambda^2
s^{(\ell-\ell')}( \lambda)|\partial^{m+2} \bar \Gamma_6^{(\ell')}|_{ \lambda} \; .
\end{eqnarray}\setcounter{equation}{\value{hran}
where $\partial^m$ stands for $m$ partial derivatives with respect to
momentum components and the factors $ \Lambda^2$ and $ \Lambda^4$ in front of
integrals come by maximizing the $p^2$ or $p^4$ factors in
\re{t1} and \re{t3} respectively. Actually $\partial^m$ in \re{e3b}-\re{e3c}
could also act on these $p$ factors. As we will show in the following all
these contributions are of the same order.

We now prove by induction that the theory is perturbatively renormalizable,
namely that the integrals in \re{e2} and \re{e3} are convergent for
$ \lambda \to \infty$.
By using the majorization in \re{e1}-\re{e3}, as one can expect, the
proof becomes very simple.

\noindent
1) {\it Assumptions at the loop $\ell$.}

\noindent
The assumptions we make are simply given by dimensional counting except for
logarithmic corrections and they involve the nine quantities above.

\noindent
a) Relevant couplings ($T=\log( \Lambda/\mu)$)
\begin{equation} \nome
 {a1}
 \gamma^{(\ell)}_2( \Lambda)= {\cal O}(T^{\ell-1}),\;\;\;\;
 \gamma^{(\ell)}_3( \Lambda)= {\cal O}( \Lambda^2 T^{\ell-1}),\;\;\;\;
 \gamma^{(\ell)}_4( \Lambda)= {\cal O}(T^{\ell})\,.
\end{equation}
\noindent
b) Irrelevant vertices
\begin{equation} \nome {a2}
|\Gamma_{2n}^{(\ell)}|_{ \Lambda}={\cal O}( \Lambda^{4-2n}T^{\ell-1}),\;\;\;\;
|\Delta_2^{(\ell)}|_{ \Lambda}={\cal O}( \Lambda^{2}T^{\ell-2}),\;\;\;\;
|\Delta_4^{(\ell)}|_{ \Lambda}={\cal O}(T^{\ell-1})\,,
\end{equation}
\noindent
c) Derivative vertices
\begin{equation} \nome {a3}
|\partial^m\Gamma_{2n}^{(\ell)}|_{ \Lambda}={\cal O}( \Lambda^{4-2n-m}T^{\ell-1}),\;\;\;\;
|\partial^m\Delta_2^{(\ell)}|_{ \Lambda}={\cal O}( \Lambda^{2-m}T^{\ell-2}),\;\;\;\;
|\partial^m\Delta_4^{(\ell)}|_{ \Lambda}={\cal O}( \Lambda^{-m} T^{\ell-1})\,.
\end{equation}
These assumptions are satisfied for $\ell=0$ and $1$.

\noindent
2) {\it Iteration to the loop $\ell+1$.}

\noindent
Notice that the powers of $ \Lambda$ in \re{a1}-\re{a3} are independent of the
number of loops since they are dictated by dimensional counting.
For the relevant (irrelevant) couplings the integrands increase (decrease)
with $ \lambda$ thus the integrals are dominated by the upper (lower) limit
$ \lambda= \Lambda$. For the irrelevant couplings we can therefore take the limit
$\L_0\to\infty\;$, removing the ultraviolet cutoff.
It is simple to see that the integrals in \re{e1}-\re{e3}
reproduce at loop $\ell+1$ the same dimensional counting behaviours.
This is just what
is needed to prove the perturbative renormalizability, since logarithmic
corrections cannot change the power counting at any finite order.
Actually it is relatively simple to control also the powers of $T$
and in the following we show that the behaviours \re{a1}-\re{a3} are
reproduced by the iteration.

Before discussing the large $ \Lambda$ behaviours at the $\ell+1$ loop we
derive from \re{a1}-\re{a3} some intermediate results for the
integrands at loop $\ell$.

\noindent
a) From the two point function and \re{esse1} we have
$$
s^{(\ell)}( \lambda)\sim t^{\ell-1} \,,
$$
where $t\equiv\log( \lambda/\mu)$.

\noindent
b) The leading term of the auxiliary vertices $ \bar \Gamma_{2n+2}$
is given by the contribution of Fig.~4 in which only four point vertices
are involved
$$
| \bar \Gamma_{2n+2}^{(\ell)}|_{ \lambda}\sim \lambda^{2-2n}\prod_1^n \gamma_4^{(\ell_i)}( \lambda)
\sim \lambda^{2-2n}t^{\ell}\,,
$$
where $\sum\ell_i=\ell$ and we have a factor $ \lambda^{-2}$ for each internal
propagator. All other contributions coming from higher vertices and from
loop corrections in the intermediate propagators give the same
power in $ \lambda^2$ but a lower power in $t$.

\noindent
c) The leading term of the derivatives of the auxiliary vertices
is again obtained from the contribution of Fig.~4
in which the derivatives act on the internal propagators
$$
|\partial^m \bar \Gamma^{(\ell)}_{2n+2}|_{ \lambda}\sim \lambda^{2-2n-m}t^{\ell}\,.
$$
Again, the contributions from derivatives of higher vertices or from
loop corrections of internal propagators give lower powers of $t$.

By using these results in \re{e1}-\re{e3} we reproduce at $\ell+1$ loop
order the behaviours in \re{a1}-\re{a3}
In all cases we have $\ell'=\ell$; \hbox{\it i.e.}{ }}      \def\etc{\hbox{\it etc.}{ } loop corrections of the
propagator in $s( \lambda)$ are not contributing to the leading terms.

\section{Infrared behaviour}
In this section we show that for the massless scalar theory,
the vertex functions at non exceptional momenta are finite order by
order in perturbation theory.
Namely we prove that the integration over $q$ in \re{inte1} and \re{inte2}
is convergent at the lower limit when we take $ \Lambda \to 0$.
As in the case of renormalizability, this is shown by induction on the
number of loops.

The integrands in \re{inte1} and \re{inte2} are given by vertices
with one pair $(q, -q)$ of exceptional momenta. Thus, by iteration,
one introduces vertices with any number of pairs of
exceptional momenta.
In general we say that a pair of momenta $p_i$ and $p_j$
in $\Gamma_{2n}(\{p_i\}; \Lambda)$ is exceptional if $p_i+p_j={\cal O}( \Lambda)$.
In the following we add an index to the vertex functions
to identify the number of pairs of exceptional momenta. We write
$$
\Gamma_{2n,s}(\{p_i\}; \Lambda) \equiv \Gamma_{2n}(\{p_i\}; \Lambda) \,,
\;\;\;\;\mbox {for}\;\;\;\;
(p_{i_1}+p_{i_2}) , \cdots ,
(p_{i_{2s-1}}+p_{i_{2s}})= {\cal O}( \Lambda)\,.
$$
where $s=0,\ldots,n-1$.
For $s=n-1$ all pairs of momenta are exceptional.
Therefore we denote by $\Gamma_{2n,0}$ the vertices  without exceptional
momenta. Similar notation will be used for the auxiliary vertices
$ \bar \Gamma_{2n+2,s+1}$ with $s+1$ pairs of exceptional momenta.

\vskip 0.3 cm \noindent
1) {\it Assumptions at the loop $\ell$}
\vskip 0.2 cm \noindent
For $ \Lambda \to 0$ we assume the following behaviours
\bminiG{asx}
\nome{as1}
\Gamma_{2n,0}^{(\ell)}( \Lambda) &\to& \mbox{finite} \,, \\
\nome{as2}
\Gamma_{2n,1}^{(\ell)}(p_1, \cdots, p_{2n} ; \Lambda) &=& {\cal O}( T^{\ell}) \,, \\
\nome{as3}
\Gamma_{2n,s}^{(\ell)}(p_1,\cdots, p_{2n};  \Lambda) &=& {\cal O}(  \Lambda^{2-2s}T^{\ell-1})
\, \;\;\;\; s=2 \cdots n-1\,, \\
\nome{as4}
\Gamma_{2}^{(\ell)}(p; \Lambda) &=& {\cal O}(  \Lambda^2\,T^{\ell-1})
\, \;\;\;\; \mbox{for} \; p^2= {\cal O}(  \Lambda^2)\, ,  \\
\nome{as5}
\frac {\partial}{\partial p_{\mu}}
\Gamma_{2n,n-1}^{(\ell)}(p,-p,p_1,\cdots, p_{2n-2};  \Lambda)
&=& {\cal O}(  \Lambda^{4-2n}T^{\ell-1} \; \frac {p_{\mu}+P_{\mu}}{ \Lambda^2} )
\, ,
\end{eqnarray}\setcounter{equation}{\value{hran}
where $T=\ln( \Lambda/\mu)$.
The first equation states the most important result.
In \re{as5} all momenta are of order $ \Lambda$ and $P_{\mu}$
stands for any combination of $p_i$'s.
These assumptions are satisfied for $\ell=0$ and $1$ (see Subsection
2.3).

\vskip 0.3 cm \noindent
2) {\it Iteration at loop $\ell+1$}
\vskip 0.2 cm \noindent
Before discussing the $ \Lambda \to 0$ behaviour at loop $\ell+1$ we derive from
\re{asx} some intermediate results for the vertices at loop
$\ell$ entering in the integrands of \re{inte1} and \re{inte2}

\noindent
a) From the two point function we have
\begin{equation}\nome{ci1}
s^{(\ell)}( \lambda) = {\cal O}(t^{\ell-1})\,,
\end{equation}
where $t=\log( \lambda/\mu)$.

\noindent
b) For the four point function we have
$$
\frac{\partial }{\partial p^2}
\Gamma_{4,1}^{(\ell)}(q,p,-p,-q; \lambda)|_{p^2=\mu^2,\,q^2= \lambda^2}
={\cal O} ( t^{\ell})\,,
$$
$$
\Delta \Gamma_4^{(\ell)}(q,p,-p,-q; \lambda)|_{q^2= \lambda^2} = {\cal O}(t^{\ell}) \,.
$$
The first equation is obtained from \re{as2} observing that the derivative is
evaluated at non vanishing momentum, while the second is
obtained from the behaviour of the single terms in
the subtracted vertex $\Delta\Gamma_4$. Actually there are cancellations
giving a less singular behaviour, but we do not need to analyse them.

\noindent
c) For the auxiliary vertices with $s=0$ we have
\begin{equation}\nome{ci6}
 \bar \Gamma^{(\ell)}_{2n+2,1}(q,p_1,\cdots,p_{2n},-q; \lambda)|_{q^2= \lambda^2}\sim
\Gamma_{2n+2,1}^{(\ell)}( \lambda)\sim t^{\ell} \,.
\end{equation}
This behaviour is given by the first contribution of Fig.~2.
Since only the pair $(q,-q)$ is exceptional, the other terms
involve vertices with non exceptional momenta which are finite.

\noindent
d) For the auxiliary vertices with $s>0$ we have
\begin{equation}\nome{ci10}
 \bar \Gamma_{2n+2,s+1}^{(\ell)}(q,p_1,\cdots,p_{2n},-q; \lambda)|_{q^2= \lambda^2}\sim
 \lambda^{-2s}t^{\ell}\,,\;\;\;\;\;\;\;s\ge 1\,.
\end{equation}
This behaviour is obtained by taking the largest number of
internal propagators at momentum $q^2= \lambda^2$.
This is given by the contribution of Fig.~5 in which the $s$
pairs of exceptional momenta
among the $p_i$'s are emitted in the four point functions to
the left (or right). In this way we have $s$ internal propagators at
momentum $q^2= \lambda^2$ giving the factor $ \lambda^{-2s}$ in \re{ci10}.
Loop corrections to the internal propagators give non leading
logarithmic powers.

\noindent
e) For the derivatives of the auxiliary vertices with all
exceptional momenta we have
\begin{equation}\nome{ci13}
\frac {\partial}{\partial p_{\mu}}
 \bar \Gamma_{2n+2,n}^{(\ell)}(q,p,-p,p_1\cdots p_{2n-2},-q;  \lambda)
={\cal O}(  \lambda^{4-2n}t^{\ell}\; \frac {p_{\mu}+q_{\mu}+ P_{\mu}}{ \lambda^4} )
\,,\;\;\;\;\;\;n>1\,.
\end{equation}
This behaviour is given by the graphs of Fig.~6. Here $P_{\mu}$ is
a combination of $p_i$'s and the derivative acts on
a propagator connecting the two vertices with $p$ or $-p$ incoming.
All other contributions with higher vertices, with loop corrections
in the propagators or with the derivatives acting on vertices lead to
lower powers of $t$.

We now deduce the $ \Lambda \to 0$ behaviour of the $\ell+1$ vertices.

\vskip 0.3 true cm \noindent
a) For the two point function we find
$$
 \gamma_2^{(\ell+1)}( \Lambda) = {\cal O}( \Lambda^2 T^{\ell}) \to 0 \, ,
$$
$$
 \gamma_3^{(\ell+1)}( \Lambda)
= {\cal O}( \Lambda^2 T^{\ell}) \to 0 \,,
$$
$$
\Delta_2^{(\ell+1)}(p; \Lambda) = {\cal O}( \Lambda^{0}) \, .
$$
One can show that the behaviour for $ \gamma_2^{(\ell+1)}$ can be
improved to $ \Lambda^2 T^{\ell-2}$.
These results show that $\Gamma_2^{(\ell+1)}$ is finite for $ \Lambda \to 0$,
thus verifying the main assumption in \re{as1} for the case $n=1$.

To study the behaviour of $\Delta_2^{(\ell+1)}$ for $p^2={\cal O}( \Lambda^2)$
we use the Taylor expansion
\begin{equation}\nome{ci51}
\eqalign{
\Delta_2^{(\ell+1)}(p; \Lambda)= &
\mbox{\small $\frac{1}{2}$}\int_{ \Lambda} \d4 q \frac{s^{(\ell-\ell')}( \lambda)}{q^2}
\cr &
\times
\int_0^1 dx (p \cdot \partial')
\Gamma_{4,1}^{(\ell')}(q,p',-p',-q; \lambda)|_{ \lambda^2=q^2,\; p'=xp}
={\cal O}(p^2T^{\ell}).
}
\end{equation}
This proves \re{as4} at loop $\ell+1$.
All these behaviours  are obtained by ignoring loop corrections in
$s( \lambda)$ since they would give lower powers of $T$.

\vskip 0.3 true cm \noindent
b) For the four point function at non exceptional momenta
we obtain from \re{ci6}
\begin{equation}\nome{ci7}
 \gamma_4^{(\ell+1)}( \Lambda) = {\cal O}( \Lambda^2 T^{\ell}) \to 0 \,,\;\;\;\;\;\;
\Delta_4^{(\ell+1)}(p_1,\cdots,p_4; \Lambda)
= {\cal O}( \Lambda^{0}) \,,
\end{equation}
where again for the leading terms we do not have contributions from
loop corrections in $s( \lambda)$.

For $\Gamma_{2n,0}^{(\ell+1)}$ with $n>2$ we obtain from \re{ci6}
\begin{equation}\nome{ci9}
\Gamma_{2n,0}^{(\ell+1)}(p_1,\cdots,p_{2n}; \Lambda)={\cal O}( \Lambda^{0})\,.
\end{equation}
We conclude from \re{ci7}-\re{ci9} that all physical vertices
at $ \Lambda=0$ and non exceptional momenta
are finite.
Moreover, as required by the condition $ \gamma_4 (0)=g$,
the relevant coupling $ \gamma_4^{(\ell+1)}( \Lambda)$ vanishes
at $ \Lambda=0$.
This verifies at loop $\ell+1$ the main assumption \re{as1}.

\vskip 0.3 true cm \noindent
c) For the vertices with pairs of exceptional momenta
we prove \re{as2}-\re{as3} at the $\ell+1$ loop order
by using the result in \re{ci10}.

\vskip 0.3 true cm \noindent
d) For the vertices with all exceptional momenta
we prove \re{as5} at the $\ell+1$ loop order.
For $n=1$ this is simply obtained by taking the derivative of
Eq.~\re{ci51} with respect to $p_{\mu}$.
For $n>1$ this can be done by writing
$$
\frac {\partial}{\partial p_{\mu}}
\Gamma_{2n,n-1}^{(\ell+1)}(p,-p,p_1,\cdots, p_{2n-2};  \Lambda)
$$
$$
=\mbox{\small $\frac{1}{2}$}\int_{ \Lambda} \d4 q \frac{s^{(\ell-\ell')}( \lambda)}{q^2}
\frac {\partial}{\partial p_{\mu}}
 \bar \Gamma_{2n+2,n}^{(\ell')}(q,p,-p,p_1,\cdots, p_{2n-2},-q;  \lambda)|_{ \lambda^2=q^2}
$$
and using \re{ci13}.

\section{Final comments}

We have used the renormalization group technique to obtain in a constructive
way equations for
the renormalized vertices of a scalar massless theory. The loop expansion
is derived from the iterative solution of these equations
and is an expansion in the physical coupling $g$.
In this approach subtractions come from the fact that we isolated
(see Eq.~\re{isol}) in the two and four point functions the three relevant
vertices $ \gamma_i( \Lambda)$ for which we have to impose the boundary conditions
\re{norm} at the physical value $ \Lambda=0$.
This procedure allows us to generate in a systematic way the subtracted
vertices $\Delta\Gamma_4$ and $\Delta \bar \Gamma_6$ in \re{inte3} which, after integration,
give $\Delta_2$ and $\Delta_4$, the ``irrelevant'' part of the two and four
point function.
These subtracted vertices and the corresponding counterterms obtained
from $ \gamma_i( \Lambda)$ evaluated at $ \Lambda= \Lambda_0$, can be expanded in the
physical coupling $g$.
This procedure of constructing the subtracted vertices is very similar to
the one of Bogoliubov $R$-operators \cite{Bo} but it seems to be based
on a more physical ground.
Once the subtracted vertices are identified, the proof of
renormalizability depends only on dimensional counting.

In this formulation the simplifications of the analysis of the
ultraviolet and infrared behaviour of the integrands of Feynman graphs
is based on the fact that, at a given loop $\ell+1$, the momentum $q$
in \re{inte1} and \re{inte2} sets the cutoff for the vertices at the
previous loop. The difference between relevant and irrelevant couplings
is in the range of $q$-integrations with respect to the cutoff $ \Lambda$.
We have then that the vertices are given by combinations of Feynman graphs
in which the internal loop momenta are ordered.
This ordering depends on whether we are generating a relevant or
irrelevant vertex. It is clear that once we have an ordering in the
momenta, the asymptotic behaviour of Feynman graphs contributions can be
studied by iterative procedure.

\vspace{3mm}\noindent{\large\bf Acknowledgements}

We have benefited greatly from discussions with C.\ Becchi and
G.\ Korchemsky.

\eject
\newpage

\begin{center}
{\Large \bf Appendix
}\end {center}

We discuss in the present formulation the role of the subtraction
point $\mu$ and deduce the beta function (see also Ref.~\cite{H}).
Denote by $\Gamma_{2n}(g,\mu)= \Gamma_{2n}(p_1,\cdots,p_{2n};g,\mu)$
the vertices defined by \re{inte1} and \re{inte2} which satisfy the
physical conditions \re{norm} with coupling $g$ at the scale $\mu$.
We assume the theory renormalizable and infrared finite so we set
$ \Lambda=0$ and $\L_0\to\infty\;$. Therefore $\mu$ is the only dimensional parameter,
apart for the external momenta.
However physical measurements should not depend on the specific
value of $\mu$.
Consider the vertices $\Gamma_{2n}(g',\mu')$ with coupling $g'$ at a
new scale $\mu'$.
The request that the two sets of vertices  $\Gamma_{2n}(g,\mu)$
and $\Gamma_{2n}(g',\mu')$ describe the same theory
implies that the corresponding effective actions
$\Gamma[\psi;g,\mu]$  and $\Gamma[\psi';g',\mu']$ are equal, where the two
fields $\psi$ and $\psi'$ are related by a rescaling,
$\psi'(p)=\sqrt{Z_2}\psi(p)$.
This implies that the two sets of vertices are related by
$$
\Gamma_{2n}(g',\mu')=Z_2^{-n} \Gamma_{2n}(g,\mu) \,.
$$
Notice that the evolution equations \re{eveq} are invariant with respect to
this rescaling, provided we rescale also the propagator
$D_{ \Lambda, \Lambda_0}(p)$ accordingly.
This implies, as usual, that $Z_2$ is fixed only by the physical
condition \re{norm}.
To obtain $Z_2$ we use the decomposition \re{isol}
$$
\Gamma_2(p;g,\mu)=p^2+\Delta_2(p;g,\mu)\,,
$$
where
$$
\Delta_2(0;g,\mu)=0\,, \;\;\;\;\;
\frac{\partial}{\partial p^2}\Delta_2(p;g,\mu)|_{p^2=\mu^2}=0\, ,
$$
as required by the physical conditions \re{norm}.
We use the corresponding decomposition and normalization for the
vertex $\Gamma_2(p;g',\mu')$.
The renormalization function $Z_2$ is obtained
by expanding $Z_2\Gamma_2(p;g',\mu')=\Gamma_2(p;g,\mu)$ at $p^2=\mu^2$ or
$p^2=\mu'^2$. We find
$$
Z_2=1+a_2(\mu';g,\mu) \,=\, \frac {1}{1+a_2(\mu;g',\mu')} \,,
$$
where $a_2$ is the dimensionless quantity
$$
a_2(\mu';g,\mu) \equiv
\frac{\partial}{\partial p^2}\Delta_2(p;g,\mu)|_{p^2=\mu'^2}\, .
$$
Similarly we use the decomposition \re{isol} for the four point function
$$
\Gamma_4(p_1,\cdots,p_4;g,\mu)=g+\Delta_4(p_1,\cdots,p_4;g,\mu)\,,
$$
where $\Delta_4( \bar p_1,\cdots, \bar p_4;g,\mu)=0$  at the symmetric point
in \re{sp}.
Again we use the corresponding decomposition and normalization for the
vertex $\Gamma_4(p_1,\cdots,p_4;g',\mu')$.
Introducing the dimensionless quantity
$$
a_4(\mu';g,\mu) \equiv \Delta_4( \bar p'_1,\cdots, \bar p'_4;g,\mu)\,,
\;\;\;\;\;
 \bar p'_i \bar p'_j=\mu'^2(\delta_{ij}-\frac 14)\;,
$$
from $Z^2_2\Gamma_4(g',\mu')=\Gamma_4(g,\mu)$ we obtain the renormalization
group relation
$$
g'= \frac{g+a_4(\mu';g,\mu)}{\left( 1+a_2(\mu';g,\mu) \right)^2}\;.
$$
The beta function is obtained by considering an infinitesimal
scale change and is given by
$$
\beta(g)=\mu'\partder{\mu'}
\left\{ a_4(\mu';g,\mu)-2g\,a_2(\mu';g,\mu) \right\}|_{\mu'=\mu}\,.
$$
Since the theory is renormalizable and infrared finite,
the two quantities $a_i(\mu';g,\mu)$ are functions of $g$ and the
ratio $\mu'/\mu$, thus and the beta function depends only on $g$.
At the one loop order we find $a^{(1)}_2$ independent of $\mu'$
and the beta function is obtained only from $a^{(1)}_4$ given in
\re{onel1}.
We find
$$
\beta^{(1)}(g)=
\frac {3}{32\pi^2}g^2 \mu'\partder{\mu'} \ln\frac{\mu'^2}{\mu^2}
=\frac {3}{16\pi^2}g^2 \,,
$$
which is the usual one loop result.

\eject
\newpage

redpajama_set_name: RedPajamaArXiv


text:
Vladimir Putin and the Household Budget
When Sebastian Coe, a multiple Olympic medalist in the 1500 metres discipline, and currently president of the International Association of Athletics Federations, announced on the 17th of June that the Russian Track and Field team would continue to be banned from international competition, my first thought was that the inner child within Vladimir Putin must have thrown a tantrum. The unanimous vote by the 27 members of the IAAF board to effectively exclude the Russian Athletics team from the 2016 Rio Olympics must have felt particularly unjust to the man who had orchestrated the spending of more than 50 billion US dollars to hold the 2014 Winter Olympics in Sochi.
This Russian recreational town, nestling the Eastern shores of the Black Sea, enjoys balmy weather throughout the year. Sochi has a humid climate with mild winters. The average temperature from December through March is 11 degrees during the day and 4 degrees at night. Owing to the relative lack of snow in Sochi itself, enormous effort was made to transport both competitors and fans between the mountain ski slopes some 40 kilometres away and then back to the infrastructure in Sochi. In Sochi, events such as curling and figure skating were held. At the slopes, snowboarding and alpine skiing were on offer.
To put this somewhat into perspective, imagine if the Winter Olympics for 2018 were to be held in Pietermaritzburg, and that the Drakensberg Mountains were used for the downhill and slalom ski events, whilst Town Hall in Maritzburg was converted into an ice rink. Then imagine a railway track being constructed between the two locations. That would be something along the lines of what Putin achieved with Sochi. In fact, Pietermaritzburg in mid-winter can be chillier than Sochi in January.
Days later after Sebastian Coe's statement, when Russia hinted at the potential sale of 20 percent of one of its crown jewels, the state energy company Rosneft, it became clear that Putin had moved past anger as his primary emotion and had migrated rapidly into the acceptance stage of the five stages that constitute the experience of loss. Whilst Putin may play the aggressor on the international stage he has been very much the granny of belt-tightening on his home turf.
One must feel, however, some sympathy for the man. Since Sochi, the oil price has collapsed, Russian gross domestic product per capita has declined from US 14,000 dollars in 2013 to just US 8,000 US dollars in 2016. Even the executive heads of state-sponsored businesses are being ferried around the traffic-infested streets of Moscow in Toyotas today rather than in the limousines of the past.
This new ascetic version of Putin and his adoption of Greek-style economics reminds me of a friend who threw an enormous party that went on all night in the penthouse suite of the Michelangelo Towers. Moet and Chandon as well as Chivas Regal 18 year-old was flowing freely and the DJ played into the early hours of the morning. By all accounts, a rollicking time was had of it. It was only a few months later that he sold a particularly cherished sports car to help fund paying off the bond on his Cape Town home.
Putin's cherished Rosneft, a firm that was begotten out of the chaos that ensued following the collapse of communism is now up for grabs by Chinese and Indian investors. According to the latest reports these are the only 'partners' that Putin desires as potential acquirers of the 20 percent Rosneft stake.
To put this sale somewhat into perspective, the 20 percent stake the Russian government is thinking of selling will bring in approximately US 11 billion dollars of cash. That is something like 20 percent of what Putin spent on the Winter Olympics in Sochi – which would suggest that he will have to find a few more state-assets to flog. Perhaps even a sports car or two.
Putin must have felt some relief though on learning this week that the Russian Judo team, in spite of much invective from sports bodies around the world, has been given the go-ahead to compete in the Rio Olympics. If I am not mistaken Vladimir Putin is a Judo fan, practicing the sport himself – on occasion without a shirt and in front of domestic cameras. Or perhaps I am mistaken. Perhaps it was on one of his hunting trips…

redpajama_set_name: RedPajamaCommonCrawl


text:
BIOSync CLS
Cardiac Device App
MRI System Check
Edora 8 DR-T/SR-T
Eluna 8 DR-T/SR-T
Itrevia 7 VR-T DX
Linox smart
Iperia 7 DR-T/VR-T
Evia HF-T
Inventra 7 HF-T QP
Iperia 7 HF-T
Itrevia 7 HF-T
Ilesto 7 HF-T
Edora 8 HF-T QP/HF-T
Guide Wires
VisionWire
Cruiser and Cruiser Hydro
Passeo-35 HP
The following cookie settings apply to the use of the BIOTRONIK websites www.biotronik.com, www.magmaris.com, www.orsiro.com, www.cortronik.com, www.vascomed.com and manuals.biotronik.com.
Cookies are small text files or comparable storage technologies that are stored on your end device by your browser and that enable your browser to be recognized. We use so-called session cookies (temporary cookies) and permanent (persistent) cookies. Session cookies are only cached for the duration of your use of our website. These cookies are required for transactions (such as logging into a user account) and are valid until the end of the browser session. This means that depending on the type and settings of your browser, the cookies are automatically removed after the end of the session or after the tab or browser is closed, unless you have configured your browser settings otherwise. The session cookies we use only contain a session ID. Persistent cookies are stored on your computer for future sessions in order to be able to recognize returning visitors even after a long period of time and to be able to offer products or services tailored to your needs. Permanent cookies are permanently saved on your device and are not deleted when you leave the website or close your browser. If permanently set cookies are to be removed, this must be done manually.
Required cookies are used to make a website usable by enabling basic functions such as page navigation and access to secure areas of the website (shopping basket cookies, cookies for log-ins, cookies that affect a country or language selection, cookies by cookie consent tools Consent). The website cannot function properly without these cookies. These cookies cannot be deactivated. You can set your browser to block these cookies or to notify you about these cookies. Some areas of the website may not work.
Statistics cookies help us understand how visitors interact with our website. We only use information without personal reference (e.g. through shortened IP addresses).
Functional cookies enable the provision of extended functionality and personalization, such as B. Videos and live chat. If you do not allow these cookies, some or all of these functions may not work properly.
Analysis cookies are used by us or third parties for tracking and analyzing user behavior, for delivering user-based content or for marketing activities. This allows us to present the website content in a target group-specific manner and to improve the content and functionality of the website. If you do not allow these cookies, you will e.g. B. experience no user-based content or targeted offers on other websites.
You can find more information in our
Save Selected Cookies Save All Cookies
Claudia (at BIOTRONIK since 2005)
I began my career at BIOTRONIK at the age of 19. After high school, I decided to pursue a "cooperative education" program, a combination of vocational training and academic study. I studied at the University of Applied Sciences in Berlin while working at BIOTRONIK for three years, which allowed me to gain valuable and practical work experience. I obtained my Bachelor of Arts in Economics with a specialization in Industry.
During the practical phases of my internship, I learned much about BIOTRONIK and the various activities of its departments. I started in Customer Sales Support, Purchasing, and Pacemaker Production Planning. After that, I moved to Human Resources, followed by the Treasury department.
In the summer of 2007, I had the great opportunity to work for the BIOTRONIK France subsidiary in Paris, where I worked on various projects in various departments. I discovered my fascination for the combination of HR issues and IT expertise. I finished my studies in September 2008 and started my job in the Human Resources Controlling department. Today I am leading this department. Some people may think Human Resources Controllers are just creative bean counters, but I find it very interesting, that a combination of lateral thinking and precision along with a significant knowledge of organizational structure and management is also required.
For me, BIOTRONIK is an innovative company that offers a variety of opportunities to develop oneself. I am fascinated by our products and the fact that they help to improve patients' quality of life. And the thing I appreciate most about BIOTRONIK is the personal contact and the good atmosphere.
360°CRT
Heart Monitoring
Do you want to leave this website?
By clicking the links below to access the news on our International website, you are leaving this website.
Cancel Go out


text:
Q: Derivative for the asymptotics of integral Let $A=[-a,a]$, with $a>2$. I have a function $g(x)$ continuous in $A$ such that
$$\lim_{s\rightarrow\infty} \int_A x^{2s}f(x)dx = \int_A x^{2s}g(x)dx.$$  Can I claim that $f(x) = g(x)$ for $x\in A$ ?
What would be the reason for that answer.

A: It is unclear what you mean by $f(x) \to g(x)$ since there is no limit being taken there; perhaps you mean $f(x) = g(x)$ instead?
In this case, the answer is no. Consider when $a < 1$. then $x^{2s} \to 0$ uniformly on $[-a,a]$ as $s \to \infty$. Thus for any bounded $f$ we see $x^{2s}f(x) \to 0$ uniformly as well and so $$\int_A x^{2s} f(x) \, dx \to 0 = \int_A x^{2s}g(x) \, dx$$ where $g(x)$ is identically zero. But it is certainly not that case that any bounded function $f$ is the zero function.
However, we can say that if $$\int_A x^{n} f(x) dx = \int_A x^n g(x) dx$$ for all $n \in \mathbb N$, then $f(x) = g(x)$ for all $x \in A$. This follows from the Stone-Weierstrass theorem.

redpajama_set_name: RedPajamaStackExchange


text:
Wednesday, 19 January 2022, 02:29
Japan Pavilion at Epcot
Japan in Gulliver's Travels
Japan (Buronson manga)
Renewable energy is generally defined as energy that is collected from resources which are naturally replenished on a human timescale, such as sunlight, wind, rain, tides, waves, and geothermal heat. Renewable energy often provides energy in four important areas: electricity generation, air and water heating/cooling, transportation, and rural (off-grid) energy services.
Based on REN21's 2014 report, renewables contributed 19 percent to humans' global energy consumption and 22 percent to their generation of electricity in 2012 and 2013, respectively. This energy consumption is divided as 9% coming from traditional biomass, 4.2% as heat energy (non-biomass), 3.8% hydro electricity and 2% is electricity from wind, solar, geothermal, and biomass. Worldwide investments in renewable technologies amounted to more than US$214 billion in 2013, with countries like China and the United States heavily investing in wind, hydro, solar and biofuels.
This page contains text from Wikipedia, the Free Encyclopedia -	https://wn.com/Renewable_energy
The incentive to use 100% renewable energy for electricity, heating and cooling and transport is motivated by global warming, pollution and other ecological as well as economic and energy security concerns. Shifting the total global primary energy supply to renewable sources requires a transition of the energy system. In 2013 the Intergovernmental Panel on Climate Change has said that there are few fundamental technological limits to integrating a portfolio of renewable energy technologies to meet most of total global energy demand. Renewable energy use has grown much faster than even advocates anticipated.
In 2014 renewables such as wind, geothermal, solar, biomass and burnt waste provided 19% of the total world final energy consumption, roughly half of it traditional use of biomass. The most important sector electricity with a renewable share of 22.8%, most of it coming from water power with a share of 16.6%, followed by wind with 3.1%. Several places run their grids almost exclusively on renewable energy. At the national level, at least 30 nations around the world already have renewable energy contributing more than 20% of energy supply. Also, Professors S. Pacala and Robert H. Socolow have developed a series of "Climate stabilization wedges" that can allow us to maintain our quality of life while avoiding catastrophic climate change, and "renewable energy sources," in aggregate, constitute the largest number of their "wedges."
This page contains text from Wikipedia, the Free Encyclopedia -	https://wn.com/100%_renewable_energy
The Japan Pavilion is a Japan-themed pavilion that is part of the World Showcase, within Epcot at Walt Disney World Resort in Florida. Its location is between The American Adventure and Moroccan Pavilions.
The Japan Pavilion is one of the original World Showcase pavilions and had been in planning since the late 1970s. Many attractions have been proposed for the pavilion and one show building was built, but left unused. Meet the World was one planned attraction and was a clone of the attraction Meet the World that was once at Tokyo Disneyland. But because management thought that the Japanese film's omission of World War II might upset many Veterans, it was dropped. The show was so close to opening that the show building and rotating platform was built, but not used.
For years, Imagineers have considered building an indoor roller coaster attraction based on Matterhorn Bobsleds from Disneyland but themed to Japan's Mount Fuji inside a replica of Mount Fuji. At one point, Godzilla or a large lizard attacking guests in their cars was considered. Fujifilm originally wanted to sponsor the ride in the early 1990s, but Kodak, a major Epcot sponsor, convinced Disney to decline the sponsorship. Luckily, the Matterhorn derived design elements survived to be incorporated into Expedition Everest at Disney's Animal Kingdom Park. Another proposed attraction was a walk-through version of "Circle-Vision", in which guests would board and walk through a Shinkansen (bullet train) and look through windows (actually film screens) that showcase Japan's changing landscapes. The train would have shaken and moved like a train traveling through the countryside.
This page contains text from Wikipedia, the Free Encyclopedia -	https://wn.com/Japan_Pavilion_at_Epcot
Japan is referred to in Gulliver's Travels, the satire by Jonathan Swift.
Part III of the book has the account of Lemuel Gulliver's visit to Japan, the only real location visited by him. It is used as a venue for Swift's satire on the actions of Dutch traders to that land. His description reflects the state of European knowledge of the country in the 17th and early 18th centuries, and the tensions due to commercial rivalry between the English and the Dutch at that time.
Japan is shown on the map at the beginning of part III, which also shows the island of "Yesso" (i.e. Hokkaido), "Stats island" (Iturup) and "Companys Land" (Urup) to the north. The map also marks the Vries Strait and Cape Patience, though this is shown on the northeast coast of Yesso, rather than as part of Sakhalin, which was little known in Swift's time. On the island of Japan itself the map shows "Nivato" (Nagato), Yedo, "Meaco" (Kyoto), Inaba and "Osacca" (Osaka)
The text describes Gulliver's journey from Luggnagg, which took fifteen days, and his landing at "Xamoschi" (i.e. Shimosa} which lies "on the western part of a narrow strait leading northward into a long arm of the sea, on the northwest part of which Yedo, the metropolis stands". This description matches the geography of Tokyo Bay, except that Shimosa is on the north, rather than the western shore of the bay.
This page contains text from Wikipedia, the Free Encyclopedia -	https://wn.com/Japan_in_Gulliver's_Travels
Japan (ジャパン) is a Japanese manga written by Buronson (author of Fist of the North Star) and illustrated by Kentaro Miura (author of Berserk). It was published in the Hakusensha magazine Young Animal in 1992 and licensed in English by Dark Horse Comics and released on August 24, 2005.
A yakuza, in love with a TV reporter, comes to Barcelona, Spain, where she's making a reportage on what's the foreigners' idea of the Japanese people, and how Japanese people see themselves; during her speech, she makes a parallel between modern day Japan and the ancient Carthage, saying that the Carthaginians were wiped out by the Romans because of the same attitude Japanese people have nowadays, and because economic superiority brings war, and in the end loses to military strength. Suddenly there's an earthquake, and the ghosts of the Carthaginians bring the group (the two yakuza, the TV reporter and some university students) to the future, when the sea level has increased and all the islands which compose the Japanese archipelago have been submerged; the Japanese people have thus emigrated in the other countries, and they're now scattered around the world, and in particular in Europe, where after the cataclysm a dictatorship has been established, they've become slaves and bandits. Japan is long gone, and Japanese people are lost and oppressed; but among the newcomers, desperate of what they learn, the yakuza, who mostly wishes to protect the woman he dearly loves, has a dream: Japan can be refounded, if the Japanese people come together to fight for it.
This page contains text from Wikipedia, the Free Encyclopedia -	https://wn.com/Japan_(Buronson_manga)
Radio Stations - Tokyo
FM Nirai Okinawa Contemporary Japan
FM Tachikawa 84.4 Varied Japan
NHK Radio 1 News Japan
FM Shiroishi WITH-s Varied Japan
FM Uruma Oldies,Folk Japan
FREEBIRD RADIO 1 (Classic Rock) Classic Rock Japan
Retro AC GAME MUSIC Streaming Radio Electronica Japan
Soryu's Stream Varied Japan
BAN-BAN Radio Ambient,Talk Japan
Retro PC GAME MUSIC Streaming Radio Experimental,Electronica Japan
ChofuFM 83.8 Varied Japan
T-WAVE (RAKIRAKI Beach Radio) Varied Japan
Ozawa Akikazu's Ozawa Akikazu teki KOKORO Varied Japan
Ohayo Michikami Youzou Sports Varied Japan
Jp-Stream TOSHIBO'S WEB RADIO Varied Japan
FM Mot.com, Fukushima Varied Japan
Radio Shonan Varied,Talk Japan
ENDLESS DREAM 1 Varied Japan
Mumix Piano Varied Japan
Old Stories Of Japan (T-WAVE) Varied Japan
MBS News Japan
Kuwahara Shohei Sui mo Amai mo Varied Japan
FM Aizu Varied Japan
NHK FM Varied Japan
FM Takamatu 81.5 Varied Japan
NOAS FM Rock,Varied,Talk Japan
OTTAVA Classical Japan
Narumi Happo Gokigensan Varied Japan
Earth Dreaming - Save the glass Earth - Varied Japan
FM GIG 81.8 Varied Japan
NHK World Portuguese News Japan
P-tomu.Comic Jack Varied Japan
Goketsu P Varied Japan
Uwasa no GIG Maizuru Varied Japan
FM Mikky 76.1 Varied Japan
Kosakin DE Waao! Varied Japan
FM CASTLE Oldies,Pop Japan
News Tantei Kyoku Varied Japan
FM NishiTokyo 84.2 Varied Japan
FM Odawara 78.7 News Talk,News,Talk Japan
NHK news News Japan
M Channel from Maizuru Varied Japan
FM senri Varied,Talk Japan
KOCO Radio 79.1 FM Varied Japan
Anime-Q8 Varied Japan
FM River, Asahikawa Varied Japan
Go J! Varied Japan
Sankakuyama 76.2 FM Varied Japan
Hardcore J Pop Japan
Hitsujikai Radio Religious,Christian Japan
source: The Bottom Line
source: Tiffany Dixon
source: Knox Pods
The renewable and non renewable energy
source: Science Maniacs!
source: Renewable energy and fossil fuels
source: Politics for the People
source: The Mike Slater Show
source: Innovation: The Environment - for iPod/iPhone
source: LabRats
source: Diogo's Words
source: Pricking the Bubble
source: The Fallon Forum
source: Making Cent$
source: It's Rocket Science
source: Renewable Energy
source: Lo show di Carla Campogiani
Spiritual Renewal Japan
source: Solo Travel Talk with Astrid
Renewable Energy & Energy Planning
source: Deep Dive with Region Five
Renewable Energies and Mediterranean Energy Cooperation
source: Policy Center for the New South Podcast
Rain Tree Crow
Rain Tree Crow released: 1992
Big Wheels in Shanty Town
Every Colour You Are
Red Earth (As Summertime Ends)
Pocket Full of Change
Boat's for Burning
New Moon at Deer Wallow
A Reassuringly Dull Sunday
Blackcrow Hits Shoe Shine City
Scratchings on the Bible Belt
Cries and Whispers
The Very Best Of released: 2006
Assemblage released: 2006
Adolescent Sex released: 2004
The Best of Japan released: 2002
The Singles: Japan released: 1996
In Vogue released: 1996
Methods of Dance released: 1991
The Other Side of Japan released: 1991
Gentlemen Take Polaroids released: 1991
Souvenir From Japan released: 1989
Quiet Life released: 1986
Exorcising Ghosts released: 1984
Oil on Canvas released: 1983
Leidenese Boy released: 1982
Visions of China released: 1982
Japan released: 1982
Live at Hammersmith Odeon released: 1981
Tin Drum released: 1981
Obscure Alternatives released: 1978
1982-12-08: Budokan Hall, Tokyo, Japan released:
Quiet Life (7" version)
Visions of China
I Second That Emotion
Life in Tokyo
European Son
All Tomorrow's Parties (7" version)
Adolescent Sex
Don't Rain on My Parade
In Vogue
The Unconventional
Suburban Berlin
Life in Tokyo (12" version)
All Tomorrow's Parties (12" version)
Ghosts (single version)
I Second That Emotion (single mix)
Gentlemen Take Polaroids
The Art of Parties (single version)
Taking Islands in Africa (Steve Nye remix)
European Son (single mix)
Cantonese Boy
Life in Tokyo, Part 1 (Special remix)
Nightporter
Methods of Dance
Canton (live)
Ghosts (album version)
...Rhodesia
European Son (John Punter 12" mix)
I Second That Emotion (12" version)
Life in Tokyo (1982 version)
Wish You Were Black
Lovers On Main Street
Suburban Love
The Best of Japan
Adolescent Sex (Re-Recorded version)
Automatic Gun
Fall in Love With Me
The Other Side of Life
A Foreign Place
Deviation (live)
Obscure Alternatives (live)
In Vogue (live)
Sometimes I Feel So Low (live)
Love is Infectious
In-Vogue
Obscure Alternatives
The Singles: Japan
Life in Tokyo (Theme)
In Vogue (live in Tokyo)
Life in Tokyo (12" extended version)
Sometimes I Feel So Low
Life in Tokyo (extended remix)
All Tomorrow's Parties (Steve Nye's 1983 remix)
I Second That Emotion (extended remix)
European Son (extended remix)
Quiet Life (extended version)
European Son (12" version)
My New Career
Taking Islands in Africa
Ain't That Peculiar
The Other Side of Japan
The Tenant
Burning Bridges
The Experience of Swimming
The Width of a Room
Souvenir From Japan
.....Rhodesia
Exorcising Ghosts
The Art of Parties
Sons of Pioneers
Voices Raised in Welcome, Hands Held in Prayer
Leidenese Boy
Still Life in Mobile Homes
Intro (Burning Bridges)
Taking Islands in Africa (remix)
Live at Hammersmith Odeon
1982-12-08: Budokan Hall, Tokyo, Japan


This is part of Lundy's Tree Series. The background is multiple layers of resin with a color-changing paint. The tree is sculpted with an acrylic gel medium and infused with color on the resin background. The piece is framed in Koa wood from Hawaii.
The painting itself measures 30" tall by 30" wide. With the frame, the piece is 35.5" tall by 35.5" wide.


text:
Heathrow Public Consultation
Heathrow Airport Limited is holding a public consultation on proposals surrounding their expansion plans. They are required to do this as part of the Development Consent Order process, which is part of the National Infrastructure planning process.
Nat West Branch Closure in Southfields
Nat West has announced a number of branch closures nationally, including two branches closing in our area in Southfields and Wandsworth, though Putney branch is unaffected.
Policing along Putney High Street and Moped Crime
This month I had the opportunity to 'Walk the Met' with members of our local police along Putney High Street to discuss ongoing crime problems in the area. We were also joined by Positively Putney, the new Putney Business Improvement District organisation, as Putney businesses identified crime -
Improved Train Services for Putney
Constituents regularly get in touch with me about the frequency and reliability of our train services, particularly at peak times into Waterloo. I recently met the Managing Director of South Western Railway, Andy Mellors, to discuss plans for improving these services.
Following concerns raised with me by residents on Dover House Road, at the Roehampton Forum meeting and in my MP inbox, I got in touch with Wandsworth Borough Council to discuss the proposed sale of the Roehampton Playing Fields.
Plans for Wimbledon Park
This month, along with the Friends of Wimbledon Park, I held a public meeting on future plans for Wimbledon Park.
EU Negotiation Update
The Prime Minister had secured an agreement earlier this month to conclude the first part of our negotiations with the European Union which now allows us to move on to the next stage talks. An update on the progress of negotiations can be seen here.
Local Transport Improvements
I met with the Managing Director of the Underground for Transport for London, Mark Wild, today to outline our concerns over delays on the District line and hear about TfL's future plans for the area.
Wimbledon Park Public Meeting
Wimbledon Park Public Meeting: 23 November at 7pm at the Wimbledon Club
The Friends of Wimbledon Park and Justine will host a public meeting on 23rd November at 7pm at the Wimbledon Club to get your views on ideas for improving Wimbledon Park.
Possible strike action on South Western Railway
As you may know, the RMT union has announced its intention to hold strikes on South Western Railway on 8 and 9 November which would affect train services in our constituency.


What Is Best Scabies Treatment – Natural Remedies Vs Permethrin Vs Lindane?
Do you have a prickly feeling after scabies treatment?
My Boyfriend Has Scabies, Will I Get Infected Too?
Will Sleeping With Scabies Infected Person Cause Me The Infection Too?
How to Protect Yourself From House Guests with Scabies Infection?


text:
Kwartalnik Historii Żydów
Jewish History Quarterly
Publishing House: Żydowski Instytut Historyczny
Subject(s): History, Jewish studies
Issue No. 04/212
Until the year 2000 Kwartalnik Historii Żydów was published as Biuletyn Żydowskiego Instytutu Historycznego. In the years 1949-1950 it appeared as a bulletin in Polish and Yiddish, carrying news about the activities of the Institute. Since 1951, Biuletyn has been published by the Jewish Historical Institute of Warsaw as a scientific journal (initially as semi-annual and from 1953 as a quarterly). Formally, the scope of its subjects was the history of the Jews from the ancient to the most recent times. In practice, however, the focus was on history of modern times and most recent history. Until 1966, the magazine's editor in chief was Bernard Mark, director of the Jewish Historical Institute. He was succeeded by Artur Eisenbach, Szymon Datner, Marian Fuks, Maurycy Horn (from 1973 to 1990), Jerzy Tomaszewski, Szymon Rudnicki, Daniel Grinberg, Alina Cała and, since 1999, Jan Doktór. Since 2001, the magazine has appeared under the present name, although neither the publisher not the general concept of it have changed. The editors seek to present various aspects of life of the Jewish community in Poland and the neighbouring countries over the ages. A significant part of the articles and source materials concerns the tragic fate of Jews during World War II. Next to articles in Polish, Kwartalnik prints texts in other languages (German and English). The following regular columns appear in each issue: "Studia i artykuły" "Z historii regionalnej" "Dokumenty i materiały" (mostly source materials) "Recenzje i artykuły recenzyjne". Editor-in-chief: Jan Doktór, Ph.D. Editorial board: Wolfgang Benz (University of Technology, Center for Anti-Semitic Research, Berlin) Dan Diner (Hebrew University, Jerusalem; Simon Dubnow Institute, Leipzig) Jan Doktór (Jewish Historical Institute, Warsaw) David Gershon Hundert (McGill University, Montreal) Gertrud Pickhan (Free University, Berlin) Antony Polonsky (Brandeis University, Waltham, Mass.) Szymon Rudnicki (University of Warsaw) Paweł Śpiewak (Jewish Historical Institute, Warsaw) Editorial office: Eleonora Bergman, Michał Czajka (associate editor), Helena Datner, Jan Doktór (editor-in-chief), Daniel Grinberg, Anna Michałowska-Mycielska, Joanna Nalewajko-Kulikov, Szymon Rudnicki, Paweł Śpiewak, Rafał Żebrowski
Editor-in-chief: Jan Doktór, Ph.D. Editorial board: Wolfgang Benz (University of Technology, Center for Anti-Semitic Research, Berlin) Dan Diner (Hebrew University, Jerusalem; Simon Dubnow Institute, Leipzig) Jan Doktór (Jewish Historical Institute, Warsaw) David Gershon Hundert (McGill University, Montreal) Gertrud Pickhan (Free University, Berlin) Antony Polonsky (Brandeis University, Waltham, Mass.) Szymon Rudnicki (University of Warsaw) Paweł Śpiewak (Jewish Historical Institute, Warsaw) Editorial office: Eleonora Bergman, Michał Czajka (associate editor), Helena Datner, Jan Doktór (editor-in-chief), Daniel Grinberg, Anna Michałowska-Mycielska, Joanna Nalewajko-Kulikov, Szymon Rudnicki, Paweł Śpiewak, Rafał Żebrowski
Abraham bar Jacob and His Copperplate Engravings in 17th and 18th Century Amsterdam Prints Abraham bar Jacob and His Copperplate Engravings in 17th and 18th Century Amsterdam Prints Year: 2017, Issue No. 01 Jewish Military Union in Warsaw Ghetto Żydowski Związek Wojskowy w getcie warszawskim Year: 2006, Issue No. 01 Multicultural Galicia. The Origin of a Myth During World War I Das multikulturelle Galizien. Die Entstehung eines Mythos waehrend des Ersten Weltkriegen Year: 2004, Issue No. 04 The Jewish Historical Institute 70 Years on – Political History and Research. Birthday Commentary 70 lat Żydowskiego Instytutu Historycznego – historia polityczna i badania. Jubileuszowy komentarz Year: 2019, Issue No. 01 Anan ben David as the Archetypal Religious Troublemaker: how Rav Natronai Gaon Contributed to Elevating the Ananite Leader to the Position of Founding Father of Karaism Anan ben Dawid jako archetyp religijnego wichrzyciela: jak raw Natronaj Gaon przyczynił się do wykreowania przywódcy ananitów na ojca założyciela karaizmu Year: 2017, Issue No. 03


text:
by Will Hindmarch | Apr 11, 2012 | Conventions, Creativity, Events, Fan Culture, Horror, Play, Social Games
PAX East obliterated Jared Sorensen's voice but he still had an event left to run. Players were counting on him. The convention schedule had the game session locked in. Sorensen, his voice already spent on conversations and events in the noisy convention venues,...
Wil Wheaton's New TableTop
by Will Hindmarch | Mar 19, 2012 | Board Games, Creativity, Events, Fan Culture, Promotion, RPGs, Websites
Writer, actor, geek, gamer, and producer Wil Wheaton has a shiny new web series coming to the shiny new YouTube channel, Geek and Sundry. The show's called TableTop and its something like Celebrity Poker meets Dinner For Five except instead of dinner or poker...
Gameplaywright at GenCon
by Jeff Tidball | Aug 2, 2011 | Conventions, Events, Hamlet's Hit Points
We hope you'll join us for two Gameplaywright events at GenCon this week: Robin Laws will be talking about Hamlet's Hit Points, presenting the system of analysis behind his ENnie-nominated book and taking questions about its theory and application....
Will @ Gen Con 2011
by Will Hindmarch | Aug 1, 2011 | Conventions, Dice, Events, Fan Culture, Freelancing
I'm sure you've been waiting with breath all bated for this. It's my Gen Con Indy schedule for 2011. If you're looking for me at Gen Con this year, you'll find me at the following places (and at sundry other places in the gaps within this...
Origins 2011 in the Rear-View
by Will Hindmarch | Jun 27, 2011 | Conventions, Events, RPGs
It's my goal to write about #Origins2011 every day this week. This is tricky. How much do you want to hear about a convention that already happened, right? Probably not a lot. I played a lot of compelling games, though, and want to share some of what I've...
Everyone's Awards
by Will Hindmarch | Jun 18, 2011 | ARGs, awards, Board Games, Books, card games, Dice, Events, Fan Culture, MMOs, RPGs, Video Games
You have a single prize to award to one noun — person, place, thing, or event — in recognition of excellence in gaming for the previous year, from July 2010 to July 2011. This is a prize of celebration and recognition only. No trophy, no money, no stickers, no medals....


text:
Money grows on tree
A Picton restaurant did more than add seasonal sparkle with its Christmas tree.
The Glenbridge encouraged its patrons to hang an ornament "in memory of Someone Special." For a donation of $1 or more, a memorial ornament was provided by the restaurant.
The proceeds from the Christmas fundraiser--$600--plus a matching donation of $600 from the Glenbridge--went to The Prince Edward County Community Care for Seniors Association.
"It has been said that money doesn't grow on trees and generally this is true, but through the generosity of a number of people, $1200 has been raised and that's remarkable," Stan Whitehouse, chair of Community Care for Seniors said.
Glenbridge Co-owners Larry Browne and Ron Girouard, who opened their restaurant in 1998, said they too were pleased with the response from the community in 1998 when $799 was realized and decided to repeat the effort in 1999.
By inviting people to "help us decorate our tree with love," the proprietors found people responded by remembering their loved ones as they placed their ornaments.
For information contact: Debbie Moynes (article was written by Pat Whittaker, Chair of Public Relations for Prince Edward Community Care)


text:
Pub Our planet Advanced online casino Distributions Bit of Decreased Refund Pay out Limitations Absolutely
I would ike to let you know about Do payday advances affect credit
Arkansas has some associated with the most challenging restrictions on low-dollar, high-cost loans into the country, but a fresh kind of company is challenging its restrictions, a national watchdog stated.
Рубрика: Underwater Metal Detectors
Top Underwater Metal Detectors Guide!
It is submersible in waters up to 250 feet and is regarded by many users as the best freshwater detector on the market. The primary function that aids in the general operation of this unit is underwater abilities.
The Garrett Infinium is one of the best metal detectors for the money and this is for both land and subaqueous digging. It is evident that you must have to travel for metal detecting, so a carry back is another useful equipment for this purpose.
With that in mind, here are three of the best diving metal detectors that can be used at least 200 feet below the surface. Metal detectors have a depth rating, so check the model is suitable before you buy. Like the AT Pro, the ATX is waterproof down to a depth of 10 feet. The standard package includes a 10″x12″ DD coil, which provides great depth and sensitivity for small items.
The metal detector case color is bright and fluorescent, due to which you won't loose track of it in muddy water. The metal detector has Koss waterproof headphones in its set and a lot of other stuff. Garrett AT Pro is an amphibian metal detector due to which you can put it completely under water up to 3 meters deep. This metal detector has 2 operation frequencies – 3 and 15 kHz which ensures its efficient functioning for various tasks and search conditions.
Unlike the ATX and SDC, the AT Gold is a very-low frequency (VLF) detector. With Adjustable Ground Blance Window, the detector ground balances properly and smooths out the ground signals, thereby allowing the user to hear the deeply buried targets. The smoothness of the threshold on the SDC 2300 is ideal for underwater hunting to eliminate the need to dig false targets.
Now that you've got everything you need, you're ready to head out SCUBA diving with your favorite underwater metal detector. To get started, be sure to select a model of underwater https://metaldetectorshub.com/underwater-metal-detectors/ metal detector that offers the deepest depth rating possible. Selection-The first factor to be considered when hunting for the best underwater metal detector is the selection.
Rapid Secrets Of Waterproof Metal Detectors — An Analysis
Because most detectors are unusable in a salt water environment, you need a dual function VLF or a PI type of metal detector. The Garrett AT Pro is a VLF metal detector that's at the lower end of the price range in this review of underwater metal detectors.
Well, in some aspect they are pretty much the same since they actually deliver the same task, get you the metal you want, but still are not the same. Users are advised to adjust the ground balance control to improve performance. These detectors are sensitive to precious metals and work by transmitting electronic pulses in the ground. Most ancient civilizations used water as a means of transport and had some bad luck crossing the waters and lost a lot of their precious stones and metals in there.
We tested each metal detector's performance finding iron nails, aluminum pull-tabs, pennies, nickels, dimes, quarters and precious metals. In addition to the optimization of the inductive system, the distance of the planar array to the ROV and the type of materials used were selected in a way so they would not create any increase in the electronic background noise of the Metal Detector. The final design of the Metal Detector Array was preceded by a series of measurements conducted on the surface with the purpose of verifying its electromagnetic environmental compatibility.
Опубликовано 29.05.2019 15.10.2019 Рубрики Underwater Metal DetectorsДобавить комментарий