%%% Translated by ErgoTeX V1.1 (c) 1990-96 by Oliver Schnetz 
\input{natren.def}
\documentstyle[12pt,epsf]{article}\input{a4head}\hyphenation{Schwin-ger}\pagestyle%l.3
{headings}\title{Natural renormalization}\author{Oliver Schnetz\thanks%l.4 {
{Institut f{\"u}r theoretische Physik III, Staudtstra{\ss}e 7, 91058 %l.5 2{
Erlangen, Germany,\newline %l.6 2{
e-mail: schnetz@pest.physik.uni-erlangen.de\newline %l.7 2{
Supported in parts by the DFG Graduiertenkolleg 'Starke Wechselwirkung' %l.8 2{
and the BMBF.\newline %l.9 2{
FAU-TP3-96/1}}\date{June 18 1996}%l.10
\begin{document} %l.11
\maketitle %l.12
\begin{abstract} %l.13 BAt
A careful analysis of differential renormalization shows that a distinguished %l.14 BAt
choice of renormalization constants allows for a mathematically %l.15 BAt
more fundamental interpretation of the scheme. With this set of %l.16 BAt
a priori fixed integration constants differential renormalization %l.17 BAt
is most closely related to the theory of generalized functions. %l.18 BAt
The special properties of this scheme are illustrated by application %l.19 BAt
to the toy example of a free massive bosonic theory. Then we apply %l.20 BAt
the scheme to the $\varphi ^{4}$-theory. The two-point function %l.21 BAt
is calculated up to five loops. The renormalization group is analyzed, %l.22 BAt
the beta-function and the anomalous dimension are calculated up %l.23 BAt
to fourth and fifth order, respectively.%l.24 BAt
\end{abstract} %l.25
\tableofcontents %l.26
 %l.27
\section{Introduction} %l.28
With the proof of renormalizability of non-Abelian gauge theories %l.29
in the early 1970's the problem of giving a perturbative definition %l.30
of a renormalizable quantum field theory was solved (eg.\ \cite{Collins}). %l.31
However explicit calculations in the commonly used dimensional regularization %l.32
are often tedious. This kept the interest in alternative prescriptions %l.33
alive.%l.34

Quite recently differential renormalization \cite{Johnson,Freedman,Latorre} %l.36
has been proposed. For practical calculations this renormalization %l.37
scheme provides two major advantages. Firstly, it allows to regularize %l.38
and renormalize in one step. No explicit regulators or counterterms %l.39
are needed. Secondly, it is possible to keep the spacetime dimension %l.40
fixed. This is particularly useful for dimension-specific theories %l.41
like the chiral electroweak sector of the standard model.%l.42

We start by analyzing differential renormalization from a purely %l.44
mathematical point of view. Differential renormalization is usually %l.45
formulated in four-dimensional coordinate space by writing divergent %l.46
amplitudes as Laplacians (we restrict ourselves to Euclidean signature, %l.47
$ x^{4}=(x^{2})^{2}$, $ \Box =\sum _{i}\partial _{i}^{2}$) of less %l.48
divergent expressions. For example, %l.49
\begin{equation}%l.50 $
\label{1}\frac{1}{x^{4}{}} =-\frac{1}{4}\Box \frac{1}{x^{2}{}} \ln%l.51 $
\left( \frac{x^{2}}{\Lambda _{0}^{2}}\right) %l.52 $
\hspace{.6ex},\hspace{2ex}\hspace*{1cm}\frac{1}{x^{6}{}} =-%l.53 ,...
\frac{1}{32}\Box \Box \frac{1}{x^{2}{}} \ln\left( \frac{x^{2}}{%l.54 ...
\Lambda _{1}^{2}}\right) \hspace*{2ex}\hbox{ for }x\neq 0%l.55 $
\hspace{.6ex}.%l.56 $
\end{equation} %l.57
$\Lambda _{0}$ and $\Lambda _{1}$ are arbitrary integration constants %l.58
which are kept for dimensional reasons.%l.59

Initially ill-defined integrals are now regularized by the convention %l.61
that the Laplacian should act on the left and the surface term is %l.62
ignored. According to this rule the singular Fourier transforms %l.63
of $ x^{-4}$ and $ x^{-6}$ can be derived from the well-defined %l.64
Fourier transform of $ x^{-2}\ln(x^{2}{} /\Lambda ^{2})$ (calculated %l.65
below, Eq.\ (\ref{I})), %l.66
\begin{eqnarray}%l.67 $$
\label{2}\int \limits _{{\rm diff.ren.\hspace{.38ex}}}\frac{d^{4%l.68 ,...
}x}{4\pi ^{2}{}} \frac{e^{ix\cdot p}}{x^{4}{}} &\equiv &-\frac{%l.69 ,...
1}{4}\int \frac{d^{4}x}{4\pi ^{2}{}} \left( \Box e^{ix\cdot p}%l.70 $$
\right) \frac{1}{x^{2}{}} \ln\left( \frac{x^{2}}{\Lambda _{0}^{%l.71 ,...
2}}\right) \hspace*{1ex}=\hspace*{1ex}-\frac{1}{4}\ln\left( %l.72 ,...
\frac{p^{2}}{\bar{\Lambda }_{0}^{2}}\right) \hspace{.6ex},\\ %l.73 $$
\label{2a}\int \limits _{{\rm diff.ren.\hspace{.38ex}}}\frac{d^{%l.74 ,...
4}x}{4\pi ^{2}{}} \frac{e^{ix\cdot p}}{x^{6}{}} &\equiv &-%l.75 ,...
\frac{1}{32}\int \frac{d^{4}x}{4\pi ^{2}{}} \left( \Box \Box e^%l.76 { (,)...
{ix\cdot p}\right) \frac{1}{x^{2}{}} \ln\left( \frac{x^{2}}{%l.77 ...
\Lambda _{1}^{2}}\right) \hspace*{1ex}=\hspace*{1ex}\frac{1}{32%l.78 ,...
}p^{2}\ln\left( \frac{p^{2}}{\bar{\Lambda }_{1}^{2}}\right) %l.79 $$
\hspace{.6ex},%l.80 $$
\end{eqnarray} %l.81
where $ \bar{\Lambda }_{0(1)}=2/(e^C\Lambda _{0(1)})$ and $C$=0.5772156{\dots} %l.82
is the Euler constant.%l.83

The central point in this paper is to exhibit the meaning of the %l.85
above prescriptions for one-dimensional integrals. To this end we %l.86
perform the convergent angular integrals in (\ref{2}), (\ref{2a}) %l.87
which leaves us with a radial integral $ \int _{0}^{\infty }dr$ %l.88
that diverges at $ r=0$. We finally split this integral into a convergent %l.89
part $ \int _{1}^{\infty }$ which can be evaluated and a singular %l.90
part $ \int _{0}^{1}$ which is kept. These entirely well-defined %l.91
manipulations lead to %l.92
\begin{eqnarray}%l.93 $$
\int \frac{d^{4}x}{4\pi ^{2}{}} \frac{e^{ix\cdot p}}{x^{4}{}} &=%l.94 $$
&-\frac{1}{2}\left( \ln\left( \frac{e^C|p|}{2}\right) -\int _{0%l.95 ,...
}^{1}\frac{dr}{r} -\frac{1}{2}\right) \hspace{.6ex},\\ %l.96 $$
\label{888a}\int \frac{d^{4}x}{4\pi ^{2}{}} \frac{e^{ix\cdot p}%l.97 ,...
}{x^{6}{}} &=&\frac{1}{16}p^{2}\left( \ln\left( \frac{e^C|p|}{2%l.98 ,...
}\right) -\int _{0}^{1}\frac{dr}{r} -\frac{5}{4}\right) +\frac{%l.99 ,...
1}{2}\left( \int _{0}^{1}\frac{dr}{r^{3}{}} +\frac{1}{2}%l.100 $$
\right) \hspace{.6ex}.%l.101 $$
\end{eqnarray} %l.102
Now we compare this result with Eqs.\ (\ref{2}) and (\ref{2a}) derived %l.103
by differential renormalization. First, notice that the second term %l.104
on the right hand side of Eq.\ (\ref{888a}) has no $ p^{2}$-dependence %l.105
at all. To make the right hand side proportional to $ p^{2}$ we %l.106
define %l.107
\begin{equation}%l.108 $
\label{A}\int _{0}^{1}\frac{dr}{r^{3}{}} \equiv -\frac{1}{2}%l.109 $
\hspace{.6ex}.%l.110 $
\end{equation} %l.111
This leads us finally to the equations %l.112
\begin{equation}%l.113 $
\label{B}\int _{0}^{1}\frac{dr}{r} =-\frac{1}{2}-\ln\Lambda _{0}%l.114 $
=-\frac{5}{4}-\ln\Lambda _{1}\hspace{.6ex}.%l.115 $
\end{equation} %l.116
Eqs.\ (\ref{A}) and (\ref{B}) can be seen as one-dimensional definitions %l.117
of differential renormalization. However Eq.\ (\ref{B}) naturally %l.118
relates the renormalization constants via %l.119
\begin{equation}%l.120 $
\label{C}\ln\left( \frac{\Lambda _{1}}{\Lambda _{0}}\right) =-%l.121 ,...
\frac{3}{4}\hspace{.6ex}.%l.122 $
\end{equation} %l.123
For a ratio $ \Lambda _{1}/\Lambda _{0}$ different from exp(-$\frac{3}{4}$) %l.124
the one-dimensional interpretation of differential renormalization %l.125
is not possible.%l.126

We will see in the next section that all ratios of renormalization %l.128
constants are fixed by consistency conditions. Differential renormalization %l.129
with these a priori fixed ratios will be called 'natural renormalization'.%l.130

It was shown \cite{Latorre} that differential renormalization provides %l.132
a self-consistent definition of renormalizable field theories, without %l.133
referring to the ratio $ \Lambda _{1}/\Lambda _{0}$ as given in %l.134
Eq.\ (\ref{C}). Moreover in some cases it is convenient to adjust %l.135
the ratios of renormalization constants according to physical requirements %l.136
\cite{Smirnov2}. In particular for gauge theories it is useful to %l.137
fix some ratios by Ward identities \cite{Johnson,Haagensen,Freedman2}. %l.138
However, depending on the gauge, some of these ratios may differ %l.139
from the prescriptions we give. The treatment of gauge theories %l.140
in natural renormalization is still under investigation, first attempts %l.141
have been successful \cite{Schnetz}.%l.142

\vspace{1ex}%l.144
\noindent{}The main advantage of allowing for the above one-dimensional %l.145
reduction and demanding Eqs.\ (\ref{A}), (\ref{B}), (\ref{C}) is %l.146
that differential renormalization can be understood on a much more %l.147
general footing. We will see in Sec.\ \ref{genfunct} that Eqs.\ %l.148
(\ref{A}) and (\ref{B}) are almost standard in the theory of generalized %l.149
functions. Thus it becomes possible to replace the recipes of differential %l.150
renormalization by mathematically more fundamental definitions.%l.151

In contrast to differential renormalization, natural renormalization %l.153
is neither connected to coordinate nor to momentum space. One has %l.154
the freedom to choose the most convenient representation for the %l.155
respective problem.%l.156

The first example where natural renormalization becomes advantageous %l.158
is the toy theory of free massive bosons discussed in Sec.\ \ref{freemass}. %l.159
The mass is treated as two-point interaction which leads by power-counting %l.160
to a non-renormalizable theory in coordinate space. With standard %l.161
differential renormalization it becomes necessary to adjust infinitely %l.162
many constants. It will turn out that these constants coincide with %l.163
the a priori fixed ratios of our approach. This makes it possible %l.164
to recover the right result immediately within natural renormalization. %l.165
This does not happen accidentally as can be shown in a general theorem.%l.166

\vspace{1ex}%l.168
\noindent{}The main application of this paper will be the $%l.169
\varphi ^{4}$-theory in Sec.\ \ref{fi4}. We focus our attention %l.170
to the calculation of the two-point Green's function. It will turn %l.171
out that the $\varphi ^{4}$-theory performs almost like made for %l.172
our renormalization scheme: Most Feynman diagrams of a given order %l.173
precisely match into a formula which allows to calculate their sum %l.174
without evaluating single graphs. This enables us to calculate the %l.175
two-point function up to five loops.%l.176

Finally the renormalization group is discussed. The $ \beta $-function %l.178
and the anomalous dimension $\gamma $ are determined up to fourth %l.179
and fifth order in the coupling, respectively.%l.180

\section{Definition of the renormalization scheme} %l.182
\subsection{Comparison with differential renormalization\label%l.183 {
{diffreg}} %l.184
We start with a generalization of the ideas presented in the introduction. %l.185
Repeated application of the equation %l.186
\begin{equation}%l.187 $
\label{3}\Box f\left( x^{2}\right) =\frac{4}{x^{2}{}} \frac{%l.188 ,...
\partial }{\partial x^{2}{}} x^{4}{} \frac{\partial }{\partial x%l.189 ...
^{2}{}} f\left( x^{2}\right) %l.190 $
\end{equation} %l.191
leads to %l.192
\begin{equation}%l.193 $
\label{4}\Box ^{n+1}{} \frac{1}{x^{2}{}} \ln\left( \frac{x^{2}}{%l.194 ...
\Lambda ^{2}}\right) =-4^{n+1}n!\left( n+1\right) !\frac{1}{x^{%l.195 ,...
2n+4}{}} \hspace*{2ex}\hbox{ for }x\neq 0%l.196 $
\hspace{.6ex},\hspace{2ex}n=0,1,{\ldots}\hspace{.6ex}.%l.197 $
\end{equation} %l.198
Note that these equations hold strictly only for $ x\neq 0$ and may %l.199
be modified by $ \delta (x)$-terms (cf.\ Eq.\ (\ref{19})). In a %l.200
renormalizable field theory one needs only a finite number of these %l.201
equations ($ n=0,1$ for the $\varphi ^{4}$-theory), however it will %l.202
turn out to be useful to look at the general case.%l.203

The function $ x^{-2n-4}$ has no well defined Fourier transform whereas %l.205
$ x^{-2}\ln(x^{2}{} /\Lambda ^{2})$ has (cf.\ Eq.\ (\ref{I})). The %l.206
differentially renormalized Fourier transform of the right hand %l.207
side of Eq.\ (\ref{4}) is now determined by the left hand side with %l.208
the Laplacian translated as $ -p^{2}$ \cite{Johnson}, %l.209
\begin{equation}%l.210 $
\label{5}\int \limits _{{\rm diff.ren.\hspace{.38ex}}}\frac{d^{4%l.211 ,...
}x}{4\pi ^{2}{}} \frac{e^{ix\cdot p}}{x^{2n+4}{}} \equiv \frac{p%l.212 ,...
^{2n}}{\left( -4\right) ^{n+1}n!\left( n+1\right) !} \ln\left( %l.213 ,...
\frac{p^{2}}{\bar{\Lambda }_{n}^{2}}\right) \hspace{.6ex}.%l.214 $
\end{equation} %l.215
We have introduced different renormalization scales $ \bar%l.216 $
{\Lambda }_{n}$ for each $n$ to stress that they are integration %l.217
constants which a priori are independent from each other and may %l.218
differ by arbitrary positive factors. The $\Lambda _{n}$ are interpreted %l.219
as renormalization scales.%l.220

Our analysis starts with the introduction of polar coordinates. %l.222
\begin{equation}%l.223 $
\label{6}\int \frac{d^{4}x}{4\pi ^{2}{}} \frac{e^{ix\cdot p}}{x^{%l.224 ,...
2n+4}{}} =\frac{1}{\pi } \int _{0}^{\pi }d\vartheta \sin^{2}%l.225 $
\vartheta \int _{0}^{\infty }drr^{-2n-1}e^{ir|p|\cos\vartheta }%l.226 $
\hspace{.6ex},%l.227 $
\end{equation} %l.228
where we have chosen the $z$-axis to be parallel to $p$. We evaluate %l.229
the convergent $\vartheta $-integral and split the $r$-integral %l.230
into a convergent part $ \int _{1}^{\infty }$ which is evaluated %l.231
and a part $ \int _{0}^{1}$ that diverges at zero and has to remain %l.232
unchanged.%l.233

The calculations are in principle straightforward but tedious \cite{Schnetz}. %l.235
The result is ($ \sum _{0}^{-1}\equiv \sum _{1}^{0}\equiv 0$) %l.236
\begin{eqnarray}%l.237 $$
\label{888}\int \frac{d^{4}x}{4\pi ^{2}{}} \frac{e^{ix\cdot p}}{x%l.238 ...
^{2n+4}{}} &=&\sum _{k=0}^{n-1}\frac{p^{2k}}{2\left( -4\right) ^{%l.239 ,...
k}k!\left( k+1\right) !} \left( \int _{0}^{1}\frac{dr}{r^{2n-2k%l.240 ,...
+1}{}} +\frac{1}{2n-2k} \right) \\ %l.241 $$
&&+\frac{2p^{2n}}{\left( -4\right) ^{n+1}n!\left( n+1\right) !} %l.242 (,)...
\left( -\int _{0}^{1}\frac{dr}{r} +\ln\left( \frac{e^C|p|}{2}%l.243 (,)...
\right) -\frac{1}{2\left( n+1\right) } -\sum _{k=1}^{n}{} %l.244 ,...
\frac{1}{k}\right) \hspace{.6ex}.\nonumber %l.245 $$
\end{eqnarray} %l.246
From a mathematical point of view we want $p$-independent integrals %l.247
to give $p$-independent results. So we are forced to make the following %l.248
definitions in order to regain the result obtained by differential %l.249
renormalization (\ref{5}), %l.250
\begin{eqnarray}%l.251 $$
\label{a}\int _{0}^{1}\frac{dr}{r^{2n-2k+1}{}} &=&-\frac{1}{2n-2%l.252 ,...
k}\hspace*{1cm}\hbox{\hspace{.38ex}and\hspace{.38ex}}\\ %l.253 $$
\label{b}\int _{0}^{1}\frac{dr}{r} &=&-\ln\Lambda _{n}-\frac{1}%l.254 ,...
{2n+2}-\sum _{k=1}^{n}{} \frac{1}{k}\hspace{.6ex}.%l.255 $$
\end{eqnarray} %l.256
Eq.\ (\ref{a}) is the analytic continuation of the formula %l.257
\begin{equation}%l.258 $
\label{10}\int _{0}^{1}drr^{n}{} =\frac{1}{n+1}%l.259 $
\end{equation} %l.260
to $ n<-1$. Eq.\ (\ref{b}) shows that within our approach we can %l.261
not equate the renormalization scales $ \Lambda _{n}$ among each %l.262
other. We find instead %l.263
\begin{equation}%l.264 $
\label{c}\ln\Lambda _{n}=\ln\Lambda -\frac{1}{2n+2}-\sum _{k=1}^{%l.265 ,...
n}{} \frac{1}{k}%l.266 $
\end{equation} %l.267
for some scale $\Lambda $. The difference $ \ln\Lambda _{i}-\ln%l.268 $
\Lambda _{j}$ for any $ i\neq j$ is a well-defined non-zero rational %l.269
number. If one violates Eq.\ (\ref{c}) one changes the definition %l.270
of convergent integrals or generates $p$-dependences from $p$-independent %l.271
divergent integrals. In the differentially renormalized $%l.272
\varphi ^{4}$-theory $\Lambda _{0}$ and $\Lambda _{1}$ are usually %l.273
equated which however does not destroy the self-consistency of the %l.274
theory since it is incorporated in the freedom of choosing the renormalization %l.275
scheme.%l.276

\vspace{1ex}%l.278
\noindent{}An overall factor in the renormalization constants is %l.279
irrelevant, so we choose a renormalization scale $\Lambda $ according %l.280
to %l.281
\begin{equation}%l.282 $
\label{11}\int _{0}^{1}\frac{dr}{r} =-\ln\Lambda \hspace{.6ex}.%l.283 $
\end{equation}%l.284

Notice that the left hand side of this equation has no explicit $%l.286
\Lambda $ dependence. One assumes $ r^{-1}$ to have the implicit %l.287
local $\Lambda $-term $ -\ln\Lambda \cdot \delta (r)$ in a similar %l.288
way as the differentially renormalized version of $ x^{-4}$ acquires %l.289
the local renormalization dependence $ -\frac{1}{4}\ln\Lambda _{%l.290 ,...
0}^{2}\cdot 4\pi ^{2}\delta ^{(4)}(x)$ (cf.\ Eq.\ (\ref{2})).%l.291

\subsection{First results\label{firstres}} %l.293
'Natural renormalization' corresponds to differential renormalization %l.294
with the $\Lambda _{n}$ defined via Eq.\ (\ref{c}). It gives a generalization %l.295
of the usual definition of integrals.%l.296

The renormalization scale $\Lambda $ is kept for 'dimensional reasons'. %l.298
If we integrate over dimensionful parameters then $\Lambda $ combines %l.299
with other ln-terms to provide a scalar argument of the logarithms. %l.300
$\Lambda $ is not a cutoff (notice that the integrals over higher %l.301
order poles (\ref{a}) are $\Lambda $-independent), it is neither %l.302
large nor small (cf.\ Sec.\ \ref{freemass}).%l.303

We summarize the above discussion by giving our definition for the %l.305
singular Fourier transform ($ \bar{\Lambda }=2/e^C\Lambda $) %l.306
\begin{equation}%l.307 $
\label{18}\int \frac{d^{4}x}{4\pi ^{2}{}} \frac{e^{ip\cdot x}}{x^{%l.308 ,...
2n+4}{}} =\frac{p^{2n}}{\left( -4\right) ^{n+1}n!\left( n+1%l.309 ...
\right) !} \left( \ln\left( \frac{p^{2}}{\bar{\Lambda }^{2}}%l.310 (,)...
\right) -\frac{1}{n+1}-2\sum _{k=1}^{n}{} \frac{1}{k} \right) %l.311 $
\hspace{.6ex}.%l.312 $
\end{equation} %l.313
Moreover we can derive this equation in the spirit of differential %l.314
renormalization by Fourier transforming and translating the Laplacian %l.315
$ \Box $ as $ -p^{2}$. But then we have to add $ \delta (x)$-terms %l.316
in Eqs.\ (\ref{1}) and (\ref{4}) which are now uniquely fixed as %l.317
\begin{equation}%l.318 $
\label{19}\Box ^{n+1}{} \frac{1}{x^{2}{}} \ln\left( \frac{x^{2}}%l.319 ...
{\Lambda ^{2}}\right)  =-4^{n+1}n!\left( n+1\right) !\frac{1}{x^{%l.320 ,...
2n+4}{}} +\left( 2\sum _{k=1}^{n}{} \frac{1}{k}+\frac{1}{n+1} %l.321 $
\right) \Box ^{n}\delta \left( x\right) \hspace{.6ex}.%l.322 $
\end{equation} %l.323
The fundamental divergent integrals (\ref{10}) and (\ref{11}) are %l.324
easily generalized to\footnote{In fact it is not possible to introduce %l.325 f
different renormalization scales $\Lambda _{m}$ in Eq.\ (\ref{13}) %l.326 f
as can e.g.\ be seen by comparing the $ (m+1)$-fold one-dimensional %l.327 f
convolution of $ |r|^{-1}$ with the $ (m+1)$st power of the Fourier %l.328 f
transform of $ |r|^{-1}$.} %l.329
\begin{eqnarray}%l.330 $$
\label{12}\int _{0}^{1}dr r^{n}\ln^{m}\left( r\right) &=&\frac{%l.331 (,)...
\left( -1\right) ^{m}m!}{\left( n+1\right) ^{m+1}{}} %l.332 $$
\hspace{.6ex},\hspace{2ex}n\neq -1\hspace{.6ex},\hspace{2ex}m%l.333 $$
\in {\ErgoBbb N}_{0}\hspace{.6ex},\\ %l.334 $$
\label{13}\int _{0}^{1}dr \frac{\ln^{m}\left( r\right) }{r} &=&-%l.335 ,...
\frac{\ln^{m+1}\left( \Lambda \right) }{m+1} %l.336 $$
\hspace{.6ex},\hspace{2ex}m\in {\ErgoBbb N}_{0}\hspace{.6ex}.%l.337 $$
\end{eqnarray} %l.338
So far we have only discussed singularities located at zero. By translation %l.339
we can shift the poles to any point of ${\ErgoBbb R}$. At infinity %l.340
however one could introduce a new renormalization scale $ %l.341 $
\Lambda _{\infty }$ according to %l.342
\begin{equation}%l.343 $
\label{20}\int _{1}^{\infty }\frac{dr}{r} =\ln\Lambda _{\infty }%l.344 $
\hspace{.6ex}.%l.345 $
\end{equation} %l.346
$\Lambda $ should be proportional to $ \Lambda _{\infty }$ for dimensional %l.347
reasons and it is very convenient\footnote{By Fourier transforms, %l.348 f
e.g., singularities at zero are mapped to singularities at infinity. %l.349 f
Eq.\ (\ref{18}) could also be obtained by an ($ n+2$)-fold convolution %l.350 f
of $ p^{-2}$ (the Fourier transform of $ x^{-2}$). In this case %l.351 f
the integrals are divergent at infinity and our result would depend %l.352 f
on $\Lambda _{\infty }$. Comparison with (\ref{18}) leads to (\ref{D}) %l.353 f
\cite{Schnetz}.} to set %l.354
\begin{equation}%l.355 $
\label{D}\Lambda =\Lambda _{\infty }\hspace{.6ex}.%l.356 $
\end{equation} %l.357
This allows us to generalize Eq.\ (\ref{20}) to %l.358
\begin{eqnarray}%l.359 $$
\label{14}\int _{1}^{\infty }drr^{n}\ln^{m}\left( r\right) &=&-%l.360 ,...
\frac{\left( -1\right) ^{m}m!}{\left( n+1\right) ^{m+1}{}} %l.361 $$
\hspace{.6ex},\hspace{2ex}n\neq -1\hspace{.6ex},\hspace{2ex}m%l.362 $$
\in {\ErgoBbb N}_{0}\hspace{.6ex},\\ %l.363 $$
\label{15}\int _{1}^{\infty }dr\frac{\ln^{m}\left( r\right) }{r} %l.364 $$
&=&\frac{\ln^{m+1}\left( \Lambda \right) }{m+1} %l.365 $$
\hspace{.6ex},\hspace{2ex}m\in {\ErgoBbb N}_{0}\hspace{.6ex},%l.366 $$
\end{eqnarray} %l.367
and together with Eqs.\ (\ref{12}) and (\ref{13}) we get %l.368
\begin{equation}%l.369 $
\label{16}\int _{0}^{\infty }drr^{n}\ln^{m}\left( r\right) =0%l.370 $
\hspace{.6ex}.%l.371 $
\end{equation} %l.372
All the integrals defined so far can be summarized by the convention %l.373
\begin{equation}%l.374 $
\label{01}0^{n}\equiv \infty ^{n}\equiv 0\hspace*{1ex}\forall n%l.375 $
\neq 0\hspace{.6ex},\hspace{2ex}\ln0\equiv \ln\infty \equiv \ln%l.376 $
\Lambda \hspace{.6ex}.%l.377 $
\end{equation} %l.378
Note that these equations are symmetric under the interchange of %l.379
zero and infinity which comes from the close connection to analytic %l.380
continuation.%l.381

\vspace{1ex}%l.383
\noindent{}We close this section with some remarks on changing variables. %l.384
Integrals that converge at infinity may be shifted by definition. %l.385
However a naive rescaling $ r\mapsto ar$ in Eq.\ (\ref{11}) leads %l.386
to %l.387
\begin{equation}%l.388 $
\label{24}\int _{0}^{1/a}\frac{dar}{ar} =\int _{0}^{1}\frac{dr}{r%l.389 ...
} +\int _{1}^{1/a}\frac{dr}{r} =-\ln\left( a\Lambda \right) %l.390 $
\neq -\ln\Lambda \hspace{.6ex}.%l.391 $
\end{equation} %l.392
To keep Eq.\ (\ref{11}) invariant under rescalings one has to treat %l.393
the lower limit zero like a variable and write $ \int _{0}^{1}dr%l.394 $
/r=\int _{0/a}^{1/a}dr/r=\ln(1/a)-\ln(0/a)=-\ln\Lambda $. Or, equivalently, %l.395
one rescales the renormalization scale $\Lambda $ according to $ %l.396 $
\Lambda \mapsto \Lambda /a$. If, like in Eq.\ (\ref{10}), the integral %l.397
does not depend on $\Lambda $, rescalings do not affect the result. %l.398
For more complicated variable substitutions it is always appropriate %l.399
to return to the original variables before one approaches the limits %l.400
(cf.\ the bipyramide graph in Sec.\ \ref{bipyramide}).%l.401

\subsection{Relation to the theory of generalized functions\label%l.403 {
{genfunct}} %l.404
We recognized already in the last section that Eq.\ (\ref{a}) can %l.405
be understood in the context of analytic continuation. In order %l.406
to include Eq.\ (\ref{11}) into this concept one has to 'care for %l.407
dimensions' and multiply the integrand by the dimensionless factor %l.408
$ (r/\Lambda )^\alpha $, $ \alpha \mapsto 0$, %l.409
\begin{displaymath}%l.410 $*
\int _{0}^{1}drr^{n}\left( \frac{r}{\Lambda }\right) ^\alpha =%l.411 ,...
\frac{\Lambda ^{-\alpha }}{n+\alpha +1} \hspace*{1ex}%l.412 $*
\hspace{.6ex},%l.413 $*
\end{displaymath} %l.414
which gives $ (n+1)^{-1}$ for $ \alpha =0$, $ n\neq -1$ and $ %l.415 $
\alpha ^{-1}-\ln\Lambda $ for $ n=-1$. If, according to Eq.\ (\ref{01}), %l.416
we replace $\alpha ^{-1}$ by zero we are back at (\ref{11}). Note %l.417
that analytic continuation is only correct if one uses the factor %l.418
$ (r/\Lambda )^\alpha $ and if there exists an $\alpha $-region %l.419
in ${\ErgoBbb C}$ where the integral converges. This prescription %l.420
differs from dimensional regularization by the absence of the surface %l.421
area $ \Omega _{\alpha +1}$. In general the $\alpha $-dependence %l.422
of $ \Omega _{\alpha +1}$ cannot be compensated by a redefinition %l.423
of the renormalization scale.%l.424

There are other contexts in which we can understand the renormalization %l.426
scheme. Such are contour integrals in the complex plane or lattice %l.427
theory which generalizes the Riemann sum prescription and eventually %l.428
provides a purely numerical definition of divergent integrals \cite{Schnetz}. %l.429
Here we present the relation to the theory of generalized functions.%l.430

Assume we are interested in an integral which contains the generalized %l.432
function $f$ that is given as derivative of another generalized %l.433
function $ F'=f$. With a test function $\varphi $ we obtain (e.g.\ %l.434
\cite{Gelfand}) %l.435
\begin{equation}%l.436 $
\label{23}\int _{-\infty }^{+\infty }dxf\left( x\right) \varphi %l.437 (,)...
\left( x\right) \equiv \left( f,\varphi \right) \equiv \left( F',%l.438 (,)...
\varphi \right) \equiv -\left( F,\varphi '\right) \equiv -\int _{%l.439 ,...
-\infty }^{+\infty }dxF\left( x\right) \varphi '\left( x%l.440 $
\right) \hspace{.6ex}.%l.441 $
\end{equation} %l.442
If $\varphi $ is sufficiently constant at the poles of $F$ the right %l.443
hand side converges and can be used to define the integral on the %l.444
left hand side.%l.445

Let us e.g.\ take $ f(x)=x^\lambda \Theta (x)$; $ \Theta (x)=1$ for %l.447
$ x>0$ and $ \Theta (x)=0$ for $ x<0$. We choose $ \varphi =%l.448 $
\Theta (1-|x|)$ where the edges at $ x=\pm 1$ may be smoothed to %l.449
be $ C^{\infty }$. In the limit where this becomes irrelevant we %l.450
have for $ \lambda \neq -1$ %l.451
\begin{equation}%l.452 $
\label{26a}\left( f,\varphi \right) =\int _{0}^{1}dxx^\lambda %l.453 $
\equiv -\int _{0}^{\infty }dx\frac{x^{\lambda +1}}{\lambda +1} %l.454 (,)...
\left( -\delta \left( x-1\right) \right) =\frac{1}{\lambda +1}%l.455 $
\end{equation} %l.456
which coincides with our renormalization rule (\ref{10}). %l.457
For $ \lambda =-1$ we may take $ F(x)=\ln(x)\Theta (x)$ and get %l.458
\begin{equation}%l.459 $
\label{27}\int _{0}^{1}\frac{dx}{x} =-\int _{0}^{\infty }dx\ln%l.460 (,)...
\left( x\right) \left( -\delta \left( x-1\right) \right) =0%l.461 $
\end{equation} %l.462
which is Eq.\ (\ref{11}) for $ \Lambda =1$. The same holds for Eqs.\ %l.463
(\ref{12}) and (\ref{13}).%l.464

To see what happened with the renormalization scale $\Lambda $ we %l.466
have to notice that the above calculation is ambiguous. There exist %l.467
several functions $F$ which have the same derivative. On the real %l.468
line they differ by a constant which is irrelevant since the test %l.469
function $\varphi $ vanishes at $ \pm \infty $.%l.470

In general however the number of undetermined parameters equals the %l.472
number of disconnected pieces of the integration domain. A singularity %l.473
of the integrand $f$ at the origin splits ${\ErgoBbb R}$ into two %l.474
disconnected parts $ {\ErgoBbb R}^-$ and $ {\ErgoBbb R}^+$. Each %l.475
of the functions $ F(x)+C+D\Theta (x)$ is with the same right an %l.476
integral of $ f(x)$ on the real line. However they give different %l.477
results to the integrals (\ref{26a}) and (\ref{27}), %l.478
\begin{eqnarray}%l.479 $$
\label{28}&&\hspace*{-0.9cm}\int \limits _{0}^{1}\!\!dxx^%l.480 $$
\lambda \equiv -\!\!\!\int \limits _{-\infty }^{\infty }\hspace%l.481 $$
{-1ex} dx\left( C_\lambda +\Theta \left( x\right) \!\left( %l.482 ,...
\frac{x^{\lambda +1}}{\lambda +1} +D_\lambda \right) \!\right) %l.483 (,)...
\left( \delta \left( x+1\right) -\delta \left( x-1\right) %l.484 $$
\right) =\frac{1}{\lambda +1} +D_\lambda ,\\ %l.485 $$
\label{29}&&\hspace*{-0.9cm}\int \limits _{0}^{1}\frac{dx}{x} %l.486 $$
\equiv -\int \limits _{-\infty }^{\infty }\hspace{-1ex} dx%l.487 (,)...
\left( C_{-1}+\Theta \left( x\right) \left( \ln x+D_{-1}%l.488 (,)...
\right) \right) \left( \delta \left( x+1\right) -\delta \left( x%l.489 (,)...
-1\right) \right) =D_{-1}\hspace{.6ex},%l.490 $$
\end{eqnarray} %l.491
where all the $ C_\lambda $ and $ D_\lambda $ can be chosen separately.%l.492

The way out of this ambiguity is to change the topology of the integration %l.494
domain. We can compactify ${\ErgoBbb R}$ to $ \overline {%l.495 ,...
{\ErgoBbb R}}$ by adding $\infty $ ($ =-\infty $). Since for $ %l.496 $
\lambda <-1$ the integral over $ x^\lambda $ is well-defined at %l.497
infinity we can define $F$ over $ \overline {{\ErgoBbb R}}%l.498 $
\backslash \{0\}$ which is again a connected domain with one integration %l.499
constant. The $ D_\lambda \Theta (x)$ term is discontinuous at infinity %l.500
and thus no longer allowed in Eq.\ (\ref{28}). The integral $ %l.501 $
\int _{0}^{1}dxx^\lambda $ acquires again the unique value $ (%l.502 $
\lambda +1)^{-1}$.%l.503

For $ \lambda \ge -1$ the integral diverges at infinity and the gluing %l.505
is not possible. However for $ \lambda >-1$ the integral is finite %l.506
at $ x=0$ and can be defined on the connected domain $%l.507
{\ErgoBbb R}$. Just for the case $ \lambda =-1$ the ambiguity remains %l.508
since the integral is divergent both at zero and infinity. The function %l.509
$ \ln |x|$ can only be defined on $ {\ErgoBbb R}^-\cup %l.510 $
{\ErgoBbb R}^+$ and one should keep the arbitrary constant $ D_{%l.511 ,...
-1}$ in Eq.\ (\ref{29}). With the more intuitive relabeling $ D_{%l.512 ,...
-1}=-\ln\Lambda $ we are back at Eq.\ (\ref{11}).%l.513

In practice the introduction of the renormalization scale $%l.515
\Lambda $ is a matter of convenience. It will turn out to be useful %l.516
to have this parameter at hand. In principle one could set $ %l.517 $
\Lambda =1$ using the standard theory of generalized functions and %l.518
recover $\Lambda $ in physical results by getting the dimensions %l.519
right in logarithmic terms.%l.520

\section{Applications} %l.522
Now we turn to physical applications. In the following we are mainly %l.523
concerned with four-dimensional integrals which we normalize according %l.524
to %l.525
\begin{equation}%l.526 $
\label{54}\int dx\equiv \int _{{\ErgoBbb R}^{4}}\frac{d^{4}x}{%l.527 (,)...
\left( 2\pi \right) ^{2}{}} \hspace*{1cm}\hbox{\hspace{.38ex}and\hspace{.38ex}}%l.528 $
\hspace*{1cm}\delta \left( x\right) =\left( 2\pi \right) ^{2}%l.529 $
\delta ^{\left( 4\right) }\left( \vec{x}\right) \hspace{.6ex}.%l.530 $
\end{equation} %l.531
This eliminates all irrelevant factors of $\pi $ from the theory. %l.532
We get e.g.\ $ \int dx\delta (x)=1$ and $ \int dxe^{ip\cdot x}=%l.533 $
\delta (p)$. Analogously $n$-dimensional integrals are normalized %l.534
by $ (2\pi )^{(-n/2)}$. The metric is always Euclidean.%l.535

\subsection{A toy example: the free massive bosonic theory\label%l.537 {
{freemass}} %l.538
As a first test let us calculate the four-dimensional free massive %l.539
boson propagator in coordinate space. The result is well known, %l.540
\begin{eqnarray}%l.541 $$
\Delta \left( x\right) &=&\int dp\frac{e^{-ip\cdot x}}{p^{2}+m^{%l.542 ,...
2}{}} =\frac{m}{|x|} K_{1}\left( m|x|\right) \nonumber \\ %l.543 $$
\label{51}&=&\frac{1}{x^{2}{}} +\frac{m^{2}}{4}\sum _{k=0}^{%l.544 ,...
\infty }\frac{\left( m^{2}x^{2}{} /4\right) ^{k}}{k!\left( k+1%l.545 ...
\right) !} \left( \ln\left( \frac{m^{2}x^{2}}{4}\right) -\Psi %l.546 (,)...
\left( k+2\right) -\Psi \left( k+1\right) \right) %l.547 $$
\hspace{.6ex}.%l.548 $$
\end{eqnarray} %l.549
The propagator $\Delta $ is perfectly well-defined. However it is %l.550
not analytic at $ m=0$ since the series contains logarithmic terms %l.551
in $m$.%l.552

Now let us treat the mass $ (-m^{2})$ as a two-point interaction %l.554
and study perturbation theory around $ m=0$. %l.555
\begin{equation}%l.556 $
\bullet {\Etcompose{\Etcompose{\lline}{\raisebox{ .5pt}{$\lline%l.557 { ,...
$}}}{\raisebox{-.5pt}{$\lline$}}}\bullet =\bullet \lline%l.558 $
\bullet +\Etcompose{\bullet \lline\bullet \lline\bullet }{%l.559 ,...
\raisebox{-1.3ex}{$\scriptstyle -m^{2}$}}\hspace{ 6.8ex} +%l.560 $
\Etcompose{\bullet \lline\bullet \lline\bullet \lline\bullet }{%l.561 ,...
\raisebox{-1.3ex}{$\scriptstyle -m^{2}\hspace{ 4ex} -m^{2}$}}%l.562 $
\hspace{ 6.8ex} +{\ldots}%l.563 $
\end{equation} %l.564
The free propagator in four dimensions is %l.565
\begin{equation}%l.566 $
\int dp\frac{1}{p^{2}{}} e^{-ip\cdot x}=\frac{1}{x^{2}{}} %l.567 $
\hspace{.6ex},%l.568 $
\end{equation} %l.569
and therefore %l.570
\begin{equation}%l.571 $
\Delta \left( x\right) =\frac{1}{x^{2}{}} -m^{2}\int dx_{1}%l.572 ,...
\frac{1}{\left( x\!-\!x_{1}\right) ^{2}{}} \frac{1}{x_{1}^{2}{}%l.573 ...
} +\left( -m^{2}\right) ^{2}\int \int dx_{1}dx_{2}\frac{1}{%l.574 (,)...
\left( x\!-\!x_{1}\right) ^{2}{}} \frac{1}{\left( x_{1}\!-\!x_{%l.575 ,...
2}\right) ^{2}{}} \frac{1}{x_{2}^{2}{}} +{\ldots}\hspace{.6ex}.%l.576 $
\end{equation} %l.577
Since the 'coupling' has mass-dimension the terms become more divergent %l.578
with every order and the expansion is non-renormalizable. However %l.579
we can treat the integrals according to our rules and obtain an %l.580
unambiguous result which contains by construction only one renormalization %l.581
scale $\Lambda $.%l.582

In order to evaluate the integrals we can use $ \Box (x-x_{1})^{%l.584 ,...
-2}=-\delta (x-x_{1})$ and Eq.\ (\ref{2}) to derive a recursive %l.585
formula. Here it is even simpler to remember that the $n$-th term %l.586
is the Fourier transform of $ p^{-2n}$. Eq.\ (\ref{18}) gives %l.587
\begin{eqnarray}%l.588 $$
\Delta _{{\rm nat.ren.\hspace{.38ex}}}\left( x\right) &=&\frac{1%l.589 ,...
}{x^{2}{}} +\sum _{k=0}^{\infty }\left( -m^{2}\right) ^{k+1}%l.590 $$
\int dp\frac{1}{p^{2k+4}{}} e^{ip\cdot x}\nonumber \\ %l.591 $$
&=&\frac{1}{x^{2}{}} +\frac{m^{2}}{4}\sum _{k=0}^{\infty }%l.592 ,...
\frac{\left( m^{2}x^{2}{} /4\right) ^{k}}{k!\left( k+1\right) !%l.593 ...
} \left( \ln\left( \frac{\Lambda ^{2}x^{2}}{4}\right) -\Psi %l.594 (,)...
\left( k+2\right) -\Psi \left( k+1\right) \right) \!.%l.595 $$
\end{eqnarray} %l.596
One could study the renormalization group by looking at rescalings %l.597
of $\Lambda $. In fact comparison with Eq.\ (\ref{51}) shows that %l.598
the situation is even simpler. We just have to equate $ %l.599 $
\Lambda =m$ to obtain precisely the correct result. This does not %l.600
happen accidentally as we will see in the next section.%l.601

\vspace{1ex}%l.603
\noindent{}The differentially renormalized result can be obtained %l.604
by using Eq.\ (\ref{5}) instead of Eq.\ (\ref{18}), %l.605
\begin{eqnarray}%l.606 $$
\Delta _{{\rm diff.ren.\hspace{.38ex}}}\left( x\right) &=&\frac{%l.607 ,...
1}{x^{2}{}} +\frac{m^{2}}{4}\sum _{k=0}^{\infty }\frac{\left( m^{%l.608 ,...
2}x^{2}{} /4\right) ^{k}}{k!\left( k+1\right) !} \left( \ln%l.609 (,)...
\left( \frac{\Lambda _{k}^{2}x^{2}}{4}\right) -2C\right) %l.610 $$
\hspace{.6ex}.%l.611 $$
\end{eqnarray} %l.612
It is necessary to adjust the infinitely many parameters $ %l.613 $
\Lambda _{k}$ precisely according to the a priori settings (\ref{c}) %l.614
of our scheme.%l.615

\vspace{1ex}%l.617
\noindent{}Dimensional regularization leads to a series in $ 4-n%l.618 $
$ with a simple pole, %l.619
\begin{eqnarray}%l.620 $$
\label{E}\Delta _{{\rm dim.reg.\hspace{.38ex}}}\left( x\right) &%l.621 $$
=&\frac{1}{x^{2}{}} +\frac{m^{2}}{4}\sum _{k=0}^{\infty }\frac{%l.622 (,)...
\left( m^{2}x^{2}{} /4\right) ^{k}}{k!\left( k+1\right) !} %l.623 (,)...
\left( \frac{2}{4-n}+\ln\left( \frac{\Lambda ^{2}x^{2}}{2}%l.624 (,)...
\right) -\Psi \left( k+1\right) \right) \hspace{.6ex}.%l.625 $$
\end{eqnarray} %l.626
Since the series is non-renormalizable, it is not possible to renormalize %l.627
by introducing a finite number of counter terms. If one nevertheless %l.628
tries to follow a minimum subtraction prescription, one misses a %l.629
term $ -\Psi (k+2)$ to obtain the correct result. In the next section %l.630
we will present a general method that allows us to calculate this %l.631
term.%l.632

\subsection{A theorem on singular expansions} %l.634
Let us summarize what we did in the last section. We started from %l.635
a well-defined integral $ \int dp\exp(-ip\cdot x)/(p^{2}+m^{2})%l.636 $
$ which we tried to expand into a series at $ m=0$. To this end %l.637
we expanded the integrand into a power series $ \int dp\exp(-ip%l.638 $
\cdot x)\sum _{k=0}^{\infty }(-m^{2})^{k}p^{-2k-2}$. The interchange %l.639
of the sum and the integral led us to the perturbation series in %l.640
coordinate space $ \sum _{k=0}^{\infty }(-m^{2})^{k}\int dp\exp%l.641 $
(-ip\cdot x)p^{-2k-2}$. This interchange is obviously illegal. Firstly, %l.642
the integrals diverge at $ p=0$. Secondly, we obtain a power series %l.643
in $m$ and we know that the correct result has no such representation %l.644
but contains logarithmic terms in $m$  (Eq.\ (\ref{51})). Although %l.645
the integrand is analytic at $ m=0$ the integral is not. So necessarily %l.646
the expansion is wrong and the diverging integrals reflect this %l.647
fact. We want to study the issue how to reconstruct the true result %l.648
from such an incorrect, singular expansion.%l.649

\vspace{1ex}%l.651
\noindent{}Let us slightly generalize the situation and look for %l.652
the expansion of an integral $ I(a)=\int dxf(x,a)$ into a series %l.653
at $ a=0$. The integrand has a Taylor (or Laurant) series $ f(x%l.654 $
,a)=\sum _{k}a^{k}f^{(k)}(x,0)/k!$, but in general we can not expect %l.655
that the series of $ I(a)$ is given by the integrals over the coefficients %l.656
$ f^{(k)}(x,0)$ since the integrals may diverge. We define %l.657
\begin{equation}%l.658 $
\Delta I\left( a\right) =\int dxf\left( x,a\right) -\sum _{k}%l.659 ,...
\frac{a^{k}}{k!} \int dxf^{\left( k\right) }\left( x,0\right) %l.660 $
\hspace{.6ex},%l.661 $
\end{equation} %l.662
and conclude that $ \Delta I(a)$ will only be zero if $ I(a)$ is %l.663
analytic at $ a=0$. So $ \Delta I(a)$ gives the part of the expansion %l.664
of $ I(a)$ that can not be reached by standard perturbation theory.%l.665

We call $ \Delta I$ the non-perturbative part of the expansion. A %l.667
priori we know almost nothing about it. However in many cases where %l.668
$ I(a)$ is not analytic at $ a=0$ one can calculate $ \Delta I(a%l.669 $
)$ by the following theorem.%l.670

\pagebreak[3]%l.672

\noindent {\bf Theorem}. Assume the integrals $ \int dxf^{(k)}%l.674 $
(x,0)$ are regular at $ x\neq 0$. If there exists a neighborhood %l.675
of $ x=0$, $ a=0$ where $f$ can be written as $ f=\sum _{\ell }f%l.676 $
_{\ell }$ with $ f_{\ell }(x,a)$ integrable at $ x=0$ and the $ f%l.677 $
_{\ell }^{(k)}(x,0)$ having the following properties %l.678
\begin{eqnarray}%l.679 $$
\label{41a}\lefteqn{f_{\ell }^{\left( k\right) }\left( |x|,0%l.680 { $$
\right) \propto |x|^{n\left( k,\ell \right) }\ln^{m\left( k,%l.681 (,)...
\ell \right) }\left( |x|\right) \hspace*{2ex}\hbox%l.682 "...
{\hspace{.38ex}with\hspace{.38ex}}}\\ %l.683 $$
\label{36a}&&\hbox{\hspace{.38ex}given }\ell :\hspace*{2ex}\lim_{%l.684 ,...
k\rightarrow \infty }n\left( k,\ell \right) /k <0\hbox{ or }f_{%l.685 ,...
\ell }^{\left( k\right) }\left( x,0\right) \equiv 0\hbox{ for almost %l.686 "...
all\footnotemark{} }k\hspace*{2ex}\hbox{\hspace{.38ex}and\hspace{.38ex}}%l.687 $$
\\ %l.688 $$
\label{36b}&&\hbox{\hspace{.38ex}given }k:\hspace*{2ex}n\left( k%l.689 (,)...
,\ell \right) \ge 0\hbox{ for almost all }\ell \hspace{.6ex},%l.690 $$
\end{eqnarray} %l.691
then %l.692
\begin{equation}%l.693 $
\Delta I\left( a\right) =\sum _{\ell }\int dxf_{\ell }\left( x,a%l.694 $
\right) \hspace{.6ex}.%l.695 $
\end{equation}\footnotetext{all up to a finite number}%l.696

\vspace{1ex}%l.698
\noindent{}Note that all the integrals may diverge and have to be %l.699
defined according to the rules given above. The range of integration %l.700
can be $ {\ErgoBbb R}^{n}$ or $ {\ErgoBbb R}^+$, subsets can be %l.701
taken into account by using step-functions.%l.702

\pagebreak[3]%l.704

\noindent {\bf Proof} (sketch). Without restriction we can assume %l.706
that the support of $f$ is a little ball $ B_\varepsilon $ around %l.707
$ x=0$ since the integrals over the remaining domain are regular %l.708
and therefore do not contribute to $ \Delta I$. Moreover we can %l.709
assume $ |a|$ to be small, so that we can write $f$ as a sum over %l.710
$ f_{\ell }$. Since the $ f_{\ell }$ are integrable at $ x=0$ one %l.711
gets for sufficiently small $\varepsilon $ %l.712
\begin{eqnarray*}%l.713 $$*
\int _{B_\varepsilon }dxf\left( x,a\right) &=&\sum _{\ell }\int _%l.714 { $$*
{B_\varepsilon }dxf_{\ell }\left( x,a\right) \\ %l.715 $$*
&=&\sum _{\ell }\int _{{\ErgoBbb R}^{n}}dxf_{\ell }\left( x,a%l.716 $$*
\right) -\sum _{\ell }\int _{{\ErgoBbb R}^{n}\backslash B_%l.717 { $$*
\varepsilon }dx\sum _{k}\frac{a^{k}}{k!} f_{\ell }^{\left( k%l.718 { $$*
\right) }\left( x,0\right) \hspace{.6ex}.%l.719 $$*
\end{eqnarray*} %l.720
In the second integral the singularity at $ x=0$ is excluded and %l.721
(\ref{36a}) assures that the sum can be interchanged with the integral %l.722
for small enough $a$, yielding %l.723
\begin{displaymath}%l.724 $*
\int _{B_\varepsilon }dxf\left( x,a\right) =\sum _{\ell }\int _{%l.725 { $*
{\ErgoBbb R}^{n}}dxf_{\ell }\left( x,a\right) +\sum _{\ell }%l.726 $*
\sum _{k}\frac{a_{k}}{k!} \left( \int _{B_\varepsilon }-\int _{%l.727 { (,)...
{\ErgoBbb R}^{n}}\right) dxf_{\ell }^{\left( k\right) }\left( x%l.728 (,)...
,0\right) \hspace{.6ex}.%l.729 $*
\end{displaymath} %l.730
Now we can use the central argument of the proof. The last integral %l.731
over the entire $ {\ErgoBbb R}^{n}$ vanishes since (Eq.\ (\ref{41a})) %l.732
it is proportional to $ \int _{0}^{\infty }dr\,r^N\ln^M(r)$ and %l.733
all those integrals are zero in our renormalization scheme (Eq.\ %l.734
(\ref{16})). We finally use Eq.\ (\ref{36b}) to interchange the %l.735
second sum over $ \ell $ with the sum over $k$ and the integral. %l.736
Therefore %l.737
\begin{displaymath}%l.738 $*
\Delta I\left( a\right) \equiv \int _{B_\varepsilon }dxf\left( x%l.739 (,)...
,a\right) -\sum _{k}\frac{a^{k}}{k!} \int _{B_\varepsilon }dxf^%l.740 { $*
{\left( k\right) }\left( x,0\right) =\sum _{\ell }\int _{%l.741 { $*
{\ErgoBbb R}^{n}}dxf_{\ell }\left( x,a\right) \hspace{.6ex}.%l.742 $*
\end{displaymath}\hfill $\Box $%l.743
\pagebreak[3] %l.744

\vspace{1ex}%l.746
\noindent{}Now let us use the theorem to derive $ \Delta I(m^{2}%l.747 $
)$ of the scalar bosonic theory. We get $ f(p,m^{2})=e^{-ip%l.748 { $
\cdot x}/(p^{2}+m^{2})$. Expanding the exponential yields $ f(p%l.749 $
,m^{2})=\sum _{\ell }(-ip\cdot x)^{\ell }{} /(\ell !(p^{2}+m^{2%l.750 ,...
}))\equiv \sum _{\ell }f_{\ell }(p,m^{2})$. Moreover $ f_{\ell %l.751 ,...
}^{(k)}(p,0)=(-ip\cdot x)^{\ell }(-1)^{k}k!p^{-2-2k}{} /\ell !$ %l.752
and $ f_{\ell }^{(k)}(|p|,0)\propto |p|^{\ell -2-2k}$ meets Eqs.\ %l.753
(\ref{41a}), (\ref{36a}), (\ref{36b}). We can apply the theorem %l.754
and obtain %l.755
\begin{equation}%l.756 $
\label{F}\Delta I_{{\rm nat.ren.\hspace{.38ex}}}\left( m^{2}%l.757 $
\right) =\sum _{\ell =0}^{\infty }\int \frac{d^{4}x}{4\pi ^{2}%l.758 ...
{}} \frac{\left( -ip\cdot x\right) ^{\ell }}{\ell !\left( p^{2}%l.759 (,)...
+m^{2}\right) } =\sum _{\ell =0}^{\infty }\frac{\left( -i|x|%l.760 ,...
\right) ^{\ell }}{\ell !} \frac{1}{\pi } \int _{0}^{\pi }d%l.761 $
\vartheta \sin^{2}\vartheta \cos^{\ell }\vartheta \int _{0}^{%l.762 ,...
\infty }\frac{dp p^{\ell +3}}{p^{2}+m^{2}{}} .%l.763 $
\end{equation} %l.764
The $\vartheta $-integral vanishes for odd $ \ell $. The divergent %l.765
$p$-integral can be reduced to fundamental integrals as follows: %l.766
$ \int _{0}^{\infty }dp p^{2\ell +3}{} /(p^{2}+m^{2})=\int _{0}^{%l.767 ,...
\infty }pdp ((p^{2}+m^{2})-m^{2})^{\ell +1}{} /(p^{2}+m^{2})$. With %l.768
Eq.\ (\ref{16}) we obtain $ (-m^{2})^{\ell +1}\int _{0}^{%l.769 ,...
\infty }pdp/(p^{2}+m^{2}) =\frac{1}{2}(-m^{2})^{\ell +1}\ln(p^{%l.770 ,...
2}+m^{2})|^{\infty }_{0}=\frac{1}{2}(-m^{2})^{\ell +1}\ln(%l.771 $
\Lambda ^{2}{} /m^{2})$ (with Eq.\ (\ref{01})\footnote{More precisely %l.772 f
$ \int _{0}^{\infty }pdp/(p^{2}+m^{2})=\int _{0}^{1}pdp/(p^{2}+m%l.773 $...
^{2})+\int _{1}^{\infty }dp(p/(p^{2}+m^{2})-1/p)+\int _{1}^{%l.774 ,...
\infty }dp/p =\frac{1}{2}\ln(1+m^{-2})-\frac{1}{2}\ln(1+m^{2})+%l.775 $...
\ln\Lambda =\ln\Lambda /m$.}). The result is proportional to $ %l.776 $
\ln(\Lambda /m)$ and vanishes thus for $ \Lambda =m$. This confirms %l.777
the explicit calculation of the last section.%l.778

The above theorem holds for any spacetime dimension. Hence it should %l.780
as well be possible to apply it to the dimensionally regularized %l.781
result and 'correct' Eq.\ (\ref{E}) by adding the non-perturbative %l.782
part. We start with the $n$-dimensional analogon of Eq.\ (\ref{F}). %l.783
All integrals are standard and one obtains %l.784
\begin{eqnarray}%l.785 $$
 \Delta I_{{\rm dim.reg.\hspace{.38ex}}}\left( m^{2}\right) &=&%l.786 $$
\sum _{\ell =0}^{\infty }\frac{\left( -i|x|\right) ^{\ell }}{%l.787 ...
\ell !} \frac{\Omega _{n-1}}{\left( 2\pi \right) ^{n/2}{}} %l.788 $$
\Lambda ^{4-n}{} \int _{0}^{\pi }d\vartheta \sin^{n-2}%l.789 $$
\vartheta \cos^{\ell }\vartheta \int _{0}^{\infty }\frac{dp p^{%l.790 ,...
\ell +n-1}}{p^{2}+m^{2}{}} \nonumber \\ %l.791 $$
&=&\sum _{\ell =0}^{\infty }\frac{m^{2}\left( -x^{2}m^{2}%l.792 ,...
\right) ^{\ell }}{2^{\ell +2}\ell !} \left( \frac{m^{2}}{2%l.793 ...
\Lambda ^{2}}\right) ^{\left( n-4\right) /2}\Gamma \left( %l.794 ,...
\frac{4-n}{2}-\ell -1\right) \\ %l.795 $$
&=&\sum _{\ell =0}^{\infty }\frac{m^{2\ell +2}x^{2\ell }}{4^{%l.796 ,...
\ell +1}\ell !\left( \ell +1\right) !} \left( -\frac{2}{4-n}-%l.797 (,)...
\Psi \left( \ell +2\right) +\ln\left( \frac{m^{2}}{2\Lambda ^{2%l.798 ,...
}}\right) \right) +{\cal O}\left( 4\hspace{-.5ex}-\hspace{-.5%l.799 (,)...
ex} n\right) .\nonumber %l.800 $$
\end{eqnarray} %l.801
Together with $ \Delta _{{\rm dim.reg.\hspace{.38ex}}}$ (Eq.\ (\ref{E})) %l.802
the renormalization scale drops out and one obtains the full propagator.%l.803

\vspace{1ex}%l.805
\noindent{}So in the example of a free massive bosonic theory we %l.806
do not have to go through the standard renormalization business. %l.807
One can use the above theorem instead. The simplest way to expand %l.808
the propagator is using natural renormalization, however dimensional %l.809
regularization leads eventually to the same result.%l.810

In a realistic field theory with dimensionless coupling the situation %l.812
is slightly different. The path integral is a priori ill-defined %l.813
and the renormalization scale an intrinsic parameter of the theory %l.814
(like e.g.\ in the integral $ \int _{0}^{1}\frac{dx}{x} =-\ln%l.815 $
\Lambda $). It makes no sense to equate the renormalization parameter %l.816
$\Lambda $ with the coupling. However it is challenging to try to %l.817
generalize the theorem to path integrals providing a non-perturbative %l.818
but analytic definition of a quantum field theory.%l.819

\vspace{1ex}%l.821
\noindent{}Anyway the theorem on its own is useful in many elementary %l.822
mathematical applications. Integrals like $ \int _{0}^bdx/(x^{n%l.823 ,...
}+a^{n})$, $ \int _{0}^{1}dx\ln^{m}(x/\Lambda )/(x+a)$, $ \int _{%l.824 ,...
0}^{\infty }dxe^{-bx}/(x+a)^{n}$, $ \int _{0}^{\infty }dxe^{-bx%l.825 { $
-a/x}$, etc.\ can be expanded at $ a=0$ by virtue of the theorem %l.826
\cite{Schnetz}.%l.827

\subsection{Fourier transforms\label{fourier}} %l.829
Before we start to study $\varphi ^{4}$-theory it is useful to discuss %l.830
Fourier transforms since many Feynman amplitudes are determined %l.831
by multiplications and convolutions.%l.832

To this end we generalize the Fourier transforms discussed in the %l.834
beginning (Eq.\ (\ref{18})). It is convenient to derive the result %l.835
by analytic continuation. A straightforward calculation gives ($ %l.836 $
\bar{\Lambda }=2/(e^C\Lambda )$) %l.837
\begin{eqnarray}%l.838 $$
\label{02}\int \!\frac{d^{4}x}{\left( 2\pi \right) ^{2}{}} %l.839 ,...
\frac{e^{ip\cdot x}}{x^{2n+4}{}} \left( \frac{x^{2}}{\Lambda ^{%l.840 ,...
2}}\right) ^{\!-\alpha }\hspace{-2.5ex}&=&\hspace{-.5ex}\frac{p^{%l.841 ,...
2n}}{2^{2n+2}{}} \left( \frac{p^{2}}{\bar{\Lambda }^{2}}%l.842 $$
\right) ^{\!\alpha }{} \frac{e^{-2C\alpha }\Gamma \left( -n-%l.843 (,)...
\alpha \right) }{\Gamma \left( n+2+\alpha \right) } \equiv %l.844 ,...
\frac{p^{2n}}{2^{2n+2}{}} \left( \frac{p^{2}}{\bar{\Lambda }^{2%l.845 ,...
}}\right) ^{\!\alpha }\sum _{\ell =-1}^{\infty }a_{n,\ell }%l.846 $$
\alpha ^{\ell },\hspace*{ 3ex}\\ %l.847 $$
\hbox{\hspace{.38ex}where\hspace{.38ex}}\hspace*{3ex}\sum _{%l.848 ,...
\ell =-1}^{\infty }a_{n,\ell }\alpha ^{\ell }\hspace{-.5ex}&=&%l.849 $$
\hspace{-.5ex}\frac{\Gamma \left( -n-\alpha \right) }{\Gamma %l.850 (,)...
\left( 1-\alpha \right) } \frac{\Gamma \left( 1+\alpha \right) %l.851 ,...
}{\Gamma \left( n+2+\alpha \right) } \exp\left( \sum _{k=1}^{%l.852 ,...
\infty }\frac{2\zeta \left( 2k+1\right) }{2k+1}\alpha ^{2k+1}%l.853 $$
\right) \hspace{.6ex}.%l.854 $$
\end{eqnarray} %l.855
Now it is easy to determine all Fourier transforms of the form $ x%l.856 $
^{-2n-4}\ln^{m}(x^{2}{} /\Lambda ^{2})$. To produce the logarithms %l.857
we divide Eq.\ (\ref{02}) by $\alpha ^{m}$ and pick up the finite %l.858
term in the $\alpha $-expansion: %l.859
\begin{equation}%l.860 $
\int dx \frac{\ln^{m}\left( x^{2}{} /\Lambda ^{2}\right) }{x^{2n%l.861 ,...
+4}{}} e^{ip\cdot x}=\left( -1\right) ^{m}m!\frac{p^{2n}}{2^{2n%l.862 ,...
+2}{}} \sum _{\ell =-1}^{\infty }a_{n,\ell }g_{\ell -m}\left( %l.863 ,...
\frac{p^{2}}{\bar{\Lambda }^{2}}\right) \hspace{.6ex},%l.864 $
\end{equation} %l.865
where $ g_{\ell }(p^{2}{} /\bar{\Lambda }^{2})$ is the finite term %l.866
of $ (p^{2}{} /\bar{\Lambda }^{2})^\alpha \alpha ^{\ell }$ at $ %l.867 $
\alpha =0$. We obtain $ g_{\ell }=\ln^{|\ell |}(p^{2}{} /\bar%l.868 $
{\Lambda }^{2})/|\ell |!$ for $ \ell \le 0$. For $ \ell $ positive %l.869
$ g_{\ell }(p^{2}{} /\bar{\Lambda }^{2})=0$ for all $ p\neq 0$. %l.870
A more detailed calculation \cite{Schnetz} shows that $ p^{-4}g_{%l.871 ,...
1}(p^{2}{} /\bar{\Lambda }^{2})=\frac{1}{4}\delta (p)$.%l.872

Let us take e.g.\ $ n=1$ yielding $ \sum _{\ell =-1}^{\infty }a_{%l.874 ,...
1,\ell }\alpha ^{\ell }=\frac{1}{2}\alpha ^{-1}{} -\frac{5}{4}+%l.875 ,...
\frac{17}{8}\alpha +(-\frac{49}{64}+\frac{1}{3}\zeta (3))%l.876 $
\alpha ^{2}+{\cal O}(\alpha ^{3})$ and therefore (cf.\ Eq.\ (\ref{18})) %l.877
\begin{eqnarray}%l.878 $$
\label{G}\int dxe^{ip\cdot x}\frac{1}{x^{6}}&=&\frac{1}{16}p^{2}%l.879 (,)...
\left( \frac{1}{2}\ln\left( \frac{p^{2}}{\bar{\Lambda }^{2}}%l.880 (,)...
\right) -\frac{5}{4}\right) \hspace{.6ex},\\ %l.881 $$
\label{H}\int dxe^{ip\cdot x}\frac{\ln\left( x^{2}{} /\Lambda ^{%l.882 ,...
2}\right) }{x^{6}}&=&\frac{1}{16}p^{2}\left( -\frac{1}{4}\ln^{2%l.883 ,...
}\left( \frac{p^{2}}{\bar{\Lambda }^{2}}\right) +\frac{5}{4}\ln%l.884 (,)...
\left( \frac{p^{2}}{\bar{\Lambda }^{2}}\right) -\frac{17}{8}%l.885 $$
\right) \hspace{.6ex}.%l.886 $$
\end{eqnarray} %l.887
$ n=-1$ gives $ \sum _{\ell =-1}^{\infty }a_{-1,\ell }\alpha ^{%l.888 ,...
\ell }=1+\frac{2}{3}\zeta (3)\alpha ^{3}+{\cal O}(\alpha ^{5})$, %l.889
thus %l.890
\begin{equation}%l.891 $
\label{I}\int dxe^{ip\cdot x}\frac{1}{x^{2}{}} =\frac{1}{p^{2}{}%l.892 ...
} \hspace{.6ex},\hspace{2ex}\hspace*{1cm}\int dxe^{ip\cdot x}%l.893 ,...
\frac{\ln\left( x^{2}{} /\Lambda ^{2}\right) }{x^{2}{}} =-%l.894 ,...
\frac{\ln\left( p^{2}{} /\bar{\Lambda }^{2}\right) }{p^{2}{}} %l.895 $
\hspace{.6ex}.%l.896 $
\end{equation} %l.897
With $ n=-2$ we finally obtain the standard formula $ \int dxe^{ip%l.898 { $
\cdot x}=\delta (p)$. Less obvious is $ \int dxe^{ip\cdot x}\ln%l.899 $
(x^{2}{} /\Lambda ^{2})=-4p^{-4}+\delta (p)$.%l.900


\subsection{The massless $\varphi ^{4}$-theory\label{fi4}} %l.903
The first serious test of the renormalization scheme is the discussion %l.904
of the $\varphi ^{4}$-theory. Note that once the Feynman rules and %l.905
the propagator are fixed the results are unique. There is no freedom %l.906
to choose a certain subtraction scheme.%l.907

We keep our integral normalization of $ (2\pi )^{-2}$ which results %l.909
in a rescaling of the coupling by $ (2\pi )^{2}$. So $g$ is related %l.910
to the usual 'irrationalized' coupling via %l.911
\begin{equation}%l.912 $
\frac{g}{4}=\frac{\lambda }{16\pi ^{2}{}} \hspace{.6ex}.%l.913 $
\end{equation}%l.914
\begin{figure}\centerline{\epsfbox{natren1.eps}}
\caption{Feynman diagrams of the $\varphi ^{4}$-theory.}
\end{figure}
The Feynman diagrams we are concerned with are depicted %l.922
in Fig.\ 1 ($ a$), {\dots}, ($ u$). The corresponding amplitudes %l.923
are labeled by $ G_{a}$, {\dots}, $ G_u$. With natural renormalization %l.924
we have the freedom to switch between coordinate and momentum space. %l.925
However, most often it is convenient to start the calculation in %l.926
coordinate space where, at least at higher loops, the Feynman rules %l.927
are more transparent. The final result is given in momentum space %l.928
to make it easier to compare it with other work.%l.929

\subsubsection{Simple results} %l.931
The free propagator ($ a$) is given by $ G_{a}=p^{-2}$.%l.932

The loop in the diagram ($ b$) gives rise to a term $ \int dpp^{%l.934 ,...
-2}$ in momentum space or a term $ (x-x)^{-2}{} =0^{-2}$ in coordinate %l.935
space. Both expressions are set to zero in our scheme: $ G_b=0$. %l.936
In this aspect it behaves like dimensional regularization.%l.937

More generally, diagrams that contain tadpole insertions give zero %l.939
and can be dropped. This remains true for any number of internal %l.940
lines the tadpole may have: $ G_c=0$. The reason is that in a massless %l.941
theory a tadpole insertion can only give rise to a number times %l.942
a momentum conserving $\delta $-function. On the other hand it has %l.943
dimension $ p^{2}$ and the only number with this scaling property %l.944
is zero. In our renormalization scheme $ \bar{\Lambda }^{2}$ occurs %l.945
only in combination with logarithms.%l.946

Moreover, due to translation invariance and Eq.\ (\ref{16}), all %l.948
vacuum bubbles vanish: $ G_u=0$. So only connected diagrams contribute %l.949
to the two-point function. Altogether this reduces the number of %l.950
relevant Feynman diagrams considerably.%l.951

Diagram ($ d$) for $ n=1$ is the sunset diagram. It was already calculated %l.953
in the last section. The triple line gives $ x^{-6}$ which transforms %l.954
into momentum space as $ (\frac{p}{4})^{2}(\frac{1}{2}\ln(p/%l.955 $
\bar{\Lambda } )^{2}- \frac{5}{4})$. Together with the two external %l.956
legs and the symmetry factor $\frac{1}{6}$ we obtain %l.957
\begin{equation}%l.958 $
G_{d,1}=\left( \frac{g}{4}\right) ^{2}{} \frac{1}{p^{2}{}} %l.959 (,)...
\left( \frac{1}{12}\ln\left( \frac{p^{2}}{\bar{\Lambda }^{2}}%l.960 (,)...
\right) -\frac{5}{24}\right) \hspace{.6ex}.%l.961 $
\end{equation}%l.962

\subsubsection{Chain graphs} %l.964
We call diagrams of type ($ d$) chain graphs. To any order there %l.965
exists one chain graph and, if we disregard the vanishing diagrams %l.966
with tadpoles, the only remaining diagrams up to three loops are %l.967
chain graphs.%l.968

It is possible to calculate chain graph amplitudes for any $n$ by %l.970
Fourier transformation. In momentum space the series of bubbles %l.971
gives rise to the $n$-th power of $ \frac{1}{4}(-\ln(p/\bar%l.972 $
{\Lambda })^{2}+1)$ (cf.\ Eq.\ (\ref{18})). A final convolution %l.973
with $ p^{-2}$ (use Eq.\ (\ref{3}) as suggested in Sec.\ \ref{freemass}) %l.974
provides the result as an $n$-th order polynomial in $ \ln(p/%l.975 $
\bar{\Lambda })^{2}$ with purely rational coefficients. Including %l.976
the symmetry factors and the external legs we obtain for $ n%l.977 $
\ge 2$ (the case $ n=1$ has an extra symmetry which changes the %l.978
symmetry factor from $ 2^{-n}$ to $\frac{1}{6}$) %l.979
\begin{equation}%l.980 $
G_{d,n}=\left( \frac{g}{4}\right) ^{n+1}{} \frac{1}{p^{2}{}} %l.981 ,...
\frac{n!}{2^{n+1}{}} \sum _{k=0}^{n}\left( -1\right) ^{k}\frac{%l.982 ,...
\ln^{n-k}\left( p/\bar{\Lambda }\right) ^{2}}{\left( n-k%l.983 ...
\right) !} \sum _{\ell =0}^{k}\frac{2-2^{\ell -k}}{\ell !} %l.984 $
\hspace{.6ex}.%l.985 $
\end{equation} %l.986
There is a nice way to compile this result by a generating function. %l.987
If we multiply $ G_{d,n}$ with the factor $ a_{n}=3^{n-2}$ for $ n%l.988 $
\ge 2$, $ a_{1}=1$, it reproduces the leading logarithms of the %l.989
full $\varphi ^{4}$ two-point function correctly. The result may %l.990
be seen as some approximation to the propagator. We get %l.991
\begin{equation}%l.992 $
\label{03}\sum _{n=1}^{\infty }\frac{a_{n}G_{d,n}}{n!} =\frac{g}%l.993 ...
{9p^{2}{}} \left( \frac{\left( p^{2}{} /e\bar{\Lambda }^{2}%l.994 ,...
\right) ^{3g/8}}{9\left( 1+g/4\right) ^{2}-1} -\frac{1}{8}%l.995 $
\right) \hspace{.6ex}.%l.996 $
\end{equation} %l.997
We easily read off %l.998
\begin{eqnarray}%l.999 $$
\label{75}\hspace{-3ex} G_{d,2}&\hspace{-1.1ex}=\hspace{-1.1ex}&%l.1000 (,)...
\left( \frac{g}{4}\right) ^{3}\!\frac{1}{p^{2}{}} \!\left( %l.1001 ,...
\frac{1}{8}\ln^{2}\!\left( \frac{p^{2}}{\bar{\Lambda }^{2}}%l.1002 (,)...
\right) -\frac{5}{8}\ln\!\left( \frac{p^{2}}{\bar{\Lambda }^{2}%l.1003 ...
}\right)  +\frac{15}{16}\right) \\ %l.1004 $$
\label{76}\hspace{-3ex} G_{d,3}&\hspace{-1.1ex}=\hspace{-1.1ex}&%l.1005 (,)...
\left( \frac{g}{4}\right) ^{4}\!\frac{1}{p^{2}{}} \!\left( %l.1006 ,...
\frac{1}{16}\ln^{3}\!\left( \frac{p^{2}}{\bar{\Lambda }^{2}}%l.1007 (,)...
\right)  -\frac{15}{32}\ln^{2}\!\left( \frac{p^{2}}{\bar%l.1008 ...
{\Lambda }^{2}}\right)  +\frac{45}{32}\ln\!\left( \frac{p^{2}}{%l.1009 ...
\bar{\Lambda }^{2}}\right)  -\frac{109}{64}\right) \\ %l.1010 $$
\label{J}\hspace{-3ex} G_{d,4}&\hspace{-1.1ex}=\hspace{-1.1ex}&%l.1011 (,)...
\left( \frac{g}{4}\right) ^{5}\!\frac{1}{p^{2}{}} \!\left( %l.1012 ,...
\frac{1}{32}\ln^{4}\!\left( \frac{p^{2}}{\bar{\Lambda }^{2}}%l.1013 (,)...
\right)  -\frac{5}{16}\ln^{3}\!\left( \frac{p^{2}}{\bar%l.1014 ...
{\Lambda }^{2}}\right)  +\frac{45}{32}\ln^{2}\!\left( \frac{p^{%l.1015 ,...
2}}{\bar{\Lambda }^{2}}\right)  -\frac{109}{32}\ln\!\left( %l.1016 ,...
\frac{p^{2}}{\bar{\Lambda }^{2}}\right) +\frac{239}{64}\right) %l.1017 $$
\!.%l.1018 $$
\end{eqnarray}%l.1019

\subsubsection{Four loops} %l.1021
Before we start with the analysis of four and five loops a word of %l.1022
caution is in order. In general it is not sufficient to define the %l.1023
integral over generalized functions for defining a field theory, %l.1024
since also products of generalized functions appear. In principle %l.1025
e.g.\ one has to consider terms like $ x^{2}\delta (x)$ since they %l.1026
might give finite contributions after multiplication with $ x^{%l.1027 ,...
-2}$.%l.1028

In the following we do not care about such terms. The main message %l.1030
of the next two subsections is to show that there is a miraculous %l.1031
matching of Feynman amplitudes in the natural renormalization scheme %l.1032
that makes calculations easy. This matching is not affected by the %l.1033
above problems nor are the leading logarithms of the results. This %l.1034
is confirmed by the existence of the renormalization group equation %l.1035
studied in Sec.\ \ref{rengroup}.%l.1036

\vspace{1ex}%l.1038
\noindent{}Diagrams ($ e$), ($ f$), ($ g$), and ($ h$) remain to %l.1039
be evaluated. $ G_e$ is basically the square of $ G_{d,1}$. %l.1040
\begin{equation}%l.1041 $
G_e=\left( \frac{g}{4}\right) ^{4}{} \frac{1}{p^{2}{}} \left( %l.1042 ,...
\frac{1}{144}\ln^{2}\left( \frac{p}{\bar{\Lambda }}\right) ^{2}%l.1043 (,)...
{} -\frac{5}{144}\ln\left( \frac{p}{\bar{\Lambda }}\right) ^{2}%l.1044 (,)...
{} +\frac{25}{576}\right) \hspace{.6ex}.%l.1045 $
\end{equation} %l.1046
$ G_f$ can be calculated with Fourier transforms. Adding propagators %l.1047
from the interior loop to the exterior lines we obtain (see Eqs.\ %l.1048
(\ref{G})--(\ref{I})) %l.1049
\begin{eqnarray*}%l.1050 $$*
&&\hspace{-0.65cm} \frac{1}{x^{6}{}} \stackrel{{\cal F}}{%l.1051 ...
\longrightarrow } \frac{p^{2}}{4^{2}{}} \left( \frac{1}{2}\ln%l.1052 (,)...
\left( \frac{p}{\bar{\Lambda }}\right) ^{2}\!\!-\frac{5}{4}%l.1053 $$*
\right)  \stackrel{p^{-4}}{\longrightarrow } \frac{1}{4^{2}p^{2%l.1054 ,...
}{}} \left( \frac{1}{2}\ln\left( \frac{p}{\bar{\Lambda }}%l.1055 (,)...
\right) ^{2}\!\!-\frac{5}{4}\right)  \stackrel{{\cal F}}{%l.1056 ...
\longrightarrow } \frac{1}{4^{2}x^{2}{}} \left( -\frac{1}{2}\ln%l.1057 (,)...
\left( \frac{x}{\Lambda }\right) ^{2}\!\!-\frac{5}{4}\right) \\%l.1058 $$*
 %l.1059 $$*
&&\hspace{-0.65cm} \stackrel{x^{-4}}{\longrightarrow } \frac{1}{4%l.1060 ...
^{2}x^{6}{}} \left( -\frac{1}{2}\ln\left( \frac{x}{\Lambda }%l.1061 (,)...
\right) ^{2}\!\!-\frac{5}{4}\right)  \stackrel{{\cal F}}{%l.1062 ...
\longrightarrow } \frac{p^{2}}{4^{4}{}} \left( \frac{1}{8}\ln^{%l.1063 ,...
2}\left( \frac{p}{\bar{\Lambda }}\right) ^{2}\!\!-\frac{5}{8}\ln%l.1064 (,)...
\left( \frac{p}{\bar{\Lambda }}\right) ^{2}\!\!+\frac{17}{16}{} %l.1065 (,)...
-\frac{5}{8}\ln\left( \frac{p}{\bar{\Lambda }}\right) ^{2}\!\!+%l.1066 ,...
\frac{25}{16}\right) .%l.1067 $$*
\end{eqnarray*} %l.1068
Together with the external lines and the symmetry factor $ %l.1069 ,...
\frac{1}{12}$ one gets %l.1070
\begin{equation}%l.1071 $
G_f=\left( \frac{g}{4}\right) ^{4}{} \frac{1}{p^{2}{}} \left( %l.1072 ,...
\frac{1}{96}\ln^{2}\left( \frac{p^{2}}{\bar{\Lambda }^{2}}%l.1073 (,)...
\right)  -\frac{5}{48}\ln\left( \frac{p^{2}}{\bar{\Lambda }^{2}%l.1074 ...
}\right)  +\frac{7}{32}\right) \hspace{.6ex}.%l.1075 $
\end{equation} %l.1076
We are left with two diagrams each of which cannot be calculated %l.1077
by Fourier transformation. Both have the same symmetry factor $\frac{1}{4}$. %l.1078
This makes it possible to use a formula which is specific to four %l.1079
dimensions.%l.1080

\vspace{1ex}%l.1082
\centerline{\epsfbox{natren2.eps}}\vspace{-1.8cm}
\begin{equation}%l.1084 $
\label{100a}%l.1085 $
\end{equation}\vspace{.6cm}

\noindent{}This equation holds up to a total derivative proportional %l.1089
to %l.1090
\begin{equation}%l.1091 $
\frac{\partial }{\partial x^{\mu }{}} \left( \frac{1}{\left( x-x_{%l.1092 ,...
1}\right) ^{2}{}} \frac{x^{\mu }-x_{2}^{\mu }}{\left( x-x_{2}%l.1093 ...
\right) ^{2}{}} \frac{1}{\left( x-x_{3}\right) ^{2}{}} -\frac{x^{%l.1094 ,...
\mu }-x_{1}^{\mu }}{\left( x-x_{1}\right) ^{4}{}} \frac{1}{%l.1095 (,)...
\left( x-x_{3}\right) ^{2}}\right) %l.1096 $
\end{equation} %l.1097
 (notice that $ \frac{\partial }{\partial x^{\mu }{}} \frac{x^{%l.1098 ,...
\mu }-x_{1}^{\mu }}{(x-x_{1})^{4}{}} =\frac{1}{2}\delta (x-x_{1%l.1099 ,...
})$). After integration over $x$ the total derivative vanishes and %l.1100
we obtain:%l.1101

\vspace{3ex}%l.1104
\epsfbox{natren3.eps}\vspace{-1.6cm}
\begin{equation}%l.1105 $
\label{100}%l.1106 $
\end{equation}\vspace{.2cm}

\noindent{}Thus we need not solve each of the complicated diagrams %l.1110
($ g$) and ($ h$) separately\footnote{We do this in the next subsection. %l.1111 f
The single results will be more complicated than the sum. Each of %l.1112 f
the amplitudes ($ g$) and ($ h$) has a $ \zeta (3)$-dependence that %l.1113 f
cancels in the sum.}. Their sum is equal to two chain diagrams.%l.1114

Eq.\ (\ref{100a}) can be interpreted as integration by parts which %l.1116
also proved to be useful within dimensional regularization \cite{Chetyrkin}. %l.1117
However only in natural renormalization it allows one to calculate %l.1118
the sum of diagrams without evaluating single graphs. One should %l.1119
take this as a hint that calculating single diagrams is in general %l.1120
not an appropriate method to evaluate higher order perturbation %l.1121
theory. All diagrams (or at least groups of diagrams) of a given %l.1122
order should be treated as a unit and calculated together. This %l.1123
strategy will be even more useful in the next section. %l.1124
\begin{equation}%l.1125 $
G_g+G_h=\left( \frac{g}{4}\right) ^{4}{} \frac{1}{p^{2}{}} %l.1126 (,)...
\left( \frac{1}{8}\ln^{3}\left( \frac{p^{2}}{\bar{\Lambda }^{2}%l.1127 ...
}\right)  -\frac{19}{16}\ln^{2}\left( \frac{p^{2}}{\bar%l.1128 ...
{\Lambda }^{2}}\right)  +\frac{65}{16}\ln\left( \frac{p^{2}}{%l.1129 ...
\bar{\Lambda }^{2}}\right)  -\frac{169}{32}\right) %l.1130 $
\hspace{.6ex}.%l.1131 $
\end{equation}%l.1132

\subsubsection{Five loops.\label{fiveloops}} %l.1134
Apart from the trivial diagram %l.1135
\begin{equation}%l.1136 $
G_{i}=\left( \frac{g}{4}\right) ^{5}{} \frac{1}{p^{2}{}} \left( %l.1137 ,...
\frac{1}{96}\ln^{3}\left( \frac{p^{2}}{\bar{\Lambda }^{2}}%l.1138 (,)...
\right)  -\frac{5}{64}\ln^{2}\left( \frac{p^{2}}{\bar{\Lambda }^{%l.1139 ,...
2}}\right)  +\frac{5}{24}\ln\left( \frac{p^{2}}{\bar{\Lambda }^{%l.1140 ,...
2}}\right) -\frac{25}{128}\right) %l.1141 $
\end{equation} %l.1142
and the five loop chain graph, Eq.\ (\ref{J}), ten diagrams have %l.1143
to be evaluated. These graphs split into three classes: (1) Diagrams %l.1144
that can be solved by Fourier transforms ($j$ -- $l$). Let us call %l.1145
such diagrams Fourier graphs. (2) Diagrams that can be reduced to %l.1146
Fourier graphs via integration by parts ($m$ -- $r$), and (3) the %l.1147
nonplanar diagram ($s$) that we call the bipyramide graph. %l.1148
\paragraph{Fourier graphs.} %l.1149
Every graph that reduces under the replacement of multiple lines %l.1150
($ \bullet \hspace{-1.3ex}=\hspace{-.5ex}=\hspace{-1.3ex}%l.1151 $
\bullet $, $ \bullet \hspace{-1.3ex}\equiv \hspace{-.5ex}%l.1152 $
\equiv \hspace{-1.3ex}\bullet $) and iterated lines ($ \bullet %l.1153 $
\hspace{-1ex}-\hspace{-1ex}-\hspace{-1ex}\bullet \hspace{-1ex}-%l.1154 $
\hspace{-1ex}-\hspace{-1ex}\bullet $) by a simple line ($ %l.1155 $
\bullet \hspace{-1ex}-\hspace{-1ex}-\hspace{-1ex}\bullet $) to the %l.1156
free propagator can be solved with Fourier transforms. The calculations %l.1157
are analogous to the evaluation of diagram ($f$\/) in the last section. %l.1158
Including the respective symmetry factors $\frac{1}{12}$, $\frac{1}{24}$, %l.1159
$\frac{1}{8}$ we get %l.1160
\begin{eqnarray}%l.1161 $$
G_{j}&\!=\!&\left( \frac{g}{4}\right) ^{5}{} \frac{1}{p^{2}{}} %l.1162 (,)...
\left( \frac{1}{96}\ln^{3}\!\left( \frac{p^{2}}{\bar{\Lambda }^{%l.1163 ,...
2}}\right)  -\frac{25}{192}\ln^{2}\!\left( \frac{p^{2}}{\bar%l.1164 ...
{\Lambda }^{2}}\right)  +\frac{97}{192}\ln\!\left( \frac{p^{2}}%l.1165 ...
{\bar{\Lambda }^{2}}\right) -\frac{269}{384}\right) %l.1166 $$
\hspace{.6ex},\\ %l.1167 $$
\label{75a}G_{k}&\!=\!&\left( \frac{g}{4}\right) ^{5}{} \frac{1}%l.1168 ...
{p^{2}{}} \left( \frac{1}{144}\ln^{3}\!\left( \frac{p^{2}}{\bar%l.1169 ...
{\Lambda }^{2}}\right)  -\frac{5}{64}\ln^{2}\!\left( \frac{p^{2%l.1170 ,...
}}{\bar{\Lambda }^{2}}\right)  +\frac{59}{192}\ln\!\left( %l.1171 ,...
\frac{p^{2}}{\bar{\Lambda }^{2}}\right) -\frac{173}{384}+\frac{%l.1172 ,...
1}{36}\zeta \left( 3\right) \right) \!,\\ %l.1173 $$
G_l&\!=\!&\left( \frac{g}{4}\right) ^{5}{} \frac{1}{p^{2}{}} %l.1174 (,)...
\left( \frac{1}{96}\ln^{3}\!\left( \frac{p^{2}}{\bar{\Lambda }^{%l.1175 ,...
2}}\right)  -\frac{5}{32}\ln^{2}\!\left( \frac{p^{2}}{\bar%l.1176 ...
{\Lambda }^{2}}\right)  +\frac{57}{64}\ln\!\left( \frac{p^{2}}{%l.1177 ...
\bar{\Lambda }^{2}}\right) -\frac{209}{128}+\frac{1}{24}\zeta %l.1178 (,)...
\left( 3\right) \right) \hspace{.6ex}.%l.1179 $$
\end{eqnarray} %l.1180
For future use we calculate the improper four loop $\varphi ^{4}%l.1181
$-diagram ($t$) where the dotted line means a '$ (-1)$-fold' propagator %l.1182
$ (x-y)^{+2}$. The result is (with a symmetry factor of $\frac{1}{32}$) %l.1183
\begin{equation}%l.1184 $
\label{75b}G_t=\left( \frac{g}{4}\right) ^{4}{} \frac{1}{p^{2}{}%l.1185 ...
} \left( \frac{1}{48}\ln^{3}\left( \frac{p^{2}}{\bar{\Lambda }^{%l.1186 ,...
2}}\right)  -\frac{5}{32}\ln^{2}\left( \frac{p^{2}}{\bar%l.1187 ...
{\Lambda }^{2}}\right)  +\frac{17}{32}\ln\left( \frac{p^{2}}{%l.1188 ...
\bar{\Lambda }^{2}}\right) -\frac{49}{64}+\frac{1}{12}\zeta %l.1189 (,)...
\left( 3\right) \right) \hspace{.6ex}.%l.1190 $
\end{equation} %l.1191
\paragraph{Integration by parts.} %l.1192
We determine the following symmetry factors: ($m$): $\frac{1}{8}$, %l.1193
($n$): $\frac{1}{8}$, ($o$): $\frac{1}{4}$, ($p$): $\frac{1}{4}$, %l.1194
($q$): $\frac{1}{8}$, ($r$): $\frac{1}{2}$.%l.1195

The idea is to use Eq.\ (\ref{100a}) to relate the above graphs among %l.1197
each other. Sometimes it will be necessary to multiply Eq.\ (\ref{100a}) %l.1198
by $ (x_{1}-x_{2})^{2}$, $ (x_{1}-x_{3})^{2}$, or $ (x_{2}-x_{3%l.1199 ,...
})^{2}$. Since these factors are independent of $x$ they do not %l.1200
affect partial integration with respect to $x$. However, if these %l.1201
factors do not combine with propagators $ (x_{1}-x_{2})^{-2}$, etc., %l.1202
one obtains improper $\varphi ^{4}$-graphs like diagram ($t$). In %l.1203
most cases it is possible to eliminate those graphs by a second %l.1204
application of Eq.\ (\ref{100a}). In the following table we denote %l.1205
first the graph we start from, then the variables which correspond %l.1206
to $ (x,x_{1},x_{2},x_{3})$ in Eq.\ (\ref{100a}) (according to Fig.\ %l.1207
1), occasionally the variables of a second application of Eq.\ (\ref{100a}), %l.1208
and finally the resulting equation including the symmetry factors. %l.1209
\begin{equation}%l.1210 $
\hbox{\hspace{.38ex}%l.1211 BT...
\begin{tabular}{cccl} %l.1212 BT...
graph&$ \left( x,x_{1},x_{2},x_{3}\right) $&$ \left( x,x_{1},x_{%l.1213 ,...
2},x_{3}\right) $&equation\\\hline  %l.1214 BT...
($p$)&(2,1,4,3)&---&$ G_p+G_o=2G_{m}-\frac{1}{2}gG_g$\\ %l.1215 BT...
($r$)&(3,4,1,2)&---&$ \frac{1}{2}G_r=G_q-\frac{1}{4}gG_h$\\ %l.1216 BT...
($r$)&(3,4,2,1)&(2,4,1,5)&$ \frac{1}{2}G_r+2G_{n}+2G_{m}=4G_{d,4%l.1217 $...
}-\frac{1}{2}gG_h-gG_{d,3}$\\ %l.1218 BT...
($m$)&(2,3,5,1)&(4,3,1,5)&$ 2G_q+G_o+2G_{m}=4G_{d,4}-\frac{1}{2}gG_h%l.1219 $...
-gG_{d,3}$\\\hline  %l.1220 BT...
($g$)&(2,1,4,3)&---&$ G_h=-G_g+2G_{d,3}-\frac{1}{2}gG_{d,2}$\hspace*{1cm}(Eq.\ %l.1221 BT...
(\ref{100}))\\ %l.1222 BT...
($g$)&(2,1,3,4)&---&$ G_g=4G_t-\frac{1}{4}gG_{d,2}$\\ %l.1223 BT...
---&---&---&$ gG_t=12G_{k}+\frac{5}{32}G_{d,2}$\hspace*{2ex}(Eq.\ %l.1224 BT...
(\ref{75}), (\ref{75a}), (\ref{75b}))%l.1225 BT...
\end{tabular}\hspace{.38ex}}%l.1226 $
\end{equation} %l.1227
The last equation can explicitly be checked by looking at the amplitudes. %l.1228
We recognize that there are only four equations to evaluate six %l.1229
five loop diagrams. However summing up the first four equations %l.1230
gives %l.1231
\begin{equation}%l.1232 $
2G_{m}+2G_{n}+2G_o+G_p+G_q+G_r=8G_{d,4}-\left( g/4\right) \cdot %l.1233 (,)...
\left( 2G_g+5G_h+8G_{d,3}\right) \hspace{.6ex}.%l.1234 $
\end{equation} %l.1235
With the last three equations in the table we can express the left %l.1236
hand side completely in terms of Fourier amplitudes of the $%l.1237
\varphi ^{4}$-theory %l.1238
\begin{eqnarray}%l.1239 $$
\lefteqn{2G_{m}+2G_{n}+2G_o+G_p+G_q+G_r\hspace*{1ex}=%l.1240 { $$
\hspace*{1ex}8G_{d,4}+36G_{k}-18\left( g/4\right) G_{d,3}+%l.1241 ,...
\frac{29}{2}\left( g/4\right) ^{2}G_{d,2}}\nonumber \\ %l.1242 $$
&&\hspace{-4ex}=\left( \frac{g}{4}\right) ^{5}{} \frac{1}{p^{2}%l.1243 ...
{}} \left( \frac{1}{4}\ln^{4}\!\left( \frac{p^{2}}{\bar%l.1244 ...
{\Lambda }^{2}}\right) -\frac{27}{8}\ln^{3}\!\left( \frac{p^{2}%l.1245 ,...
}{\bar{\Lambda }^{2}}\right) +\frac{299}{16}\ln^{2}\!\left( %l.1246 ,...
\frac{p^{2}}{\bar{\Lambda }^{2}}\right) -\frac{809}{16}\ln\!%l.1247 (,)...
\left( \frac{p^{2}}{\bar{\Lambda }^{2}}\right) +\frac{1853}{32}%l.1248 (,)...
+\zeta \left( 3\right) \right) .\nonumber \\ %l.1249 $$
&&%l.1250 $$
\end{eqnarray} %l.1251
The graphs ($m$), ($n$), ($o$) are not symmetric under interchange %l.1252
of the external legs. Therefore we have to count them twice in the %l.1253
two-point function and the left hand side becomes exactly the combination %l.1254
we want to calculate. %l.1255
\paragraph{The bipyramide graph.\label{bipyramide}} %l.1256
The bipyramide graph ($s$) is the first non-planar two-point graph %l.1257
and commonly regarded as the most complicated five-loop diagram. %l.1258
It was first calculated in 1981 within dimensional regularization %l.1259
by K.G. Chetyrkin and F.V. Tkachov \cite{Chetyrkin}. Recently it %l.1260
was analyzed within differential renormalization by V.A. Smirnov %l.1261
\cite{Smirnov3}.%l.1262

So it is a good candidate to test the power of our calculation scheme. %l.1264
We work in coordinate space. It is convenient to introduce a quaternionic %l.1265
notation. The inversion of a quaternion $x$ is given by $ x%l.1266 $
\mapsto \frac{1}{x}$ which can be understood in the four vector %l.1267
language as inversion of the length of $x$ ($ |x| \mapsto |x|^{%l.1268 ,...
-1}$) and a reflection at the $z$-axis (the direction of the unit %l.1269
quaternion 1). The square $ (x-y)^{2}$ becomes the square of the %l.1270
absolute $ |x-y|^{2}$, however we stick to the brackets in the following %l.1271
calculation to keep the notation more transparent.%l.1272

The variables $ (x,a,b,c,y)$ correspond to $ (1,2,3,4,5)$ in Fig.\ %l.1274
1.\ We have to calculate the following integral %l.1275
\begin{equation}%l.1276 $
\label{78}\int dadbdc\frac{1}{\left( x-a\right) ^{2}\left( x-b%l.1277 ...
\right) ^{2}\left( x-c\right) ^{2}{}} \frac{1}{\left( a-b%l.1278 ...
\right) ^{2}\left( a-c\right) ^{2}\left( b-c\right) ^{2}{}} %l.1279 ,...
\frac{1}{\left( a-y\right) ^{2}\left( b-y\right) ^{2}\left( c-y%l.1280 ...
\right) ^{2}{}} \hspace{.6ex}.%l.1281 $
\end{equation} %l.1282
The external legs are amputated, they can easily be added in the %l.1283
end.%l.1284

The integral is convergent at infinity (it is logarithmically divergent %l.1286
at $ a=b=c=y$ and $ a=b=c=z$) and therefore the integration variables %l.1287
$ a,b,c$ can be shifted by $y$. With $ z=x-y$ we have %l.1288
\begin{equation}%l.1289 $
I\left( z\right) =\int dadbdc \frac{1}{a^{2}b^{2}c^{2}{}} \frac{%l.1290 ,...
1}{\left( a-b\right) ^{2}\left( a-c\right) ^{2}\left( b-c%l.1291 ...
\right) ^{2}{}} \frac{1}{\left( z-a\right) ^{2}\left( z-b%l.1292 ...
\right) ^{2}\left( z-c\right) ^{2}{}} \hspace{.6ex}.%l.1293 $
\end{equation} %l.1294
With the inversions $ a'=\frac{1}{a}$, $ b'=\frac{1}{b}$, $ c'=%l.1295 ,...
\frac{1}{c}$ one obtains ($ d^{4}a=a'^{-8}d^{4}a'$) %l.1296
\begin{eqnarray*}%l.1297 $$*
&&\int da'db'dc' \frac{1}{a'^{6}b'^{6}c'^{6}{}} \frac{1}{\left( z%l.1298 (,)...
-\frac{1}{a'}\right) ^{2}\left( z-\frac{1}{b'}\right) ^{2}%l.1299 (,)...
\left( z-\frac{1}{c'}\right) ^{2}{}} \frac{1}{\left( \frac{1}{a'} %l.1300 (,)...
-\frac{1}{b'}\right) ^{2}\left( \frac{1}{a'} -\frac{1}{c'}%l.1301 ...
\right) ^{2}\left( \frac{1}{b'} -\frac{1}{c'}\right) ^{2}{}}\\ %l.1302 $$*
&&\frac{1}{z^{6}{}} \int da'db'dc' \frac{1}{\left( a'-\frac{1}{%l.1303 ,...
z}\right) ^{2}\left( b'-\frac{1}{z}\right) ^{2}\left( c'-\frac{%l.1304 ,...
1}{z}\right) ^{2}{}} \frac{1}{\left( b'-a'\right) ^{2}\left( c'-a'%l.1305 ...
\right) ^{2}\left( c'-b'\right) ^{2}{}} \hspace{.6ex}.%l.1306 $$*
\end{eqnarray*} %l.1307
A shift $ a''=a'-\frac{1}{z}$, $ b''=b'-\frac{1}{z}$, $ c''=c'-%l.1308 ,...
\frac{1}{z}$ yields %l.1309
\begin{equation}%l.1310 $
\label{79b}\frac{1}{z^{6}{}} \int da''db''dc'' \frac{1}{a''^{2}b''%l.1311 ...
^{2}c''^{2}{}} \frac{1}{\left( a''-b''\right) ^{2}\left( a''-c''%l.1312 ...
\right) ^{2}\left( b''-c''\right) ^{2}{}} \hspace{.6ex}.%l.1313 $
\end{equation} %l.1314

\vspace{1ex}%l.1316
\noindent{}It seems that we have lost the $z$-dependence in the integral. %l.1317
However, since the integral is still divergent, this is not the %l.1318
case as we will see soon.%l.1319

We finally use the rescaling $ b''=a''u$, $ c''=a''v$ to obtain %l.1321
\begin{equation}%l.1322 $
\label{79}\frac{1}{z^{6}{}} \int \frac{da''}{a''^{4}{}} \int du%l.1323 $
\int dv\frac{1}{\left( 1-u\right) ^{2}u^{2}\left( u-v\right) ^{%l.1324 ,...
2}v^{2}\left( v-1\right) ^{2}{}} \hspace{.6ex}.%l.1325 $
\end{equation} %l.1326
The $u$- and $v$-integral is finite and gives a positive number. %l.1327
It can be evaluated using Gegenbauer polynomial techniques (e.g.\ %l.1328
\cite{Chetyrkin2}, \cite{Johnson}) with the result\footnote{Most %l.1329 f
efficiently one uses the identity $ |uv|/(u-v)^{2}{} =\frac{1}{%l.1330 ...
\pi } \int _{-\infty }^{\infty }dP (|u|/|v|)^{iP}\sum _{n=1}^{%l.1331 ,...
\infty }nC_{n-1}(\hat{u}\cdot \hat{v})\cdot (n^{2}+P^{2})^{-1}$ %l.1332 f
and the orthogonality relation $ \frac{1}{2\pi ^{2}{}} \int d%l.1333 $...
\hat{x}nC_{n-1}(\hat{y}\cdot \hat{x})mC_{m-1}(\hat{x}\cdot \hat%l.1334 $...
{z})=\delta _{n,m}nC_{n-1}(\hat{y}\cdot \hat{z})$ to obtain $ %l.1335 ,...
\frac{1}{\pi } \hspace{-1ex}\int _{-\infty }^{\infty }dP\sum _{%l.1336 ,...
n=1}^{\infty }n^{2}(n^{2}+P^{2})^{-3}$.} $ \frac{3}{8}\zeta (3)%l.1337 $
$. %l.1338
The angular integral in $ \int da'' /a''^{4}$ gives $ 2\pi ^{2}$. %l.1339
Including the normalization $ \frac{1}{4\pi ^{2}}$ (Eq.\ (\ref{54})) %l.1340
one is left with the radial integral $ \frac{1}{2}\int _{0}^{%l.1341 ,...
\infty }d|a''|/|a''|=\frac{1}{2}\ln(|a''|)|_{a''=0}^{a''=%l.1342 { $
\infty }$. If we would have started with an integral in the $ a''$-variable %l.1343
this integral would give zero. However as discussed at the end of %l.1344
Sec.\ \ref{firstres} it is now essential to reintroduce the original %l.1345
variable $a$ before approaching the limits. Since $ a''=\frac{1%l.1346 ,...
}{a}-\frac{1}{z}$, %l.1347
\begin{equation}%l.1348 $
\int \limits _{0}^{\infty }\frac{d|a''|}{2|a''|} =\left. \frac{1%l.1349 ,...
}{2}\ln\!\left( \left| \frac{1}{a}-\frac{1}{z}\right| \right) %l.1350 $
\right| _{a=z}^{a=0}\!\!=\left. \frac{1}{2}\ln\!\left( \frac{|z%l.1351 ,...
-a|}{|az|}\right) \right| _{a=z}^{a=0}\!\!=\frac{1}{2}\left( \ln%l.1352 (,)...
\!\left( \frac{1}{\Lambda }\right)  -\ln\!\left( \frac{\Lambda %l.1353 ,...
}{z^{2}}\right)  %l.1354 (,)...
\right) =\frac{1}{2}\ln\!\left( \frac{z^{2}}{\Lambda ^{2}}%l.1355 $
\right) .%l.1356 $
\end{equation} %l.1357
Collecting all pieces we have finally found %l.1358
\begin{equation}%l.1359 $
I\left( z\right) =\frac{3}{16}\zeta \left( 3\right) \frac{1}{z^{%l.1360 ,...
6}{}} \ln\left( \frac{z^{2}}{\Lambda ^{2}}\right)  %l.1361 $
\hspace{.6ex}.%l.1362 $
\end{equation} %l.1363
The transformation to momentum space is given by Eq.\ (\ref{H}). %l.1364
Including the external legs and the symmetry factor $\frac{1}{6}$ %l.1365
we obtain %l.1366
\begin{equation}%l.1367 $
G_{m}=\left( \frac{g}{4}\right) ^{5}{} \frac{1}{p^{2}{}} \left( %l.1368 ,...
\frac{1}{2}\zeta \left( 3\right) \ln^{2}\left( \frac{p^{2}}{%l.1369 ...
\bar{\Lambda }^{2}}\right) - \frac{5}{2}\zeta \left( 3\right) \ln%l.1370 (,)...
\left( \frac{p^{2}}{\bar{\Lambda }^{2}}\right)  +\frac{17}{4}%l.1371 (,)...
\zeta \left( 3\right) \right) \hspace{.6ex}.%l.1372 $
\end{equation} %l.1373
Comparing with dimensional regularization \cite{Chetyrkin} gives %l.1374
the minimum coincidence that both results are proportional to $%l.1375
\zeta $(3). It is not possible to be more precise since in \cite{Chetyrkin} %l.1376
only the singular part was calculated. Note that the techniques %l.1377
we used can not be generalized to dimensions different from four.%l.1378

It is also hard to compare our result with the one gained by differential %l.1380
renormalization in \cite{Smirnov3} since the author restricted himself %l.1381
to regularize the amplitude and did not evaluate the rather complicated %l.1382
integrals over the internal variables.%l.1383

\subsubsection{The two-point Green's function} %l.1385
Collecting all results from the last sections we obtain for the full %l.1386
propagator of the $\varphi ^{4}$-theory %l.1387
\begin{eqnarray}%l.1388 $$
G\left( p\right) &=&\frac{1}{p^{2}{}} \left( 1+\left( \frac{g}{4%l.1389 ,...
}\right) ^{2}\left( \frac{1}{12}\ln\left( \frac{p^{2}}{\bar%l.1390 ...
{\Lambda }^{2}}\right) -\frac{5}{24}\right) +\left( \frac{g}{4}%l.1391 (,)...
\right) ^{3}\left( \frac{1}{8}\ln^{2}\left( \frac{p^{2}}{\bar%l.1392 ...
{\Lambda }^{2}}\right) -\frac{5}{8}\ln\left( \frac{p^{2}}{\bar%l.1393 ...
{\Lambda }^{2}}\right) +\frac{15}{16}\right) \right. \nonumber %l.1394 $$
\\ %l.1395 $$
&&+\left( \frac{g}{4}\right) ^{4}\left( \frac{3}{16}\ln^{3}%l.1396 (,)...
\left( \frac{p^{2}}{\bar{\Lambda }^{2}}\right) -\frac{59}{36}\ln%l.1397 (,)...
^{2}\left( \frac{p^{2}}{\bar{\Lambda }^{2}}\right)  +\frac{1535%l.1398 ,...
}{288}\ln\left( \frac{p^{2}}{\bar{\Lambda }^{2}}\right) -\frac{%l.1399 ,...
121}{18}\right) \nonumber \\ %l.1400 $$
&&+\left( \frac{g}{4}\right) ^{5}\left( \frac{9}{32}\ln^{4}%l.1401 (,)...
\left( \frac{p^{2}}{\bar{\Lambda }^{2}}\right) -\frac{1045}{288%l.1402 ,...
}\ln^{3}\left( \frac{p^{2}}{\bar{\Lambda }^{2}}\right) +\left( %l.1403 ,...
\frac{3733}{192}+\frac{1}{2}\zeta \left( 3\right) \right) \ln^{%l.1404 ,...
2}\left( \frac{p^{2}}{\bar{\Lambda }^{2}}\right) \right. %l.1405 $$
\nonumber \\ %l.1406 $$
&&\label{994}-\left. \left( \frac{1643}{32}+\frac{5}{2}\zeta %l.1407 (,)...
\left( 3\right) \right) \ln\left( \frac{p^{2}}{\bar{\Lambda }^{%l.1408 ,...
2}}\right) +\frac{3697}{64}+\frac{383}{72}\zeta \left( 3%l.1409 (,)...
\right) \right) +{\cal O}\left( g^{6}\right) \hspace{.6ex}.%l.1410 $$
\end{eqnarray} %l.1411
Comparison with differential renormalization \cite{Johnson} %l.1412
\begin{equation}%l.1413 $
\label{995}G_{{\rm diff.ren.\hspace{.38ex}}}\left( p\right) =%l.1414 ,...
\frac{1}{p^{2}{}} \left( 1+\left( \frac{g}{4}\right) ^{2}{} %l.1415 ,...
\frac{1}{12}\ln\left( \frac{p}{\bar{\Lambda }}\right) ^{2}\!+%l.1416 (,)...
\left( \frac{g}{4}\right) ^{3}\left( \frac{1}{8}\ln^{2}\left( %l.1417 ,...
\frac{p}{\bar{\Lambda }}\right) ^{2}\!-\frac{3}{8}\ln\left( %l.1418 ,...
\frac{p}{\bar{\Lambda }}\right) ^{2}\right) +{\ldots}\right) %l.1419 $
\end{equation} %l.1420
shows that only the leading logarithms coincide. Notice that one %l.1421
never gets $ \ln$-independent terms in \cite{Johnson}.%l.1422

If $ \bar{\Lambda }$ is rescaled according to $ \ln\bar%l.1424 $
{\Lambda } \mapsto \ln\bar{\Lambda }+1$ in Eq.\ (\ref{995}) the %l.1425
logarithmic terms coincide with that of Eq.\ (\ref{994}). The $ %l.1426 $
\ln$-independent terms can be adjusted via a momentum independent %l.1427
rescaling by $ 1-\frac{3}{24}(g/4)^{2}{} +\frac{11}{16}(g/4)^{3%l.1428 ,...
}$. However, at this point it is not clear whether the differences %l.1429
disappear after appropriate redefinitions also at higher orders.%l.1430

\subsubsection{The renormalization group\label{rengroup}} %l.1432
It is possible to extract the $ \beta $-function and the anomalous %l.1433
dimension $\gamma $ from the two-point function alone if one assumes %l.1434
that $ \beta $ and $\gamma $ are independent of $ \bar%l.1435 $
{\Lambda }$. Moreover the existence of a renormalization group equation %l.1436
is a non-trivial test for the renormalization scheme. Comparing %l.1437
the coefficients in %l.1438
\begin{equation}%l.1439 $
\left( \frac{\partial }{\partial \ln\bar{\Lambda }} +\beta %l.1440 (,)...
\left( \frac{g}{4}\right) \frac{\partial }{\partial \left( g/4%l.1441 ...
\right) } +2\gamma \left( \frac{g}{4}\right) \right) G\left( %l.1442 ,...
\frac{g}{4},\bar{\Lambda },p\right) =0%l.1443 $
\end{equation} %l.1444
yields %l.1445
\begin{eqnarray}%l.1446 $$
\beta \left( \frac{g}{4}\right) &=&3\left( \frac{g}{4}\right) ^{%l.1447 ,...
2}{} -\frac{17}{3}\left( \frac{g}{4}\right) ^{3}+\left( \frac{7%l.1448 ,...
9}{4}+12\zeta \left( 3\right) \right) \left( \frac{g}{4}%l.1449 $$
\right) ^{4}+{\cal O}\left( g^{5}\right) \hspace{.6ex},\\ %l.1450 $$
\gamma \left( \frac{g}{4}\right) &=&\frac{1}{12}\left( \frac{g}%l.1451 ,...
{4}\right) ^{2}{} -\frac{5}{96}\left( \frac{g}{4}\right) ^{4}{} %l.1452 $$
+\frac{191}{192}\left( \frac{g}{4}\right) ^{5}+{\cal O}\left( g^{%l.1453 ,...
6}\right)  \hspace{.6ex}.%l.1454 $$
\end{eqnarray} %l.1455
The first two terms of $ \beta $ and the first term of $\gamma $ %l.1456
are standard. The coefficient in front of the $ \zeta (3)$-term %l.1457
also coincides with other schemes \cite{Johnson}, \cite{Smirnov}. %l.1458
However e.g.\ the vanishing third order and the $\zeta $(3)-independent %l.1459
fifth order term of $\gamma $ is specific to our scheme. In differential %l.1460
renormalization \cite{Johnson} one obtains $ \beta (g/4)=3(g/4)^{%l.1461 ,...
2}{} -\frac{17}{3}(g/4)^{3}+(31+12\zeta (3))(g/4)^{4}+{\ldots}$, %l.1462
$ \gamma (g/4)=\frac{1}{12}(g/4)^{2}{} -\frac{3}{8}(g/4)^{3}+%l.1463 $
{\ldots}$.%l.1464


\section{Results and outlook} %l.1467
A new renormalization scheme was proposed. It provides all amplitudes %l.1468
fully renormalized, it has no explicit cutoff or counterterms and %l.1469
allows to keep the spacetime dimension fixed. The scheme defines %l.1470
all integrals in an unambiguous way, it thus corresponds to a definite %l.1471
choice of a subtraction prescription.%l.1472

The renormalization scheme emerges from differential renormalization %l.1474
by an a priori fixing of all integration constants at their mathematically %l.1475
most natural values. It is closely related to the theory of generalized %l.1476
functions.%l.1477

We demonstrated how to use this renormalization scheme if applied %l.1479
to the toy problem of a two-point (mass) interaction in coordinate %l.1480
space. Although this theory is non-renormalizable by power-counting %l.1481
it was possible to recover the correct result within our scheme. %l.1482
A theorem was presented that allowed us in a more general framework %l.1483
to reconstruct the full result from such a singular expansion. With %l.1484
this theorem it was possible (but more complicated) to regain the %l.1485
true result even for the dimensionally regularized toy model which %l.1486
failed to give the correct perturbation series.%l.1487

\vspace{1ex}%l.1489
\noindent{}The main application of our scheme was the $\varphi ^{%l.1490 ,
4}$-theory. Equations that are very special to four dimensions and %l.1491
to our renormalization prescription enabled us to calculate the %l.1492
two-point Green's function up to five loops (Eq.\ (\ref{994})). %l.1493
Most remarkable was the observation that at (four) five loops the %l.1494
diagrams are organized in such a way that a (one-) two-fold underdetermined %l.1495
system of linear equations could be solved for the sum over certain %l.1496
diagrams. This made the evaluation of many single graphs needless.%l.1497

We were left with the nonplanar five-loop graph which could as well %l.1499
be calculated analytically in our renormalization scheme. It is %l.1500
obvious that the dimension of spacetime plays a crucial role in %l.1501
the calculation of the bipyramide graph (as it does for the matching %l.1502
of diagrams via integration by parts). Only in four dimensions the %l.1503
coupling becomes dimensionless. The resulting conformal symmetry %l.1504
was used via the inversion $ a\mapsto a^{-1}$ as the most essential %l.1505
step in the evaluation of the integrals.%l.1506

The two-point function was compatible with the renormalization group %l.1508
and it was possible to extract the $ \beta $-function up to fourth %l.1509
and the anomalous dimension $\gamma $ up to fifth order in the coupling.%l.1510

\vspace{1ex}%l.1512
\noindent{}%l.1513

For future work the idea of grouping certain classes of diagrams %l.1515
and calculating their sum without referring to single graphs appears %l.1516
especially promising to us. We expect that the matching of amplitudes %l.1517
persists to some extent at higher orders. In this way perturbation %l.1518
theory could be simplified and even analytical results beyond the %l.1519
fifth order may be possible. (Recent calculations confirm this for %l.1520
the sixth order of $\varphi ^{4}$-theory.) Most desirable would %l.1521
be to find the general structure that organizes the amplitudes to %l.1522
groups that can be evaluated via integration by parts. General questions %l.1523
of renormalizability and the problems related to the multiplication %l.1524
of generalized functions have to be investigated more carefully.%l.1525

A goal of obvious importance is the application of natural renormalization %l.1527
to gauge theories. In general one has to avoid conflicts between %l.1528
Ward identities (reflecting gauge symmetry) and the renormalization %l.1529
scheme. This problem is already present in two dimensions and can %l.1530
be solved by using the transverse (Landau) gauge. The Schwinger %l.1531
model can be solved within this framework by summing up the whole %l.1532
perturbation series (e.g.\ the fermion correlation function) \cite{Schnetz}. %l.1533
The key tool is, similar to the $\varphi ^{4}$-theory, to calculate %l.1534
whole classes of Feynman diagrams without evaluating single amplitudes. %l.1535
In four dimensions first results are promising, however for QED %l.1536
we have not yet found how to group diagrams to simplify the calculations.%l.1537

\section*{Acknowledgement} %l.1539
I thank Prof.\ M. Thies and Prof.\ F. Lenz for the pleasant and fruitful %l.1540
cooperation. Further I am grateful to Prof.\ K. Johnson and Prof.\ %l.1541
A. Bassetto for interesting and helpful discussions.%l.1542

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\end{document}%l.1571

