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\noindent
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\centerline{\sffamily\bfseries\Large Field Theory of Interacting Crumpled
  Manifolds}
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\vfill
\centerline{\large Dissertation zur Erlangung des Grades}
\vfill
\centerline{\bfseries\large Doktor der Naturwissenschaften (Dr. rer. nat.)}
\vfill
\centerline{vorgelegt am}
\vfill
\centerline{Fachbereich Physik der Universit\"at Essen}
\vspace{2cm}
\centerline{von}
\vspace{2cm}
\centerline{\Large Henryk A.\ Pinnow}
\vspace{2cm}
\centerline{aus Hameln}
\vspace{4cm}
\centerline{Essen 2002}
\smallskip\smallskip

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\begin{abstract}%\noindent
The effect of a $\delta $-interaction on a
polymerized membrane of arbitrary internal dimension $D$ has been studied. Depending on
the dimensionality of membrane and embedding space, different physical
scenarios are observed: The delocalization of a membrane from an attractive
defect as well as steric repulsions. The difference of polymers
from membranes is emphasized. For the latter, non-trivial contributions appear at
the 2-loop level. 
Furthermore, a ``massive scheme'' inspired by
calculations in fixed dimensions for scalar field theories has been exploited.
Despite the fact that these calculations are only amenable numerically,
it has been found that in the limit of $D\to 2$ each diagram can be evaluated 
{\em analytically}. This property extends in fact to any order in
perturbation theory, allowing for a summation of all orders. An analytically continued
expression for the effective coupling of membranes in the scaling limit of
large sizes as compared to the microscopic cutoff is obtained. Finally,
the construction of an expansion of the effective coupling about $D=2$ is presented. Applications to the case
of self-avoiding membranes are mentioned. 


\end{abstract}
\vfill
1. Gutachter: Dr. habil. K.J. Wiese\\
 \\
2. Gutachter: Prof. Dr. L. Sch\"afer\\
 \\
Tag der Disputation: 19. Juli 2002

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\newpage \section{Introduction}\label{introduction}
\subsection{Physics of fluctuating manifolds}
Interacting lines and more generally manifolds play an important role
in many areas of modern physics. Examples of lines are self-avoiding
polymers \cite{Schaefer}, vortex-lines in super-conductors
\cite{BlatterFeigelmanGeshkenbeinLarkinVinokur1994}, directed polymers
in a disordered environment, also equivalent to surface growth
\cite{KPZ}, diffusion of particles and many more. Generalizing results
to membranes often poses severe problems, but also new insight into
physics. Recently, a lot of work has been devoted to self-avoiding
membranes  (see \cite{WieseHabil} for a review).\\
%Applications reach as far
%as high-energy physics, where strings and M-branes have been proposed
%as a general framework for unifying all fundamental interactions.\\
From the point of view of statistical physics one major problem is to study
the effect of interactions on the thermodynamical properties of extended
fluctuating geometric objects. Generally, there are multi-particle
 attractive or  repulsive interactions involved. One may divide these into two
 classes: Either (i) one may study the interaction of a single fluctuating object
 with itself as for instance the well known two-particle excluded volume
 interaction between any two monomers in a long chain polymer in a good
 solvent.  There, the interaction leads to universal long-distance
 properties of chains as for example the anomalous scaling of the mean squared
 end-to-end distance. Or (ii), the interaction may act between different
 manifolds or between a single manifold and a fixed non-fluctuating object. It
 is then interesting to study how thermal fluctuations affect the
 depinning of the manifold from an attractive substrate as well as the steric
 repulsions from a wall.\\
Both cases have in common that they are fairly
well understood as long as the fluctuating objects are
one-dimensional \cite{Schaefer,DesCloizeauxJannink,DeGennes,Eisenriegler}. Referring to the example mentioned above, the long-distance
properties of self-avoiding polymers, which are always
crumpled on large scales, can be analyzed with renormalization group
techniques \cite{Fixman1955,SchaferWitten1977,DesCloizeaux1981}, either in the continuous Edwards Hamiltonian \cite{Edwards1965},
\begin{equation}
  \label{I.0}
  \mathcal{H}[\vec r]=\frac{1}{2}\int\limits_{x\in\mathcal{M}}\left(\nabla
  \vec
  r(x)\right)^{2}+\frac{b_{0}}{2}\int\limits_{x\in\mathcal{M}}\int\limits_{y\in\mathcal{M}}\delta^{d}(\vec
  r(x)-\vec r(y))\ ,
\end{equation}
or by mapping this model on a local $O(N)$ symmetric $\varphi^{4}$-theory in
the limit of $N=0$ components \cite{DeGennes1972,DeGennes,Schaefer}. The critical exponents describing the
long-distance properties are related to the critical exponents of the
corresponding $N$-vector model at the critical point. What makes
(\ref{I.0}) a
non-standard theory is that the
interaction is nonlocal. The integration is over the internal volume of the
manifold, that is, in
the case of polymers along the chain.\\
Obtaining  the corresponding results for membranes poses considerable
challenges. The generalization of polymers to $2D$-surfaces are crystalline
fixed-connectivity membranes as they appear for instance in the spectrin
network of cell membranes. Considering "phantom" membranes, which can freely fold into itself, the existence of a
bending rigidity induced phase transition  separating a
high rigidity, low temperature flat phase from a low rigidity, high temperature
crumpled phase is well established \cite{KantorNelson1987a,KantorNelson1987b,KantorKardarNelson1986a,KantorKardarNelson1986b,PaczuskiKardarNelson1988,PaczuskiKardar1989}. This is in contrast to polymers, which are
always crumpled on large scales. The scaling properties of the
crumpled phase of phantom membranes are described by the $2D$ generalization of the free field
part in (\ref{I.0}). Taking self-avoidance into account, which is modeled in
(\ref{I.0}) through the short range two-body interaction, we expect
more open manifolds than those predicted by the free theory, which will be
expressed in a non-trivial radius of gyration exponent. The squared radius of
  gyration $R_{g}$ is defined as the mean squared distance of the manifold
  constituents from the center of mass. In the case of polymers $R_{g}$ scales 
  like the end-to-end distance.  Much effort has been spent on calculating
corrections to the radius of gyration exponent within an expansion in the deviation $\varepsilon$ from
the critical space dimension \cite{DavidWiese1996,WieseDavid1997}. These calculations can not be performed directly
in $D=2$, since the naive scaling dimension of the coupling in (\ref{I.0})
equals
\begin{equation}
  \label{I.1}
  \varepsilon(D,d):=[b_{0}]=2D-\frac{2-D}{2}d\ ,
\end{equation}
where $d$ denotes the dimension of the embedding space, and is always finite as
$D\to 2$. Equivalently, the critical embedding dimension defined through
$\varepsilon(D,d_{c}(D))=0$ becomes $\infty$ in this limit. The reason is that
the non-self-avoiding membrane densely fills out the embedding space, such that
it always "sees" the interaction. A way to circumvent this
problem is to set up the expansion about any point $(D<0,d_{c}(D))$ and to
extrapolate along an appropriate path in the $(D,d)$-plane to the physically
interesting point $(D,d)=(2,3)$ \cite{WieseHabil,KardarNelson1987,AronovitzLubensky1988,DDG3,DDG4,Hwa1990,WieseDavid1995}. To second order in $\varepsilon$ one then
finds a radius of gyration exponent of $\nu\approx 0.86$ \cite{DavidWiese1996,WieseDavid1997} , which is a strong
correction with respect to the only logarithmic dependence in the non-interacting
theory, but nevertheless still indicates existence of a crumpled phase.\\
However, there is no unambiguous evidence for a crumpled phase in
experiments
\cite{ChianelliPrestridgePecoradoPecoradodDeNeufville1979,HwaKokufutaTanaka1991,WenGarlandHwaKardarKokufutaLiOrkiszTanaka1992,SpectorNaranjoChiruvoluZasadzinski1994}.
Latest Monte-Carlo simulations on plaquette-models \cite{Baumgaertner1991,BaumgaertnerRenz1992,KrollGompper1993,BowickTravesset2001} starting from a
discretization of the $2D$ generalized Hamiltonian (\ref{I.0}) with system
sizes of up to $\approx 17000$ plaquettes shows considerable evidence for a
vanishing of the above mentioned crumpling transition in the presence of
self-avoidance, such that even on large scales fixed-connectivity membranes
stay always flat with a radius of gyration exponent of $\nu\approx 1$. This
outcome might be reconciliated with the field theoretic calculation by going
to higher loop orders.\\
\begin{figure}[b]
  \begin{center}
    \leavevmode
    \raisebox{0mm}{\includegraphics[scale=0.45]{./eps/crump.eps}} 
\end{center}
    \caption{A fluctuating membrane interacting by excluded volume with a
    surface-like defect.}
    \label{f1.0}
\end{figure}
The aim of this work is to develop techniques which allow to go
beyond the two-loop result. We developed such techniques for a simplified model,
which replaces the non-local interaction in (\ref{I.0}) by an interaction with
a
single point in the embedding space and which is therefore related to case
(ii). The corresponding physical situation we think of is the binding and
unbinding
of a long chain as e.g.\ a polymer  or a membrane from a
wall or the wetting of an interface. More precisely, we study 
 the interaction of a single freely
fluctuating manifold with another non-fluctuating, fixed object. 
Depending on whether the interaction is attractive or repulsive, one
can distinguish two different scenarios: One may either observe a
delocalization transition % of a polymer or a membrane
 from an
attractive substrate as in wetting phenomena  or
steric repulsions by a fluctuating manifold. Both cases have in common
that excluded volume effects become important. 
The situation for polymers or a 1-dimensional interface is
 relatively simple: One knows that in this case polymers interacting with a
defect or a wall (excluding or not half of the space) as well as short-range critical wetting
are in the same universality class \cite{Abraham1980,FisherME1984,Upton1999}.
For two-dimensional interfaces, the situation is more complicated,
and  a lot of effort has been spent to understand e.g.\ the delocalization
transition
\cite{NakanishiFisher1982,BrezinHalperinLeibler1983,LipowskyKrollZia1983,KrollLipowskyZia1985,FisherHuse1985,LipowskyFisher1987,DavidLeibler1990,ForgasLipowskyNieuwenhuizenInDombGreen}.
Particularly, one is interested in critical wetting for the case of a
short-ranged interaction potential. Then, it can not be approximated by a
polynomial in the field $\vec r$ and the conventional
field-theoretic approach known from $\phi^{4}-$theory fails. This led
to a couple of different ans\"atze 
\cite{BrezinHalperinLeibler1983,LipowskyKrollZia1983,FisherHuse1985,%
LipowskyFisher1987,DavidLeibler1990}, among others the functional
renormalization group approach of \cite{LipowskyFisher1987,DavidLeibler1990}.


Here we follow a different route:  We start by constructing  a
{\em systematic perturbative}  renormalization
group treatment of the delocalization transition as well as the
universal repulsive force exerted by a membrane on a point, line or
more generally hyper-plane like defect.
We therefore introduce a 
flexible ``phantom'' manifold of internal dimension $0\leq D \leq 2$.
By introducing a 
$\delta $-potential  in a  subspace $\mathcal{E}$
of dimension $d'$,  part of the
embedding space $\R^{d+d'}$  (see fig.\ \ref{f1.0}) we punish 
configurations crossing $\mathcal{E}$. Neglecting the
effect of self-avoidance between distinct 
points within the fluctuating manifold, the free energy of a given
configuration   reads:
\begin{equation}
  \label{1.1}
  \mathcal{H}[\vec r] = \int_{x\in\mathcal{M}}\rmd ^{D}x\,\frac{1}{2}(\nabla \vec
  r(x))^{2}+g_{0}\int_{x\in \mathcal{M}}\rmd ^{D}x\int_{\vec{y} \in
\mathcal{E}}\rmd ^{d'}\!\vec{y}\ \, \delta^{d+d'}(\vec
  r(x){-}\vec y)\ ,
\end{equation}
where $\vec{r} (x) $ is the position of monomer $x\in \mathcal{M}$ in
embedding space  $\R^{d+d'}$,
and $\mathcal{M}$ denotes the $D$-dimensional coordinate space of the
manifold. In the case of 
a polymer this is simply the internal chain length. $g_{0}$ is the
attractive or repulsive interaction, of dimension (in inverse length-units)
\begin{equation}
\E (D,d):= \left[g_{0} \right] = D - \frac{2-D}{2} d\ .
\end{equation}
The interaction is naively relevant in the infrared for $\E$ positive, and irrelevant
for $\E$ negative. In a perturbative treatment, $\E$ is the natural
expansion parameter. The situation is similar to self-avoiding
membranes and thus (\ref{1.1}) has been studied as
 a ``toy-model'' for the analysis of
the renormalizability of the more complicated interaction in the generalized
Edwards model (\ref{I.0}) for self-avoiding
polymerized membranes \cite{Duplantier1989,DDG1,DDG2}. It was shown to
be renormalizable for arbitrary 
manifold dimensions $0{<}D{<}2$ \cite{DDG1,DDG2}. This means that the
large scale properties are universal, i.e.\ do neither depend on the
regularization scheme used in these calculations, nor on the form of
the contact interaction, as long as it is short-ranged. 
Universal quantities have thus  been obtained to
one-loop order. They are related, as we will be shown in section \ref{Model and physical observables}, to the correction
to scaling exponent $\omega$, which as usually in critical phenomena contains
deviations from the long-distance scaling behavior.\\ 


The aim of this work is two-fold: First, we present the necessary
techniques to treat the model (\ref{1.1}) beyond the leading order.
We  explicitly perform a two-loop calculation which gives the
correction to scaling exponent $\omega $ at order $\E^2$. Specializing
to membranes one finds that the 2-loop result naively diverges in the limit 
of $D\to 2$. This is a problem of the 
$\E$-expansion,  since there, diagrams have to be evaluated at $\E
(D,d_{c} (D)) =0$, and taking  $D \to 2$ implies 
$d_{c}(D)\to \infty$  causing the result to diverge. Following the above
mentioned strategy which has been applied successfully to the calculation of the anomalous dimension
of self-avoiding membranes we set up the expansion in $\varepsilon$ about
some point $(D<0,d_{c}(D))$ and  extrapolate along some appropriate path to
physically interesting points $(D,d)$. It will turn out that the result always depends strongly on
the expansion point. Thus, the correction at the two-loop level is
ambiguous as long as one is only interested in the limit $D\to 2$.
This motivated us to try a ``massive scheme'' in
fixed dimension, i.e.\ at finite $\varepsilon$. It turns out that the limit
$D \to 2$ can then be taken and is regular. Even more, the $2$-loop diagram,
which for $D < 2$ can only be calculated numerically, can now be evaluated
{\em analytically}. This striking property even  holds for higher
orders, and we are 
able to give an explicit -- quite simple -- expression for the perturbation
series. In a second step the whole series can be summed and the strong
coupling limit  analyzed. This is one of the very few cases where one
can  indeed obtain the {\em exact} relation
between bare and renormalized coupling in the limit of $D\to2$, and
thus the {\em exact} $\beta$-function.  This result does not depend on the
explicit regularization and renormalization prescriptions, and is also 
obtained for a membrane of spherical or toroidal topology. Furthermore, it
turns out that the behavior of the effective coupling, which is logarithmically
diverging for large membrane sizes, together with the value of the
correction-to-scaling exponent $\omega$, which is therefore forced to be zero, can not be extrapolated from any finite
loop expansion.
In a final step, we lay the foundations for an 
expansion about $D=2$.  In contrast to the leading order, the first
order corrections already depend on the cut-off procedure. We studied  
one specific procedure already in \cite{PinnowWiese2001}, which turns out to  
reproduce results for  polymers at leading order approximately, and
even exactly in $d=0$. In order to obtain exact results, we decided to set up the expansion in $2-D$ on a manifold of
toroidal internal topology; this corresponds to periodic boundary conditions. Our
aim is to expand the exponent $\omega$ about $D=2$ and this way to prove the smoothness of the limit $D\to 2$. This will be further
discussed in the sections 6 and 7. The following introductory sections are a short
expos\'e on tethered manifolds and the crumpling transition motivating
Hamiltonians like (\ref{I.0}) and (\ref{1.1}). In section 2 we further discuss
the model (\ref{1.1}) and show, how the correction-to-scaling exponent
$\omega$ is related to interesting physical observables, which might be also
accessible to Monte-Carlo simulations.






\subsection{Tethered surfaces and their realizations}\label{tethsurf}
Tethered surfaces are systems of particles (atoms or monomers) that are
connected to form a regular two-dimensional array embedded in $d$-dimensional
space \cite{JerusalemWinterSchool1989} . Independently from the precise type of the $2D$ lattice as well as from
the form of the bonding potential between the neighboring particles of the
network, it is essential that the bonds between the adjacent atoms or
monomers of the array can not be broken. This is in contrast to fluid
membranes, where molecules making up the membrane can freely circulate within
the surface. The distances between
adjacent particles are subject to thermal fluctuations which are controlled by
the next-nearest-neighbor bonding potentials. This is, too, in contrast to fluid
membranes, whose particle density stays constant,
while they undergo pronounced shape fluctuations. This property is the
reason, why the question is
still under debate whether the flat phase of fluid membranes becomes
unstable with respect to thermal fluctuations or is rather stabilized,  \cite{Helfrich1973,Helfrich1985,PelitiLeibler1985,PinnowHelfrich2000,GompperKroll1996}.\\
For tethered surfaces the situation is by far clearer: As long as no
self-avoidance is taken into account, they are known to undergo a bending
rigidity induced continuous phase transition separating a stable flat phase from a
crumpled fractal phase. Due to the fixed connectivity tethered surfaces closely
resemble linear polymers. The spatial conformation of a linear polymer can be
described by a set of position vectors $\vec r(x)$, where $x$ is the internal
index and $\vec r(x)$ the location in embedding space of the indexed
monomer. Then, a natural generalization to $2D$ surfaces is made by use of a
two-dimensional vector $\vec x$ as an index denoting the internal position of
the atom. Such use of a fixed (flat) internal space and its clear separation
from the external variable $\vec r$, on which the Hamiltonian of the system
depends, enables an unambiguous definition of the statistical measure for the
configurational space, thus avoiding problems regarding a proper measure as
they are addressed in \cite{PinnowHelfrich2000} in the case of fluid surfaces.\\
\\
A naturally occurring physical example of tethered membranes are the
cytoskeletons of cell membranes, which is essential to their stability as well
as functionality. The simplest and most thoroughly studied example is the
cytoskeleton of mammalian erythrocytes (red blood cells). The red blood cell
cytoskeleton is a tethered network of triangular plaquettes formed primarily by
the proteins spectrin and actin (see fig. (\ref{tethsurff.0}) from \cite{WieseHabil,LiuDerick1987,FalkSpeth1}).\\
\begin{figure}[t]
  \begin{center}
    \leavevmode
    \raisebox{0mm}{\includegraphics[scale=0.3]{./eps/spect3.eps}}
    \hspace{1cm}
    \raisebox{0mm}{\includegraphics[scale=0.3]{./eps/spect2.eps}}
\end{center}
    \caption{Extended view (right) of the crystalline spectrin/actin network which
    forms the cytoskeleton of the red blood cell membrane (left) .}
    \label{tethsurff.0}
\end{figure}
Inorganic realizations of crystalline membranes are graphitic oxide membranes,
which are micron size sheets of solid carbon. They are formed by exfoliating
carbon with a strong oxidizing agent \cite{HwaKokufutaTanaka1991,WenGarlandHwaKardarKokufutaLiOrkiszTanaka1992,SpectorNaranjoChiruvoluZasadzinski1994}. Furthermore, metal
dichalcogenides such as $\mbox{MoS}_{2}$ have also been observed to form
rag-like sheets \cite{ChianelliPrestridgePecoradoPecoradodDeNeufville1979}. 
\begin{comment}
\begin{figure}[t]
  \begin{center}
    \leavevmode
    \raisebox{0mm}{\includegraphics[scale=0.2]{./eps/sheet_mol.eps}}  
\end{center}
    \caption{The sheet molecule $\mbox{B}_{2}\mbox{O}_{3}$.}
    \label{tethsurff.1}
\end{figure}
\end{comment}
\subsection{Landau theory of the crumpling transition}\label{LT}
In this section let us discuss an order parameter theory of the crumpling
transition \cite{PaczuskiKardarNelson1988}. Any $D$- dimensional manifold embedded in $d$- dimensional
space can be parametrized by
\begin{equation}
  \label{LT.0}
  \vec r:\ \vec x\in \mathbb{R}^{D}\ \longrightarrow \vec r(\vec x)\in
  \mathbb{R}^{d}\ ,
\end{equation}
where $\vec r(\vec x)$ points to the position in embedding space of the
manifold constituent labeled by $\vec x$. For $D=1$, this represents a
polymer, for $D=2$ a membrane. Starting from a $D$- dimensional lattice with
nearest neighbor interactions, a statistical description is developed by
coarse graining this lattice so that the vector $\vec x$ becomes a continuous
variable, which labels a ``bloc'' of lattice points. Translational invariance
delimits the form of the free energy functional to depend only on the coarse
grained tangent vectors $\vec t_{\alpha}=\partial \vec r(\vec x)/\partial
x_{\alpha},\ \alpha=1,\dots, D$ and its derivatives. An expansion \`a la
Ginzburg-Landau for small tangents then leads to an effective free energy
\cite{Landau7}:
\newpage
\begin{eqnarray}
  \label{LT.1}
  \mathcal{H}[\vec r]&=&\int\mbox{d}^{D}x\
  \left[\frac{\kappa}{2}\left(\partial_{\alpha}\partial_{\alpha}\vec
  r\right)^{2}+\frac{t}{2}\left(\partial_{\alpha}\vec
  r\right)^{2}+u\left(\partial_{\alpha}\vec r\ \partial_{\beta}\vec
  r\right)^{2}+\tilde v\left(\partial_{\alpha}\vec r\ \partial_{\alpha}\vec
  r\right)^{2}\right]\nn\\
   &&\qquad +\frac{b}{2}\int\mbox{d}^{D}x\ \int\mbox{d}^{D}y\ \delta^{d}(\vec
  r(\vec x)-\vec r(\vec y))\ .
\end{eqnarray}
The last term is a nonlocal excluded volume term, which represents the effects
of self-avoidance at large length scales. Identifying the tangents $\vec
t_{\alpha}$ with a set of order parameters $\phi_{\alpha},\ \alpha=1,\dots,D$,
an analogy with the normal $\phi^{4}$ theory becomes apparent. However, this
analogy is only valid at the mean-field level.\\
At high temperatures one expects a crumpled phase, where $t$ is positive for
entropic reasons, while the other terms are irrelevant on large
scales. $\kappa$ denotes the bending rigidity, and the other elastic constants

$t,\ u$ and $\tilde v$ are related to the Lam\'e-coefficients of Landau
elastic theory.\\
At low temperatures, the microscopic surface tangents tend to align in order
to minimize the microscopic bending energy, driving the manifold to form a
flat phase ($t<0$). The manifold is stabilized by the anharmonic terms,
provided that $u>0$ and $v:=\tilde v+u/D>0$. There is a continuous transition
between these phases when $t\equiv \alpha(T-T_{c})= 0$. This can be seen
within mean-field theory. In the absence of self-avoidance we make the ansatz
\begin{equation}
  \label{LT.2}
  \vec r(\vec x)=\sum_{i=1}^{D}\xi x_{i}\vec \mathbf{e}_{i}\ ,
\end{equation}
where the sum is over two orthogonal cartesian unit vectors spanning a flat
surface. Minimizing the corresponding mean-field effective potential,
\begin{equation}
  \label{LT.3}
  V(\xi)=D\xi^{2}\left(\frac{t}{2}+(u+\tilde v D)\xi^{2}\right)\ ,
\end{equation}
leads to the following solution:
\begin{eqnarray}
  \label{LT.4}
  \xi^{2}=\left\{ 
  \begin{array}[h]{c}
    0\ ,\ t\geq 0\nn\\
    -\frac{t}{4(u+\tilde v D)}\ ,\ t<0
  \end{array}\right.
\end{eqnarray}
Consequently, there is a flat phase for $t<0$ and a crumpled phase for $t>0$,
separated by a crumpling transition at $t=0$. Figure (\ref{LTf.0}) shows a
``phantom'' membrane in typical states below, above and at the crumpling
transition (Monte-Carlo samples from \cite{BowickCatterallFalcioniThorleifssonAnagnostopoulos1997}).
\begin{figure}[t]
  \begin{center}
    \leavevmode
    \raisebox{0mm}{\includegraphics[scale=0.8]{./eps/confs3.eps}}  
\end{center}
    \caption{Crumpled phase (a), crumpling transition (b) and flat phase
    (c). No self-avoidance.} 
    \label{LTf.0}
\end{figure}
\subsection{Properties of the flat phase}
Fluctuational states around the ordered state can be parametrized as 
\begin{equation}
  \label{LT.5}
  \vec r(\vec x)=\xi\left[(x_{\alpha}+u_{\alpha})\vec
  \mathbf{e}_{\alpha}+h_{\beta}\vec \mathbf{e}_{\beta}\right]\ ,
\end{equation}
where the summation over $\alpha$ is as in (\ref{LT.2}), while the $\vec
\mathbf{e}_{\beta}$ are orthogonal to the latter. Specializing to membranes in $d=3$ there
is a single orthogonal space dimension. Inserting (\ref{LT.5}) into
(\ref{LT.1}) leads to lowest non-trivial order to
\begin{equation}
  \label{LT.6}
  \mathcal{H}[u,h]=\int\mbox{d}^{D}x\ \left[\frac{\kappa}{2}\left(\Delta
  h\right)^{2}+\mu u_{ij}^{2}+\frac{\lambda}{2}u_{kk}^{2}\right]\ ,
\end{equation}
where $\mu=4u\xi^{4},\ \lambda=8\tilde v\xi^{4}$ and
$u_{ij}=\frac{1}{2}[\partial_{i}u_{j}+\partial_{i}u_{j}+\partial_{i}h_{\beta}\
\partial_{j}h_{\beta}]$ denotes the strain matrix.\\
It was shown in \cite{NelsonPeliti1987} that the renormalized bending rigidity grows at large
distances and stabilizes the stretched phase against undulations. For small
wavevectors $\vec \mathbf{q}$ one finds an effective rigidity $\kappa_{R}$ with
\begin{equation}
  \label{LT.7}
  \kappa_{R}(\mathbf{q})\sim \sqrt{k_{B}T\ K_{0}}\ \mathbf{q}^{-1}\ ,\qquad
  K_{0}=\frac{4\mu(\mu+\lambda)}{2\mu+\lambda}\ .
\end{equation}
Qualitatively, this is
easily understood intuitively: Consider a low energy state of the nearly flat
membrane, which is for instance a single low-wavevector undulation \footnote{
  Out-of-plane fluctuations called undulations are to be distinguished from
  internal phonon modes.}. It does not necessitate any stretching of bonds
between adjacent molecules. However, adding to the long-wavelength deformation
with wavevector $\vec \mathbf{q}_{1}$ another short-wavelength deformation
$\vec \mathbf{q}_{2}$ with $|\vec \mathbf{q}_{1}|\ll |\vec \mathbf{q}_{2}|$
and $\vec \mathbf{q}_{1}\perp \vec \mathbf{q}_{2}$, this will induce a space
varying gaussian curvature, which inevitably forces to locally stretch and
squeeze bonds. Thus, the free energy of the short wavevector undulation is
raised up with respect to the same undulation on the flat
background. Correspondingly to (\ref{LT.7}), the normal-normal correlation
function stays
always finite:
\begin{equation}
  \label{LT.8}
  \lim_{|x|\to \infty}\left<\vec \mathbf{n}(x)\vec
  \mathbf{n}(0)\right>\ <\ \infty\ ,
\end{equation}
where $\vec n(x)$ denotes the local unit normal vector on the
surface. Therefore, the membrane stays always oriented.
The symmetry is broken, and the membrane is flat. This seems to be a violation
of the Mermin-Wagner theorem: In fact, the fluctuations in the membrane give
rise to long-range interactions, to which the Mermin-Wagner theorem does not
apply.\\
\subsection{Properties of the crumpled phase}
In the crumpled phase the free energy is governed by an entropic elasticity \cite{KantorKardarNelson1986a}
such that (\ref{LT.1}) reduces to
\begin{equation}
  \label{LT.9}
  \mathcal{H}_{0}[\vec r]=\frac{t}{2}\int\mbox{d}^{D}x\ (\nabla \vec
  r(x))^{2}+\textrm{irrel. terms}\ .
\end{equation}
The spatial extend of the manifold is characterized by its radius of gyration,
which is defined as
\begin{equation}
  \label{LT.10}
  R_{g}^{2}=\frac{1}{2N^{2}}\sum_{i,j}\left<(\vec r_{i}-\vec
  r_{j})^{2}\right>_{\mathcal{H}_{0}}\ .
\end{equation}
In the case of polymers $N$ denotes the number of segments, while for $2D$
surfaces, correspondingly, it is the number of plaquettes. The radius
of gyration is the mean squared distance of the surface constituents from
the center of mass of the manifold. It scales with the linear size $L$ of the manifold like
\begin{equation}
  \label{LT.11}
  R_{g}\sim L^{\nu}\ ,\qquad 0 \leq \nu \leq 1\ 
\end{equation}
and is related to the plaquette density correlation function, which is easily
accessible experimentally by light scattering, small-angle $X$-ray scattering
etc. \cite{Doi}.\\
For large manifolds $R_{g}$ scales like the two-point function
\begin{equation}
  \label{LT.12}
  \frac{1}{2d}\left<(\vec r(\vec x)-\vec r(\vec y))\right>_{\mathcal{H}_{0}}=
  \frac{|\vec x-\vec y|^{2-D}}{(2-D)S_{D}}\ ,
\end{equation}
where $a$ is a microscopic cutoff. One then finds:
\begin{eqnarray}
  \label{LT.13}
  R_{g}\sim \left\{
    \begin{array}[h]{c}
      L^{\frac{2-D}{2}}\ ,\quad D<0\nn\\
      \left(\ln \frac{L}{a}\right)^{1/2}\ ,\quad D=2
    \end{array}\right.
\end{eqnarray}
Taking self-avoidance into account gives rise to corrections to the exponent
$\nu$. This can be understood within the Flory approximation: It consists in
replacing $\vec r(\vec x)$ in (\ref{LT.1}) by the radius of gyration and
derivatives with respect to $x_{\alpha}$ by $1/L$ as well as the integration
over the internal volume by $L^{D}$, leading to
\begin{equation}
  \label{LT.14}
  \mathcal{H}\simeq \kappa\ L^{D-4}R_{g}^{2}+t\ L^{D-2}R_{g}^{2}+(u+D\tilde
  v)\ L^{D-4}R_{g}^{4}+b\ L^{2D}R_{g}^{-d}\ .
\end{equation}
The contribution from the bending rigidity $\kappa$ can always be neglected with
respect to the terms proportional to $t$ and $u+D\tilde v$,
respectively. Minimizing (\ref{LT.14}) provides:
\begin{eqnarray}
  \label{LT.15}
  R_{g}\sim \left\{
    \begin{array}[h]{c}
      L\ ,\quad \textrm{as}\ D\leq d\ \textrm{and}\ t<0\nn\\
      L^{\nu_{\tiny{\mbox{Flory}}}}\ ,\quad
      \nu_{\tiny{\mbox{Flory}}}=\frac{2+D}{2+d}\ ,\quad t>0
    \end{array}
\right.
\end{eqnarray}
For polymers the Flory approximation makes predictions,
which are sufficient for experiments. The prediction for membranes in $d=3$, $\nu=\frac{4}{5}$, is
close to what has been found in field theoretic calculations to two loops
($\nu\approx 0.86$).



 


\newpage \section{Model and physical observables}
\label{Model and physical observables}
\subsection{The model}
We start from the manifold Hamiltonian (\ref{1.1}). We split
the total embedding space $\R^{d+d'}$ into $\mathcal{E}$ and its
orthogonal complement 
$\mathcal{E}_{\perp}$ of dimension $d$. Each $x\in
\mathcal{V}$ points to a point $\vec r(x){=}(\vec r_{\perp}(x),\vec
r_{\parallel}(x))$, with $\vec r_{\parallel}(x)\in \mathcal{E}$ and
$\vec r_{\perp}(x)\in \mathcal{E}_{\perp}$.  The integration over
the subspace $\mathcal{E}$ is then trivial and gives
\begin{equation}
  \label{2.0}
  \int_{\vec{y} \in \mathcal{E}}\delta^{d+d'}(\vec
  r(x){-}\vec y)=\delta^{d}(\vec r_{\perp}(x))\ .
\end{equation}
In the partition function
\begin{equation}
{\cal  Z}_{\ind{total}}  = \int {\cal D}\left[\vec r \right] \, \exp (-{\cal H}\left[\vec r \right])
\end{equation}
the contributions from the parallel and orthogonal components of
$\vec r(x)$  factorize as
\begin{equation}
{\cal Z}_{\ind{total}}  = {\cal Z}_0 \times {\cal Z} (g_{0})
\end{equation}
with
\begin{eqnarray}
  {\cal Z}_0 &=& \int \mathcal{D}[\vec r_{\parallel }]\ \exp\left(-\half
(\nabla_{\parallel }\vec{ r} (x))^{2} \right) \\
\label{2.1a}
\mathcal{Z}(g_{0})&=&\int \mathcal{D}[\vec r_{\perp }]\ \exp(-\mathcal{H}_{\perp}[\vec r_{\perp}])\\
  \label{2.1}
  \mathcal{H}_{\perp}[\vec r_{\perp}]&=& \int_{\mathcal{M}}\ \mbox{d}^{D}\! x \left( \frac{1}{2}(\nabla
 \vec r_{\perp}( x))^{2}+g_{0}\ \delta^{d}( \vec r_{\perp}( x)) \right)\ .
\end{eqnarray}
Since ${\cal Z}_{0}$ is trivial, we will only consider 
$\mathcal{H}_{\perp}[\vec r_{\perp}]$ and shall drop the subscript  
 $_{\perp}$ for notational simplicity. We keep in mind 
that cases with $d<D$ make sense, for instance a polymerized (non-selfavoiding)
membrane interacting with a wall is described by (\ref{2.1})
setting $D=2$ and $d=1$.


Let us discuss (\ref{2.1}) in more detail:
The first term is the elastic energy of the manifold which is entropic in
origin. We have scaled elasticity and temperature to unity.
The second term
models the interaction of the manifold with a 
single point at the origin
 in the external $d$-dimensional space, but we remind that the
physical interpretation may well be that of a line or surface.
 The coupling constant $g_{0}$ may either be positive (repulsive
interaction) or negative (attractive interaction).
We now give the dimensional analysis. 
In  coordinate-space units, the engineering dimensions are
\begin{eqnarray*}
  &&\mbox{dim}[x]=1  \\
  \nu&:=&\mbox{dim}[\vec r]=\frac{2-D}{2}
\end{eqnarray*}
\begin{equation}
  \label{2.2}
\varepsilon :=\mbox{dim}\left[ \int_{{\mathcal{M}}}\mbox{d}^{D}\!
  x\ \delta^{d}(\vec r(x))\right] =D-\nu d
\end{equation}
\[g_{0}\ \sim \ \mu^{\varepsilon} \]
\[ \mbox{dim}[\mu]\ =\ -1 \ ,\]\\
where 
\begin{equation}\label{mufromL}
\mu\equiv \frac{1}{L}
\end{equation}
is an inverse length scale. The interaction is naively
relevant for $\E>0$, i.e. $d<d_{c}$ with (see figure \ref{f2.0})
\begin{equation}
  \label{2.3}
  d_{c}=\frac{2D}{2{-}D}\ ,
\end{equation}
irrelevant for $\E<0$ and marginal for $\E=0$.
It has been  shown \cite{DDG1,DDG2} that the model is
renormalizable for $0<D<2$ and $\E\ge0$. Results for negative $\E$ 
are obtained via analytical continuation. 
\begin{figure}[t]
  \begin{center}
    \leavevmode
    \raisebox{0mm}{\includegraphics[scale=0.45]{./eps/cl-data.eps}}  
\end{center}
    \caption{Critical line defined through $\varepsilon=0\Leftrightarrow
    d_{c}(D)=\frac{2D}{2{-}D}$. The interaction is relevant for points
that lie above that line.}
    \label{f2.0}
\end{figure}%
\begin{figure}[b]
  \begin{center}
    \leavevmode
    \raisebox{0mm}{\includegraphics[scale=0.35]{./eps/rgflow.eps}}
  \end{center} 
    \caption{RG-function and flow for increasing manifold size $L$ for
the dimensionless renormalized coupling $g$: (a) in the case
$\varepsilon{>}0$, (b) in the case $\varepsilon{<}0$, (c) in the case
$\varepsilon=0$. }
    \label{f2.1}
\end{figure}
One can define the renormalized coupling $g$ as
\begin{equation}
  \label{2.3.1}
  g:=\frac{\cal N}{\mathcal{V}_{\mathcal{M}}}\left[\mathcal{Z} (0) -\mathcal{Z} (g_{0})\right]L^{\E}\ ,
\end{equation}
%{\Huge g war noch nicht definiert}
where the normalization $\cal N$ depends on the definition of the
path-integral (but not on $L$)  and is
chosen such that 
\begin{equation}\label{2.11p}
g = g_{0}L^{\E}+ O (g_0^{2})\ . 
\end{equation} Universal quantities are obtained
at fixed-points of the $\beta$-function, which is defined as 
\begin{equation}
  \label{2.4}
  \beta(g):=\mu \left.\frac{\partial g}{\partial \mu}\right|_{g_0}\ .
\end{equation}
The $\beta$-function thus  describes, how the effective coupling $g$
changes under scale 
transformations, while  keeping the bare coupling $g_{0}$ fixed. 
Let us already anticipate the 1-loop result, which we derive later. 
It reads
\begin{equation}
  \label{2.5}
  \beta(g)=-\varepsilon g+\frac{1}{2}g^{2}+O(g^{3})\ ,
\end{equation}
where $g$ is the dimensionless renormalized coupling. Apart from the
trivial solution, $g=0$, the flow equation given by (\ref{2.4}) and
(\ref{2.5}) has a  non-trivial fixed point at the zero of the
$\beta$-function
\begin{equation}
  \label{2.6}
  g^{*}=2\varepsilon+O(\varepsilon^{2})\ .
\end{equation}
We shall show below that the scaling behavior is
described by the  slope of the RG-function at the fixed
point, which is universal as a consequence of renormalizabilty.
The long-distance behavior is then governed by the
$\delta$-interaction as considered in our model (\ref{2.1}), which is
the most relevant operator at large scales.  
Let us now discuss  possible physical situations (see fig.\ \ref{f2.1}):
\begin{itemize}
\item [(a)] $\varepsilon{>}0$: The RG-flow has an infrared stable
fixed point at $g^{*}>0$ and an IR-unstable fixed point at $g=0$. The
latter describes an unbinding transition whose critical properties are
given by the non-interacting system, while the non-trivial IR stable
fixed point determines the long-distance properties of the delocalized
state, the long-range repulsive force exerted by the fluctuating
manifold on the origin -- which we remind may be a point, a line or a
plane.
\item [(b)] $\varepsilon{<}0$: Now, the long-distance behavior is
Gaussian, while the unbinding transition occurs at some finite value
of the 
attractive potential,  $g^{*}{<}0$, which corresponds to an infrared
unstable fixed point of the $\beta $-function.  Below $g^{*}$ the
manifold stays always attracted.
\item [(c)] $\varepsilon=0$: This is the marginal situation, where the
transition takes place at $g^{*}=0$; we expect logarithmic corrections
to scaling.
\end{itemize}
Note that in the presence of an impenetrable wall constraining the
configurational space strictly to half of the embedding space, the above considerations should
still apply, when shifting the interaction strength
appropriately. We shall discuss that in \ref{Unbinding transition}. 

\subsection{Repulsive force exerted by a membrane on a wall}\label{wall-force}
Since we are mainly interested in the long-distance properties of membranes
for which always $\varepsilon>0$ (this is (a) on fig.\ \ref{f2.1}),
let us try to calculate the repulsive force exerted by the membrane
on the origin in the case that this point is strictly forbidden. We will
derive a universal expression for this force \cite{DDG2}. 
We need the (not normalized) membrane density at position $\vec{r}$
\begin{equation}\label{2.8}
  \mathcal{Z}(\vec r,g_{0}) :=
\frac{1}{\mathcal{V}_{\mathcal{M}}}\int \mathcal{D}[\vec r]
  \int_{\mathcal{M}}\rmd^{D}x\ \delta^{d}(\vec
r(x)-\vec{r} )\exp[-\mathcal{H}]\ .
\end{equation}
Since the $\delta$-interaction also appears in $\cal H$, we can 
relate the density at the origin to the derivative of the partition
function with respect to $g_{0}$: 
\begin{equation}
  \label{2.9} %%%%%%%% argh, the sign seems to have been wrong !!! %%%%%%%
  \mathcal{Z}(\vec 0,g_{0})=- \left. {}\frac{1}{\mathcal{V}_{\mathcal{M}}}
\frac{\partial}{\partial
  g_{0}}\right|_{L}\mathcal{Z}(g_{0})=
\frac{1}{\mathcal{N}} \left.\frac{\partial
(g L^{-\varepsilon})}{\partial 
  g_{0}}\right|_{L}\ ,
\end{equation}
where $g$ is the renormalized or effective coupling defined in
(\ref{2.3.1}). 
Since $\mathcal{Z}(\vec 0,g_{0})$  is dimensionless, it depends 
on the dimensionless combination $g_{0} L^{\E}$ (and
$\varepsilon$) only. Using $\mu =1/L$,  we obtain
\begin{equation}
  \label{2.10}
{\mathcal{N}}  \mathcal{Z}(\vec 0,g_{0})
=\left. L ^{-\E} 
  \frac{\partial g}{\partial  g_{0}}\right|_L
\equiv  -\frac{\beta(g)}{\varepsilon g_{0} L^{\E}}\ .
\end{equation}
For the further considerations, it is sufficient to know the behavior
of the $\beta $-function close to the nontrivial zero. Expanding about
$g{=}g^{*}$, we have
\begin{equation}
  \label{2.11}
  \beta(g) = (g-g^{*})\omega(g^{*})+O ((g-g^{*})^{2})\ .
\end{equation}
Combining (\ref{2.9}), (\ref{2.10}) and (\ref{2.11}), we get
\begin{equation}
\left.\frac{\partial g }{\partial
  ( g_{0} L^{\varepsilon})}\right|_{L} =  -\frac{\beta(g)}{\varepsilon g_{0} L^{\E}}
=-\frac{(g-g^{*})\omega(g^{*}) }{\varepsilon g_{0} L^{\E}} \ .
\end{equation}
The solution of this differential equation is
\begin{equation}
  \label{2.12}
  g-g^{*}\sim  (g_{0}L^{\E})^{-\omega(g^{*})/\varepsilon} + \dots \ 
\ .
\end{equation} 
Using  (\ref{2.10}), this implies  the scaling law
\begin{equation}
  \label{2.13}
  \mathcal{Z}(\vec 0,g_{0})\sim (g_{0}^{1/\varepsilon}L)^{-\omega(g^{*})-\varepsilon}\ .
\end{equation}
We now use this result to derive a scaling law for $\mathcal{Z}(\vec
r,g_{0})$, with $\vec{r} \neq \vec{0} $. In addition to $g_{0}L^{\E}$
(and $\E$), $\mathcal{Z}(\vec r,g_{0})$ depends on $\vec{r} $; taking
into account rotational symmetry and dimensionality, it depends
on $\vec r$ only through the dimensionless combination $r/L^{\nu }$, with $r=|\vec{r}| $:
\begin{equation}
  \label{2.16}
  \mathcal{Z}(\vec r,g_{0}) = 
  \mathcal{Z}(r/L^{\nu},g_{0}L^{\E})\ .
\end{equation}
For  $\mathcal{Z}(\vec r,g_{0})$, the most interesting limit is that
of $g_{0}\to \infty $. Physically, this corresponds to strictly forbidding
monomers to be at the origin. Therefore it is clear that this limit is
well-behaved, and  
\begin{equation}
  \label{2.17} \mathcal{Z}_{\infty}(r/L^{\nu}):=
  \lim_{g_{0}\to \infty}\mathcal{Z}(r/L^{\nu},g_{0}L^{\E})
\end{equation}
is finite. In order to be consistent with (\ref{2.13}) it has to 
obey a power law in the scaling regime $r\ll L^{\nu}$
\begin{equation}
  \label{2.18}
  \mathcal{Z}_{\infty}(r/L^{\nu})\sim (r/L^{\nu})^{\theta}\ .
\end{equation}
Comparing the $L$-dependence of $\mathcal{Z}_{\infty } (r/L^{\nu })$
and $\mathcal{Z} (\vec{0},g_{0} )$, we obtain the  exponent identity
\begin{equation}
  \label{2.19}
  \theta=\frac{\varepsilon+\omega(g^{*})}{\nu}\ .
\end{equation}
Finally, from (\ref{2.18}) we derive the repulsive force between the
origin and the manifold 
\begin{equation}
  \label{2.20}
  \vec f(\vec r)= \nabla_{\vec r}\ln \mathcal{Z}_{\infty }(| \vec{r}
|/L^{\nu }) = \theta \frac{\vec r}{r^{2}}\ .
\end{equation}
Note that to derive this result, $k_{B}T$
has been set to 1. Reestablishing the temperature-dependence, we find
\begin{equation}
  \vec f(\vec r)= k_{B}T \theta \frac{\vec r}{r^{2}}\ .
\end{equation}
Also note that this argument gives $\theta =0$ at the
Gaussian fixed point, which is necessary since for $g_{0}=0$ no force
is exerted on the membrane.
\subsection{Monomer density}\label{density} 
Let us once again turn to the ensemble as defined in (\ref{2.9}). The
expectation value of the monomer density at the potential, $\left<n(0)\right>$, where also the
manifold is pinned, is obtained from
\begin{eqnarray}
  \label{de.1}
  \left<n(0)\right>&=&\frac{1}{\mathcal{Z}(\vec 0,g_{0})}\int\mathcal{D}[\vec
  r]\int_{\mathcal{V}_{M}}\delta^{d}(\vec r(x))\ \delta^{d}(\vec r(s))\
  \mathrm{e}^{-\mathcal{H}[\vec
  r]}=-\frac{\partial}{\partial g_{0}}\ln Z(\vec 0,g_{0})
%-\frac{\frac{1}{\mathcal{V}_{M}}\frac{\partial^{2}}{\partial
%  b_{0}^{2}}\mathcal{Z}}{\mathcal{Z}_{f}}=-\frac{\frac{1}{\mathcal{V}_{M}}\frac{\partial^{2}}{\partial
%  b_{0}^{2}}\mathcal{Z}}{\frac{1}{\mathcal{V}_{M}}\frac{\partial}{\partial b_{0}}\mathcal{Z}}\ ,
\end{eqnarray}
%where $\mathcal{Z}_{f}\equiv \mathcal{Z}(\vec 0,g_{0})$. 
Since
$g=\frac{\mathcal{N}}{\mathcal{V}_{\mathcal{M}}}[\mathcal{Z}(0)-\mathcal{Z}(g_{0})]L^{\varepsilon}$
and $\frac{\partial}{\partial g_{0}}=L^{\varepsilon}\frac{\partial}{\partial
  z}$,
\begin{eqnarray}
  \label{de.1}
  \left<n(0)\right>=\frac{1}{g_{0}}\left(1+\frac{\omega(z)}{\varepsilon}\right)\ .
  %&=&\frac{-L^{\varepsilon}\frac{\partial}{\partial
  %z}\left(\frac{1}{z}z\frac{\partial}{\partial z}g(z)\right)}{\frac{\partial
  %g}{\partial
  %z}}=\frac{L^{\varepsilon}\left(\frac{1}{z}\beta(z)-\frac{\partial}{\partial
  %z}\beta(z)\right)}{\beta(z)}\nn\\
  %&=&L^{\varepsilon}\left(\frac{1}{z}-\frac{\frac{\partial}{\partial
  % z}\beta(z)}{\beta(z)}\right)=\frac{1}{g_{0}}\left(1+\frac{\omega(z)}{\varepsilon}\right)\ ,
\end{eqnarray}
where $\omega(z)=-\varepsilon z(\partial{\beta}/\partial z)/\beta(z)$ and
 $\beta(z)=-\varepsilon z\partial{g}/\partial z$.\\
 In
particular, $\lim\limits_{z\to \infty}\omega(z)=\omega(g^{*})>0$.
The monomer density can be measured in a Monte-Carlo simulation
and therefore offers a possibility also to check predictions for membranes
$(D=2)$. Let us point out that starting from the natural assumption
 that as long as $\varepsilon>0$ the
monomer density at the potential stays finite in the scaling limit of large manifolds 
one may directly deduce the existence of a strong coupling expansion of the
effective coupling $g(z)$:
\begin{equation}
  \label{de.2}
  g(z)=g^{*}-s(\ln z)z^{-\omega/\varepsilon}+O(z^{-w})\ ,
\end{equation}
where $s$ denotes some function that increases at most subexponentially and $w>\omega/\varepsilon>0$.




\subsection{Unbinding transition}\label{Unbinding transition}
Let us discuss the physical situation at the UV-stable fixed point in figure
\ref{f2.1}. The fixed point corresponds to a {\em delocalization
transition} of the manifold, which is at vanishing coupling $g^{*}{=}0$ for $\varepsilon{>}0$
and  at some finite attractive coupling $g^{*}{<}0$ for
$\varepsilon{<}0$.\\
In the localized phase $g{<}g^{*}$, correlation functions such as
$\left<[\vec r(x)-\vec r(y)]^{2}\right>$ and the associated correlation length
$\xi_{\parallel}$ (in the $D$-dimensional internal space)  should be finite, as
well as the radius of gyration $\xi_{\perp}$. Approaching the transition point
 these quantities diverge as \cite{ForgasLipowskyNieuwenhuizenInDombGreen}
\begin{equation}
  \label{u.1}
  \xi_{\parallel}\ \sim \ (g^{*}{-}g)^{-\nu_{\parallel}}\quad ,\quad
  \xi_{\perp}\ \sim \ (g^{*}{-}g)^{-\nu_{\perp}}\ .
\end{equation}
Since $\xi_\perp\sim\xi_\parallel^{\nu }$, the exponents $\nu_{\parallel}$ and $\nu_{\perp}$ are related through
\begin{equation}
  \label{u.2}
  \nu_{\perp}\ =\ \nu_{\parallel}\nu\ ,
\end{equation}
$\nu$ being the dimension of the field (\ref{2.2}).
To relate $\nu_{\parallel }$ to $\omega $, we first observe that $\xi
_{\parallel }$ depends          only on $g_0$
% {\Huge "Wo bleibt Universalitaet?"} haehh???
and the membrane size $L$, or its inverse $\mu =1/L$.  Writing $\xi
_{\parallel }$ as a function of the renormalized coupling $g$ and the
 scale $\mu $, we thus have 
\begin{eqnarray}
0&=&\mu          \frac{\partial }{\partial \mu } \xi _{\parallel } (g,\mu ) \nn\\&=&
\mu      \frac{\partial }{\partial \mu } \left(\frac{1}{\mu }f (g)
\right)\nonumber  \\
&=&  \frac{1}{\mu }\left(-f (g) +\beta (g)f' (g) \right)\ ,
\end{eqnarray}
where the dimension-full factor  $1/\mu $ has been factored from the
dependence on the dimensionless renormalized coupling $g$.
 Using that $\frac{\partial }{\partial g}\xi _{\parallel }
(g)=\frac{1}{\mu }f' (g)$, this can be rewritten as
\begin{eqnarray}
\xi _{\parallel } (g) &=& \beta (g)\frac{\partial }{\partial g}\xi
_{\parallel } (g)\nn\\
&\approx & \omega (g^{*}) (g-g^{*})\frac{\partial }{\partial g}\xi
_{\parallel } (g)
\ .
\end{eqnarray}
The solution to the above equation is 
\begin{equation}
\xi _{\parallel } \sim  |g-g^{*}| ^{1/\omega (g^{*})}\ .
\end{equation}
This leads to the identification 
\begin{equation}
\nu _{\parallel } = -\frac{1}{\displaystyle \omega (g^{*})} \ , \qquad
\nu _{\perp  } = -\frac{\nu }{\displaystyle \omega (g^{*})} \ .
\end{equation}
Note that $\omega(g^{*}){<}0$ at the transition. Specializing to
$(D,d){=}(1,1)$, we find
\begin{equation}
  \label{u.6}
  \nu_{\perp}\ =\ 1\quad ,\quad \nu_{\parallel}\ = \ 2\ .
\end{equation}

These exponents are also valid for  the delocalization transition of a
$1$-dimensional interface from an attractive hard wall in $2$-dimensional bulk
space \cite{BrezinHalperinLeibler1983,ForgasLipowskyNieuwenhuizenInDombGreen,Upton1999}. This can be understood as follows: The  partition function of a
fluctuating polymer interacting with a $\delta$-defect at the
boundary of a hard wall can be written as
\begin{eqnarray}
  \label{u.7}
  \mathcal{Z}\ =\
  \frac{1}{2}\sum_{n=0}^{\infty}\mathcal{Z}_{n}\left(\frac{1}{2}\right)^{n}\
  e^{-n\beta E}\ ,
\end{eqnarray}
\begin{figure}
$\parbox{0.45\textwidth}{\epsfxsize=0.45\textwidth\epsfbox{./eps/wall1.eps}}
\hfill \longrightarrow \hfill 
\parbox{0.45\textwidth}{\epsfxsize=0.45\textwidth\epsfbox{./eps/wall2.eps}}$
\caption{A polymer (here for simplicity directed) in interaction with
a wall (thick dots). Left before and right after flapping up. The grey
area is impenetrable.}
\label{flap}
\end{figure}
where the sum runs over the number of contacts with the defect hyper-plane. $E$
denotes the contact energy, $\beta=(k_{B}T)^{-1}$ and $\mathcal{Z}_{n}$
is the constrained partition function of the free interface having exactly
$n$ points of contact with the defect. The powers of $\frac{1}{2}$  only
appear in the presence of an impenetrable defect and reflect the fact
that all configurations in the presence of a hard wall can be obtained
by flapping up those parts of the polymer which have penetrated the
line (see figure \ref{flap}). This accounts for a factor $\frac{1}{2}$
for each possible flap.
 Thus, the hard wall
constraint translates  into a shift in the binding energy according to 
\begin{equation}
  \label{u.8}
  E_{\mathrm{no\;wall}}\ \longrightarrow\ E_{\mathrm{wall}} =
  E_{\mathrm{no\;wall}}+k_{B}T\ \ln 2\ .
\end{equation}
The delocalization transition occurring for $(D,d){=}(1,1)$ at vanishing
potential strength is shifted to an attractive interaction dependent on the
temperature. Due to the correspondence (\ref{u.8}) the exponents
characterizing the transition remain unchanged.


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\newpage
$ $
\newpage \section{Operator product expansion and one-loop result}
The partition function of the model has been  defined in (\ref{2.1a}).
Analogously, we define the expectation value of an observable $
\mathcal{O}(r)$ as (we denote $r$ for $\vec{r} $ from now on)
\begin{equation}
  \label{3.1a}
\left< \mathcal{O}(r)   \right>_{g_{0}}  :=\frac{\int \mathcal{D}[r]\ \exp(-\mathcal{H}[r])}{Z (g_{0})}\ .
\end{equation}
Most physical observables can be derived from expectation values of general
$M$-point vertex operators like
\begin{equation}
  \label{3.2}
  \mathcal{O}(r)=\prod_{j=1}^{M}\rme^{ip_{j}r(s_{j})}\ .
\end{equation}
Before we go on, let us make some changes in normalizations which will
be helpful in the following. First, we rescale the field and the coupling
constant according to \footnote{At this stage this is to be undertood only as
  a formal procedure, that simplifies calculations. All loop integrals are
  normed with respect to the one loop integral. The precise meaning in the
  limit $D\to 2$ is discussed in section (\ref{analyticcont}).}
\begin{equation}
  \label{3.8}
  r \longrightarrow (2{-}D)^{1/2}r\ ,\quad g_{0} \longrightarrow
 (2{-}D)^{d/2}g_{0}\ .
\end{equation}
Second, we change the integration over internal coordinates to 
\begin{equation}
  \label{3.8.1}
  \int_{x}:=\frac{1}{S_{D}}\int \mbox{d}^{D}x\ ,\quad S_{D}=2\frac{\pi^{D/2}}{\Gamma(D/2)}
\end{equation}
being the volume of the $D$-dimensional unit-sphere. Third, the
normalization of the $\delta$-distribution is changed to 
\begin{equation}
  \label{3.8.2}
  \tilde \delta^{d}(r(x)):=(4\pi)^{d/2}\delta(r(x))=\int_{k}\rme^{ikr(x)}
\end{equation}
with
\begin{equation}
  \label{3.8.3}
  \int_{k}:=\pi^{-d/2}\int \mbox{d}^{d}k
\ .
\end{equation}
The  advantage of these normalizations is that 
\begin{equation}
\int_{k} \rme^{-k^{2}} = 1\ , \quad \int_{x} \Theta (|x|<1) =1 \ .
\end{equation}
The model in the new normalizations now reads
\begin{equation}\label{newmodel}
{\cal H}[r] = \frac{1}{2-D}\int_{x}\half \left(\nabla r (x)  \right)^{2}+
g_{0} \int_{x}\tilde{\delta }^{d} (r (x))\ ,
\end{equation}
and (due to the factor of $1/ (2-D)$ in the above definition) the
two-point correlator is 
\begin{equation}
  \label{3.7}
  C(x_{i}{-}x_{j}):=\frac{1}{2d}\left<[r(x_{i})-r(x_{j})]^{2}\right>_{0}=|x_{i}-x_{j}|^{2\nu}
\ .
\end{equation}

We now proceed with the calculation of physical observables. 
As an explicit example, let us consider the perturbation expansion of the
$1$-point vertex operator
\begin{equation}
  \label{3.3}
  \mathcal{Z}^{(p)}:=\mathcal{Z} (g_{0}) \langle \rme^{ipr(s)}\rangle_{g_{0}}=\sum_{N=0}^{\infty}\frac{(-g_{0})^{N}}{N!}\mathcal{Z}^{(p)}_{N},
\ ,
\end{equation}
where
\begin{equation}
  \label{3.4}
  \mathcal{Z}^{(p)}_{N}=\EXP{ \rme^{ipr(s)} \prod^{N}_{i=1}\int_{x_{i}}\
  \tilde\delta^{d}(r(x_{i}))}{\ 0}\ .
\end{equation}
This can be written as
\begin{equation}
  \label{3.5}
  \mathcal{Z}^{(p)}_{N}=
  \prod_{i=1}^{N}\int_{x_{i}}\int_{k_{i}}\tilde\delta^{d}(p+\sum
k_{i})\times \EXP{ \exp\left[ i ( \sum_{i=1}^{N}k_{i}r
  (x_{i})+pr(s))\right]}{\ 0}\ ,
\end{equation}
where we have already integrated out a global translation of the field $r$.
The Gaussian average is:
\begin{equation}
  \label{3.6}
    \EXP{\exp \left[i(\sum_{i=1}^{N}k_{i}{r}
  (x_{i})+pr(s))
  \right]}{\ 0}=\exp\left[\frac{1}{2}\sum_{i,j=0}^{N}k_{i}k_{j}C(x_{i}{-}x_{j})\right]\ \
  ,\ k_{0}=p
\ .
\end{equation}
The $\mathcal{Z}^{(p)}_{N}$ in (\ref{3.4}) posses short distance
singularities for $N{\geq}2$. In order to analyze these singularities we will
use  the techniques of normal ordering and operator product
expansion in the sequel.
In our problem the procedure of normal ordering a vertex operator
$\rme^{ik_{j}r(x_{j})}$ turns out to be quite simple:  In any
expectation value, we factorize out the  contractions between the 
 vertex operators. At one-loop order or
equivalently at second order in (\ref{3.3}) this means that
\begin{equation}
  \label{3.9}
  \rme^{ik_{1}r(x_{1})}\rme^{ik_{2}r(x_{2})}\ =\
  \bn \rme^{i(k_{1}r(x_{1})+k_{2}r(x_{2}))}\en\ \rme^{k_{1}k_{2}C(x_{1}{-}x_{2})}\ .
\end{equation}
This can be understood as a definition of the normal ordered product $\bn
\rme^{i(k_{1}r(x_{1})+k_{2}r(x_{2}))}\en$. Note, that equality in the above equation is meant in the sense of
operators, that is when inserted into expectation values within the free
theory.\\
Let us now turn to an explicit derivation of the OPE of two
$\delta$-interactions, i.e. we study (\ref{3.9}) for small distances
$x_{1}{-}x_{2}$. Since the short-distance singularities appear only in the
internal contractions, the normal ordered product in the r.h.s. of (\ref{3.9})
is regular for small distances $x_{1}{-}x_{2}$ and thus can be expanded
therein. We then project the resulting terms on the corresponding 
operators. For that purpose we make a change in the momentum variables
according to
\begin{equation}
  \label{3.10}
  \left(
  \begin{array}[h]{c}
    k_{1} \\
    k_{2}
    \end{array}
    \right) \longrightarrow 
    \left(
      \begin{array}[h]{c}
        \frac{k_{1}}{2}-k_{2} \\
        \frac{k_{1}}{2}+k_{2}
      \end{array}
    \right)\ .
\end{equation}
In internal space we change coordinates to the center of mass system:
\begin{equation}
  \label{3.11}
  \bar x:=\frac{1}{2}(x_{1}+x_{2})\ ,\quad
  y:=x_{2}-x_{1}\ .
\end{equation}
Thus, the r.h.s. of (\ref{3.9}) takes the form
\begin{eqnarray}
  \label{3.12}
  &&\bn \rme^{i[(\frac{k_{1}}{2}{-}k_{2})r(\bar
  x{-}y/2){+}(\frac{k_{1}}{2}{+}k_{2})r(\bar x{+}y/2)]}\en\,
\rme^{(k_1^{2}/4-k_{2}^{2})C (y)}
\ =\ \nonumber\\
  &&\bn \rme^{i k_{1}r(\bar x) }\ \left(1+ik_{2}y\nabla
  r(\bar x)+ik_{1}\frac{1}{2!}\frac{y^{\alpha}y^{\beta}}{4}\nabla_{\alpha}\nabla_{\beta}r(\bar x)-\frac{(k_{2}y\nabla r(\bar x))^{2}}{2!}+\dots\right)\en\times\nonumber\\
  &&\hphantom{\bn \rme^{ik_{1}r(\bar x)}\ }\left(1+\frac{k_{1}^{2}}{4}C(y)+\dots\right)\
  \rme^{-k_{2}^{2}C(y)}\ .
\end{eqnarray}
Integration over  $k_{1},k_{2}$ yields
 the OPE of $\tilde\delta^{d}(r(x_{1}))$ with $ \tilde\delta^{d}(r(x_{2}))$:
\begin{eqnarray}
  \label{3.13}
 \tilde \delta^{d}(r(\bar x-\frac{y}{2}))\tilde \delta^{d}(r(\bar x+\frac{y}{2}))
  &=& y^{-\nu d} \tilde \delta^{d}(r(\bar x))-y^{-\nu
d+2\nu}\frac{1}{4}\ \partial_{r}^{2}
\tilde  \delta^{d}(r(\bar x))\nonumber\\
  &&-y^{-\nu d-2\nu+2}\frac{1}{4D}\tilde \delta^{d}(r(\bar x)) (\nabla
  r(\bar x))^{2}\nonumber \\
&& -y^{-\nu d+2}\frac{1}{16D} (-\Delta_{r})\tilde \delta^{d}(r(\bar x))
  (\nabla r(\bar x))^{2}\nonumber\\
  &&+y^{-\nu d+2}\frac{1}{8D}\ \partial_{r_{i}}
\tilde\delta^{d}(r(\bar x))  \Delta
  r_{i}(\bar x)+\dots\  .
\ .
\end{eqnarray}
Restricting ourselves to the most relevant next to leading operators for
internal dimension $D$ smaller than, but  close to $2$, the operator product expansion reads:
\begin{equation}
  \label{3.14}
\tilde\delta^{d}(r(\bar x-\frac{y}{2}))\tilde \delta^{d}(r(\bar x+\frac{y}{2}))\
  =\ y^{-\nu d}\tilde\delta^{d}(r(\bar x))+\sum_{n=1}^{\infty}y^{-\nu
  d+n(D-2)}\frac{(-\partial_{r}^{2})^{n}\tilde\delta^{d}(r(\bar x))}{4^{n}n!}
\ .
\end{equation}
The OPE-coefficient in front of the leading operator, which is the
$\delta$-interaction itself, carries the leading short distance singularity. In the
following we introduce some diagrammatic notation.
 Denoting $\GA :=\tilde\delta^{d}(r(x))$ we abbreviate the projection of two nearby
$\delta$-interactions on the $\delta$-interaction itself as
\begin{equation}
  \label{3.15}
  \MOPE{\GB}{\GA}\ =\
  y^{-\nu d}\ .
\end{equation}
We now analyze short-distance divergences of the perturbation expansion using
the OPE in (\ref{3.13}). Being interested in the leading scaling behavior, we
drop all subdominant terms for $0<D<2$, such that we only need the leading
OPE-coefficient (\ref{3.15}).
The integral over the relative distance in $\int_{y}\MOPE{\GB}{\GA}\GA$ is
logarithmically divergent for $\varepsilon{=}0$. In order to define a counter
term we use dimensional regularization, i.e.\ we set
$\varepsilon{>}0$. The integral is then UV-convergent, but IR-divergent. An IR regulator or equivalently a subtraction scale
$\mu{=}L^{-1}$ has to be introduced to define the subtraction
operation. Generally, we integrate over all distances bound by the
inverse subtraction scale $L$. The OPE-coefficient in (\ref{3.15}) then
reduces to a number as
\begin{equation}
  \label{3.16}
  \DIAGind{\GB}{\GA}{L}:=\int_{|y|{<}L}\MOPE{\GB}{\GA}=L^{\varepsilon}f(\varepsilon,D)\ .
\end{equation}
Let us use the scheme of minimal subtraction (MS). The internal dimension of the
membrane is fixed and (\ref{3.16}) is expanded in a Laurent-series in
$\varepsilon$, starting here at order $\varepsilon^{-1}$. Denoting the term of
order $\varepsilon^{p}$ in $\left.\DIAGind{\quad}{\quad}{L}\right|_{L=1}$ by
$\DIAGind{\quad}{\quad}{\varepsilon^{p}}$, the simple pole in (\ref{3.16}) is
\begin{equation}
  \label{3.17}
  \DIAGind{\GB}{\GA}{\varepsilon^{-1}}=\frac{1}{\varepsilon}\ .
\end{equation}
Of course, in this case, due to our normalizations, this is exact.\\
Let us rewrite the Hamiltonian in (\ref{2.1}) in terms of the renormalized
dimensionless coupling $g$. For perturbative calculations it is custom
to write
\begin{equation}
  \label{3.18}
  g_{0}=gZ_{g}\mu^{\varepsilon}\ ,
\end{equation}
where $Z_{g}$ is the renormalization factor, which is fixed by
demanding that observables remain finite in the limit of $\E\to0$. The
Hamiltonian becomes
\begin{equation}
  \label{3.19}
  \mathcal{H}[r]= \frac{1}{2{-}D}\int_{x}\, \frac{1}{2}(\nabla
 r(x))^{2}+\int_{x} gZ_{g}\mu^{\varepsilon}\,\tilde \delta^{d}(r(x)) \ .
\end{equation}
We find to one-loop order
\begin{equation}
  \label{3.20}
  Z_{g}=1+\frac{g}{2}\DIAG{\GB}{\GA}{_{\varepsilon^{-1}}}+O(g^{2})\ .
\end{equation}
There is no field-renormalization since the elastic part has also to
describe the manifold far away from the origin, for which the
interaction term can be neglected.
 Formally, of course, this follows from the OPE.
The renormalization group $\beta$-function is defined as
\begin{equation}
  \label{3.21}
  \beta(g):=\left.\mu \frac{\mbox{d}}{\mbox{d}\mu}\right|_{g_{0}}g\ ,
\end{equation}
from what follows together with (\ref{3.18}) that
\begin{equation}
  \label{3.22}
  \beta(g)=\frac{-\varepsilon g}{1+g\frac{\partial}{\partial g}\ln Z_{g}}\ .
\end{equation}
To one-loop order we obtain from (\ref{3.22}) and (\ref{3.20}) as
anticipated earlier
\begin{equation}
  \label{3.23}
  \beta(g)=-\varepsilon g+\frac{g^{2}}{2}+O(g^{3})\ .
\end{equation}
The long-distance behavior of the theory is governed by the IR--stable
fixed point of the RG--flow. Fixed points are zeros of the
$\beta$--function:
\begin{equation}
  \label{3.24}
  \beta(g^{*})=0\quad \Rightarrow\quad  g^{*}=0\quad \textrm{or} \quad
  g^{*}=2\varepsilon\ .
\end{equation}
In section \ref{wall-force}, we had defined the correction-to-scaling 
exponent $\omega$. It is obtained from the slope-function $\omega
(g)$, defined as
\begin{equation}
\omega (g):= \frac{\rmd }{\rmd g} \beta (g)\ .
\end{equation}
The correction to scaling exponent is $\omega (g)$ evaluated at the
fixed points, and is found to be 
\begin{equation}
  \label{3.25}
  \omega(g^{*}{=}0)=-\varepsilon\ , \qquad \omega(g^{*}{=}2\varepsilon)=\varepsilon\ .
\end{equation}
Of course, this result  is apart from the very existence of a fixed
point  rather trivial, since it is determined
through the leading 
term of the $\beta$-function, which is always $-\varepsilon g$. In order
to obtain any non-trivial information from the perturbation series
 it is
thus necessary to go beyond the leading order.

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\newpage $ $
\newpage \section{Two-loop calculation in a MS-scheme}
\subsection{Operator product expansion}
Let us now continue with the calculation at the two-loop order. At
each order of perturbation theory there is only one new diagram. At
two-loop order this comes from three coalescing
$\delta$-interactions. Again, let us rewrite the product of the three
vertex operators as its expectation value times the normal ordered
product:
\begin{eqnarray}
  \label{4.1}
  &&\rme^{ik_{1}r(x_{1})}\rme^{ik_{2}r(x_{2})}\rme^{ik_{3}r(x_{3})}\nonumber\\
  &&=\
  \bn \rme^{i[k_{1}r(x_{1})+k_{2}r(x_{2})+k_{3}r(x_{3})]}\en \ \rme^{k_{1}k_{2}C(x_{1}{-}x_{2})}\rme^{k_{1}k_{3}C(x_{1}{-}x_{3})}\rme^{k_{2}k_{3}C(x_{2}{-}x_{3})}\ .
\end{eqnarray}
We use the following change of coordinates
\begin{eqnarray}
  \label{4.2}
  \left.
    \begin{array}{r@{\ :=\ }l}
      \bar x & \frac{1}{3}(x_{1}+x_{2}+x_{3}) \\
      a & x_{2}-x_{1} \\  
      b & x_{3}-x_{2} \\
      c & x_{1}-x_{3}
    \end{array}\right\} \Leftrightarrow \left\{ 
    \begin{array}{r@{\ =\ }l}
      x_{1} & \bar x+\frac{1}{3} (c-a)\rule{0mm}{1.ex} \\
      x_{2} & \bar x+\frac{1}{3} (a-b)\rule{0mm}{2.5ex} \\
      x_{3} & \bar x+\frac{1}{3} (b-c)\rule{0mm}{2.5ex} \\
    \end{array}\right. \ ,
\ .
\end{eqnarray}
As at 1-loop order, $ \bn
\rme^{i[k_{1}r(x_{1})+k_{2}r(x_{2})+k_{3}r(x_{3})]}\en$ is free of
divergences upon approaching the points $x_{i}$, whereas the UV-divergence
comes from the factor $\rme^{k_{1}k_{2}C(x_{1}{-}x_{2})}
\rme^{k_{1}k_{3} C(x_{1}{-}x_{3})}\rme^{k_{2}k_{3}C(x_{2}{-}x_{3})}$.
The leading term is thus obtained upon setting $x_{1}=x_{2}=x_{3}=\bar
x$ in  $ \bn \rme^{i[k_{1}r(x_{1})+k_{2}r(x_{2})+k_{3}r(x_{3})]}\en$,
 dropping the gradient terms. Further shifting 
$k_{1}\to k_{1}-k_{2}-k_{3}$, we obtain 
\begin{eqnarray}
\rme^{i k_{1}r (\bar x)} \, \rme^{( k_{1}-k_{2}-k_{3})k_{2}C (a)+k_{2}k_{3}C (b)+ ( k_{1}-k_{2}-k_{3})k_{3}C ( c)}
\ .
\end{eqnarray}
Integrating over $k_{1}$, $k_{2}$ and $k_{3}$ yields the final result
\begin{eqnarray} \label{4.7}
\tilde \delta ^{d} (r (x_{1}))\tilde \delta ^{d} (r (x_{2}))
\tilde \delta ^{d} (r (x_{3}))=
\MOPE{\GC}{\GA}\tilde  \delta^{d}(r(\bar x))+\MOPE{\GC}{\GA^{\prime\prime}}\
    (-\partial_{r}^{2})\tilde \delta^{d}(r(\bar x))+\dots\ ,\nonumber \\
\end{eqnarray}
where the OPE--coefficients are given by
\begin{eqnarray}
  \label{4.8}
  \MOPE{\GC}{\GA}&=&
  \left[C(a)C(c)-\frac{1}{4}(C(a)+C(c)-C(b))^{2} \right]^{-d/2}\nonumber \\
&=&
  \left[\frac{1}{4}(a^{\nu}{+}b^{\nu}{+}c^{\nu})(b^{\nu}{+}c^{\nu}{-}a^{\nu})(a^{\nu}{+}c^{\nu}{-}b^{\nu})(a^{\nu}{+}b^{\nu}{-}c^{\nu})\right]^{{-}d/2}
\ ,
\end{eqnarray}
which contributes to the renormalization factor at two-loop order.
The first subleading term is (denoting $\GA'':=(-\partial_r^{2})\tilde
\delta ^{d} (r (\bar x))$)
\begin{eqnarray}
  \label{4.9}
  \MOPE{\GC}{\GA^{\prime\prime}}=-\frac{1}{4}C (a)C (b)C (c) \left[\frac{1}{4}(a^{\nu}{+}b^{\nu}{+}c^{\nu})(b^{\nu}{+}c^{\nu}{-}a^{\nu})(a^{\nu}{+}c^{\nu}{-}b^{\nu})(a^{\nu}{+}b^{\nu}{-}c^{\nu})\right]^{{-}d/2{-}1}\nonumber \\
\end{eqnarray}
\subsection{Numerical calculation in $0<D<2$}
Let us now turn to the explicit calculation of the second order contribution to
the coupling constant renormalization. To two-loop order the
renormalization group $Z_{g}$-factor for the coupling
constant reads
\begin{eqnarray}
  \label{4.11}
  Z_{g} = 1+ \frac{g}{2}
  \DIAG{\GB}{\GA}{_{\varepsilon^{-1}}}&-&\frac{g^{2}}{6}\left[\DIAG{\GC}{\GA}_{\varepsilon^{-2},
  \varepsilon^{-1}}-\frac32\DIAG{\GB}{\GA}_{\varepsilon^{-1}}^{2}\right]\nonumber\\
  &+&\frac{g^{2}}{4}\DIAG{\GB}{\GA}_{\varepsilon^{-1}}^{2}+O(g^{3})
\ .
\end{eqnarray}
Note that this $Z_{g}$-factor can either be obtained from
(\ref{2.3.1}) and (\ref{3.18}) or by expanding the perturbative series and checking
that divergent contributions proportional to $\tilde \delta (r (x))$
cancel.

The formula is arranged such that in the brackets the second order pole
in $\varepsilon$ cancels. This is, because to leading order in $1/\E$,
the two-loop diagram factorizes into two one-loop diagrams. The
combinatorial factor of $3/2 $ is composed of a factor of 3 for the
number of possible subdivergences, and a factor of $1/2$ for the nested
integration: The subdivergence in the distance say $a$ only appears in
the sector 
 (i.e.\ part of the domain of integration) in which $a$ is  smallest. 


Let us define:
\begin{equation}
  \label{4.12}
  I_{2}(D,L) :=\DIAG{\GC}{\GA}_{L}{-}\frac{3}{2}\DIAG{\GB}{\GA}_{L}^{2}\ .
\end{equation}
Let us already note that we will 
later calculate in a ``massive'' scheme in fixed
dimension $d$. In order to use the results derived here, we keep $d$ arbitrary
instead of setting  $d=d_{c}$. We now want to derive an expression
which can be integrated numerically. Since the bounds on the
integrations in (\ref{4.12}) are asymmetric (remind that in
$\DIAG{\GC}{\GA}_{L}$ all 3 distances $a$, $b$, and $c$ are bounded by
$L$,
whereas $ \DIAG{\GB}{\GA}_{L}^{2}$ is an integral with two bounds only)
we treat both terms separately. We start with the first one, which can
be written as 
\begin{eqnarray}
  \label{4.16}
  I_2^{\ind{M}}(D,L)&=&\int_{a,b,c}\Theta(a\leq L)\Theta(b\leq
L)\Theta(c\leq L) \MOPE{\GC}{\GA}
\ .
\end{eqnarray}
The divergence when integrating over the global scale is eliminated 
by noting that $ L\frac{\partial}{\partial L}I_{2}(D,L)=2\varepsilon\
I_{2}(D,L)$. There remains of course the sub-divergences, which will
be subtracted later by the counter-term.
 Applying this procedure to (\ref{4.16}), we obtain
\begin{eqnarray}
  \label{4.17}
  2\E   I_2^{\ind{M}}(D,L)&=& \int_{a,b,c}\Big[\delta(a-L)\Theta(b\leq L)\Theta(c\leq
  L)+\delta(c-L)\Theta(b\leq L)\Theta(a\leq L) \nonumber\\
  && \qquad +\delta(b-L)\Theta(a\leq L)\Theta(c\leq L)\Big] \MOPE{\GC}{\GA}\ .
\end{eqnarray}
Since the integral is symmetric in $a$, $b$, and $c$, we can replace
it by 3 times the first term. Introducing proper normalizations,
an explicit representation for the measure and factoring out the
explicit $L$-dependence, we obtain
\begin{eqnarray}
  \label{4.18}
  2\E   I_2^{\ind{M}}(D,L)&=& 3 L^{2\E}\frac{S_{D-1}}{S_{D}}\int_{-
  \infty}^{\infty}\mbox{d}c_{\parallel}\int_{0}^{\infty}\mbox{d}c_{\perp}
c_{\perp}^{D-2}\Theta(b\leq 1)\Theta(c\leq
  1)\left. \MOPE{\GC}{\GA} \right|_{a=1,b,c}
\ .
\end{eqnarray}
We also have to derive an expression for the counter-term.
It can be written as
\begin{eqnarray}
  \label{4.20}
  I_{2}^{\ind{C}}(D,L)&=&-\frac{3}{2}\int_{a}\int_{c}a^{-\nu d}c^{-\nu d}\,\Theta(a\leq
  L)\Theta(c\leq L)
\ .
\end{eqnarray}
Using the same trick to derive w.r.t.\ the IR-regulator as
above, we obtain
\begin{eqnarray}
  \label{4.21}
2 \E  I_{2}^{\ind{C}}(D,L)&=&-\frac{3}{2}L^{2\E}\int_{\mathcal{A},\mathcal{B},\mathcal{C}}a^{-\nu d}c^{-\nu
  d}\,\Big[\delta(a-1)\Theta(c\leq a)+\delta(c-1)\Theta(a\leq c)\Big]\
  .
\end{eqnarray}\begin{figure}[t]
  \begin{center}
    \setlength{\unitlength}{1cm}
    \begin{picture}(7,5)
    \put(0,0){\raisebox{0mm}{\includegraphics[scale=0.5]{./eps/sectheta.eps}}} 
    \put(3.8,1.5){$\mathcal{A}$}
    \put(1.8,3.5){$\mathcal{B}$}
    \put(5.8,3.5){$\mathcal{C}$}
    \put(-0.8,3.0){$\Theta(c \leq a)$}
      \end{picture}
    \end{center} 
\caption{Area of integration (dashed) of the first integral in
    (\ref{4.21}). The intersection with $\mathcal{B}$ is to be mapped onto $\mathcal{A}$}
    \label{ftwol.0}
\end{figure}%
Using the technique of conformal mapping, which is presented in
appendix \ref{conformal},
both integrals above can be transformed into one having  support in the 
single sector $\mathcal{A}$ defined  through $b,c \leq a$, (where here $a$ is set
 to $1$). Analogously, we define sectors $\mathcal{B}$ and
$\mathcal{C}$, as the subset of domains where $b$ respectively $c$ is
the longest distance. Note that the first term in (\ref{4.21}) only has contributions
from $\mathcal{A}$ and $\mathcal{B}$,  while
the second can be restricted to  $\mathcal{A}$ and $\mathcal{C}$ after
renaming $a$ into $b$ and $c$ into $a$.\\
Performing explicitly the mapping of both $\mathcal{B}$ and $\mathcal{C}$
onto $\mathcal{A}$ one obtains 
\begin{eqnarray}
  \label{4.24}
  \int_{\mathcal{B}}(a^{-\nu d}c^{-\nu
  d}\Theta(c\leq a))\ts_{ a=1}&=&\int_{\mathcal{A}}b^{-\nu d-2\varepsilon}c^{-\nu
  d}\Theta(c\leq b)\nonumber\\
  \int_{\mathcal{C}}(a^{-\nu d}b^{-\nu
  d}\Theta(b\leq a))_{\mid a=1}&=&\int_{\mathcal{A}}c^{-\nu
  d-2\varepsilon}b^{-\nu d}\Theta(b\leq c) 
\ .
\end{eqnarray}
Combining the contributions from all sectors, we arrive at
\begin{equation}
  \label{4.25}
  2 \E I_2^{\ind{C}}(D,L)=-\frac{3}{2}L^{2\E}\int_{\mathcal{A}}\left(b^{-\nu d}{+}c^{-\nu
  d}{+}b^{-\nu d}c^{-\nu d}\mbox{max}(b,c)^{-2\varepsilon}\right)\ .
\end{equation}
Combining (\ref{4.18}) and (\ref{4.25}), we obtain an
expression for the complete diagram
\begin{eqnarray}
  \label{4.18.1}
  2\E   I_2(D,L)&=&3 L^{2\E}\frac{S_{D-1}}{S_{D}}\int_{-
  \infty}^{\infty}\mbox{d}c_{\parallel}\int_{0}^{\infty}\mbox{d}c_{\perp}
c_{\perp}^{D-2}\Theta(b\leq 1)\Theta(c\leq
  1)\nn\\
&& \times\Bigg\{
\left[\frac{1}{4}(1{+}b^{\nu}{+}c^{\nu})(b^{\nu}{+}c^{\nu}{-}1)(1{+}c^{\nu}{-}b^{\nu})(1{+}b^{\nu}{-}c^{\nu})\right]^{{-}d/2} \nn\\
&&   
\qquad -\half \left[b^{-\nu d}{+}c^{-\nu d}{+}b^{-\nu
d}c^{-\nu d}\mbox{max}(b,c)^{-2\varepsilon} \right] \Bigg\} \ .
\end{eqnarray}%
\begin{figure}[h,b]
\begin{center}
  \begin{minipage}[h]{4cm}
    \hspace{-20mm}
    {\raisebox{0mm}{\includegraphics[scale=0.4]{./eps/data_epsba.eps}}}
  \end{minipage}
\begin{minipage}[h]{4cm}
\begin{tabbing}
  \hspace{40mm}
  \begin{tabular}[hc]{|c|c|}
    \hline
    D &  \\
    \hline
    0.1 & 6.71 \\
    0.2 & 2.11\\
    0.3 & 0.847\\
    0.4 & 0.325\\
    0.5 & 0.0807 \\
    0.6 & -0.0415\\
    0.7 & -0.0949 \\
    \hline
  \end{tabular} 
 
 \begin{tabular}[hc]{|c|c|}
    \hline
    D &  \\
    \hline
    0.8 & -0.102\\
    0.9 & -0.0712 \\
    1.0 & 0\\
    1.1 & 0.123\\
    1.2 & 0.321\\
    1.3 & 0.644\\
    1.4 & 1.20\\
    \hline
  \end{tabular} 
 
 \begin{tabular}[hc]{|c|c|}
    \hline
    D &  \\
    \hline
    1.5 &  2.24\\
    1.6 & 4.45\\
    1.7 & 10.4\\
    1.8 & 36.7\\
    1.9 & 658\\
    2.0 & $\infty$ \\
        &     \\
    \hline
  \end{tabular}
\end{tabbing}
\end{minipage}
\end{center}
     \caption{Numerically obtained results for the two-loop diagram in the MS-scheme. Some values are
    tabulated.}
    \label{f4.1}
\end{figure}%%%%%
This expression will also be used later in a massive
scheme. Since here we are only interested in the $\E$-expansion, we
take the  limit of $\E\to 0$  in the integrand. We have made the
(non-trivial) check that
the integrand remains a finite integrable function, which can be 
integrated numerically:
\begin{eqnarray}
  \label{4.18.2}
   I_2(D,L)&=&3 \frac{L^{2\E}}{2\E} \frac{S_{D-1}}{S_{D}}\int_{-
  \infty}^{\infty}\mbox{d}c_{\parallel}\int_{0}^{\infty}\mbox{d}c_{\perp}
c_{\perp}^{D-2}\Theta(b\leq 1)\Theta(c\leq
  1)\nn\\
&& \times\Bigg\{
\left[\frac{1}{4}(1{+}b^{\nu}{+}c^{\nu})(b^{\nu}{+}c^{\nu}{-}1)(1{+}c^{\nu}{-}b^{\nu})(1{+}b^{\nu}{-}c^{\nu})\right]^{{-}d_{c}/2} \nn\\
&&   
\qquad -\half \left[b^{-\nu d_{c}}{+}c^{-\nu d_{c}}{+}b^{-\nu
d_{c}}c^{-\nu d_{c}} \right] \Bigg\} \nn\\
&& + O (\E^0)
\ .
\end{eqnarray}
Let us now proceed with the explicit numeric evaluation  of 
(\ref{4.18.2}). There are integrable singularities both in $c_{\parallel}$ and
$c_{\perp}$. To disentangle them \cite{WieseDavid1995,WieseDavid1997} we make a
change of variables from Cartesian coordinates $(c_{\parallel},c_{\perp})$ to 
radial and angular coordinates $(c,\alpha)$ according to 
\begin{equation}
  c_{\parallel} =c \cos(\alpha)  \ , \qquad 
  c_{\perp} = c \sin(\alpha)\ .
\end{equation}
such that
\begin{equation}
  \label{4.26}
  \int_{-\infty}^{\infty}\mbox{d}c_{\parallel}\int_{0}^{\infty}\mbox{d}c_{\perp}c_{\perp}^{D-2}\longrightarrow\
  \int_{0}^{1}\mbox{d}c\, c^{D-1}\int_{0}^{\pi/2}\mbox{d}\alpha\
 (\sin(\alpha))^{D-2}\ ,
\end{equation}
where the upper bounds of $\pi /2$ on $\A$ and $1$ on $c$ is due to the bound of $1$ on 
$b$ and $c$. We can further restrict integration to the half-sector
with $c<b$. In this sector, 
there remain singularities for small $\A$ and small $c$. 
The reader is invited to verify that they 
 are eliminated by a second change of variables
\begin{eqnarray*}
  \alpha&=&\frac{\pi}{2}\beta^{\frac{1}{D-1}}\\
  c&=&\tilde c^{\frac{1}{2-D}}\ .
\end{eqnarray*}
The above method works for $D>1$ only. For $D<1$, one has to
analytically continue the integration over $c_{\perp } $ in
(\ref{4.18}).  This can be done by partial integration, since the
integrand does not explicitly depend on $c_{\perp }$ but on
$c=\sqrt{c_{\perp }^{2}+c_{\parallel }^{2}}$. To have no boundary
terms, one should partially integrate before mapping everything onto
the sector $\cal A$.
 We have
explicitly checked consistency with the earlier used measure for $1{<}D{<}2$. Note finally,
that in the special case of $D=1$ in (\ref{4.18.2}) the integration measure
becomes a distribution reducing the integration measure to a line along
$c_{\parallel}$.\\

The diagram is shown on figure (\ref{f4.1}). It grows as $D^{-1}$ for
$D\to 0$ and  exponentially for $D\to 2$:
\begin{equation}
  \label{4.31}
  I_{2}(D)\sim \left(\frac{3}{4}\right)^{-d_{c}/2}\sim  \left(\frac{3}{4}\right)^{-D/(2{-}D)}\ .
\end{equation}
In the case of polymers $D{=}1$ it vanishes. The reason is that 
 the diagrams factorize, and thus the 1-loop result
is correct to all orders when working in an appropriate scheme. (See
e.g. \cite{WieseHabil}.)



\subsection{RG-function and extrapolation}%
\begin{figure}[t]
  \begin{center}
    \hspace{-2.2cm}
    \begin{minipage}[h]{3.2cm}
      \raisebox{0mm}{\includegraphics[scale=0.35]{./eps/o11dD.epsi}} 
    \end{minipage}
    \hspace{2cm}
    \begin{minipage}[h]{3.2cm}
      \raisebox{0mm}{\includegraphics[scale=0.35]{./eps/o11Deps.epsi}} 
    \end{minipage}
    \hspace{2cm}
\begin{minipage}[h]{3.2cm}
      \raisebox{0mm}{\includegraphics[scale=0.35]{./eps/o11DDc.epsi}} 
    \end{minipage}
  \end{center}
  \begin{center}
  \caption{Correction to scaling exponent $\omega$ for $(D{=}1,d{=}1)$
with different extrapolation parameters $(D,d),
(D,\varepsilon(D,d)),(D,D_{c}(d))$ (from left to right). The abscissa
labels the corresponding internal dimension $D_{0}$ of the departure
point on the critical line $\varepsilon(D_{0},d_{0}){=}0$.}
  \label{f4.2}
  \end{center}
\end{figure}%
\begin{figure}[htbp]
  \begin{center}
    \hspace{-2.2cm}
    \begin{minipage}[h]{3.2cm}
      \raisebox{0mm}{\includegraphics[scale=0.35]{./eps/o21DDc.epsi}} 
    \end{minipage}
    \hspace{2cm}
    \begin{minipage}[h]{3.2cm}
      \raisebox{0mm}{\includegraphics[scale=0.35]{./eps/o22DDc.epsi}} 
    \end{minipage}
    \hspace{2cm}
    \begin{minipage}[h]{3.2cm}
      \raisebox{0mm}{\includegraphics[scale=0.35]{./eps/o23DDc.epsi}} 
    \end{minipage}

    \hspace{-2.2cm}
     \begin{minipage}[h]{3.2cm}
      \raisebox{0mm}{\includegraphics[scale=0.35]{./eps/o21dD.epsi}} 
    \end{minipage}
    \hspace{2cm}
    \begin{minipage}[h]{3.2cm}
      \raisebox{0mm}{\includegraphics[scale=0.35]{./eps/o22dD.epsi}} 
    \end{minipage}
    \hspace{2cm}
    \begin{minipage}[h]{3.2cm}
      \raisebox{0mm}{\includegraphics[scale=0.35]{./eps/o23dD.epsi}} 
    \end{minipage}
    \hspace{-3cm}
    \begin{minipage}[h]{2cm}
      \begin{tabbing}
        %\hspace{40mm}
        \begin{tabular}[h]{|c|c|c|c|c|}
          \hline
          $(x,y)\ ; (D,d)$ & (1,1) & (2,1) & (2,2) & (2,3)  \\
          \hline
          $(D,Dc)\ ,\ \mbox{max}$ & 0.55 & 2.80 & 2.43 & 2.10  \\
          \hline
          $(D,d)\ ,\ \mbox{max}$ & 0.55 & 2.80 & 2.43 & 2.20  \\
          \hline
          $(D,\varepsilon)\ ,\ \mbox{max}$ & 0.55 & & &  \\
          \hline
        \end{tabular} 
      \end{tabbing}
    \end{minipage}

 \end{center}

  \begin{center}
  \caption{Correction to scaling exponent $\omega$ for different
extrapolation points $(D,d)$: Extrapolation parameters are
$(D,D_c(d))$ (top) and $(D,d)$ (bottom). The abscissa labels the
corresponding internal dimension $D_{0}$ of the departure point on the
critical line $\varepsilon(D_{0},d_{0}){=}0$. Furthermore, the
two-loop values with minimal sensitivity with respect to the departure
point are tabulated.}  \label{f4.21}
  \end{center}
\end{figure}

From (\ref{3.22}) the renormalization $\beta$-function in second order of
perturbation theory reads
\begin{equation}
  \label{4.32}
  \beta(g)=-\varepsilon g+\frac{g^{2}}{2}-\frac{g^{3}}{3}I_{2}(D)+O(g^{4})\ .
\end{equation}
The long-distance behavior of the theory is governed by the IR--stable
fixed point of the RG--flow. Fixed points are given by zeros of the
$\beta$--function:
\begin{equation}
  \label{4.33}
  \beta(g^{*})=0\qquad \Rightarrow \quad g^{*}=0\quad \vee \quad
  -\varepsilon+\frac{g^{*}}{2}-\frac{g^{*2}}{3}I_{2}(D)+O(g^{*3})=0\ .
\end{equation}
The physically interesting nontrivial fixed point is the one that is 
next to zero. In an $\varepsilon$--expansion it reads:
\begin{equation}
  \label{4.35}
  g^{*}=2\varepsilon+\frac{8}{3}I_{2}(D)\varepsilon^{2}+O(\varepsilon^{3})\ .
\end{equation}
The correction to scaling exponent $\omega$ at this fixed point is found in an
$\varepsilon$-expansion to be
\begin{equation}
  \label{4.36}
  \omega(g^{*})=\varepsilon-\frac{4 }{3}I_{2}(D)\varepsilon^{2}+O(\varepsilon^{3})\ .
\end{equation}
The question now arises of how to calculate $\omega $ for a given physical
situation. 
The general strategy that we follow in the MS-scheme is that we make
use of the freedom to choose any point on the critical line
(\ref{f2.1}) around which to start our expansion
\cite{Hwa1990,WieseDavid1995}.  To expand towards some physically
interesting point $(D,d)$ we furthermore dispose of the freedom to
follow any path starting on the curve $\varepsilon{=}0$. Since our
expansion (\ref{4.36}) is exact in $D$ but only of second order in
$\E$, we can equally well expand it to second order in $D$ around any
point on the critical curve.  Now we can change our extrapolation path
through an invertible transformation $(x,y)=(x(D,d),y(D,d))$. One
expresses $D$ and $\varepsilon$ as functions of $x$ and $y$ and
reexpands $\omega$ to second order in $x$ and $y$ around the point
$(x_{0},y_{0})=(x_{0}(D_{0},d_{c}(D_{0})),y_{0}(D_{0},d_{c}(D_{0})))$,
where we recall that $d_c (D)$ is defined such that $\E(D,d_c(D))=0$.
The aim is to find an optimal choice of variables $(x,y)$. The
guidelines for such a choice have been discussed in
\cite{WieseDavid1997}, where this procedure has been successfully
applied to extrapolate results for the anomalous dimension of
self-avoiding membranes.  As a general rule, we ``trust'' most a
result that is the least sensitive to the starting point of the
extrapolation. Therefore, we would like to find a plateau for some
suitable chosen variables $(x,y)$. This procedure works well for
polymers, or more generally for membranes with inner dimension close
to 1.  However it turns out that for membranes it works much less well
than in the self-avoiding case: we could not find a plateau, but at
best an extremum.

Some extrapolations are shown in the following figures (\ref{f4.2},
\ref{f4.21}). We start with polymers
interacting with a $\delta$-wall corresponding to the point $(D,d)=(1,1)$.
(Note that for $d<2$ the interaction becomes relevant for polymers.) This can
be seen on fig.\ \ref{f4.2}. The values obtained from the plateaus appearing
at the two-loop level $(\omega{\approx} 0.55)$ are quite close to the exact
value $\omega=1/2$, known from the fact that the result in $D=1$ at
one-loop is exact. From the latter it follows that it is best to use
$(D,\varepsilon)$ as extrapolation parameters and to stay in $D=1$.


Let us now turn to the results obtained for $\omega$ extrapolated to points
$D=2$ (fig.\ \ref{4.21}): The extrapolated value depends strongly on
the starting-point on the critical line. We can identify a least sensitive value,
which lies below $D=1$, reflecting the $D$-dependence of the two-loop
diagram. We obtain identical results for different extrapolation
paths. Note, that if we fix $d=1$ and perform an
$\varepsilon$-expansion in $D$ as is commonly done \cite{LassigLipowsky1993}, then, this
corresponds to starting from $D_{c}(d{=}1)=2/3<1$. The $d$-dependence of
the extrapolated values is recovered in the massive scheme, which we discuss
in the next section.

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\newpage \section{Calculation in fixed dimension: The massive scheme}
\label{massive}
\subsection{Phantom manifolds}
An alternative scheme to an $\varepsilon$-expansion is a calculation
in fixed dimension. This way we hope to circumvent the difficulty that
we faced when calculating the two-loop diagram in the previous section
in the limit of $D\to 2$: Since the diagram behaves like
$\left(\frac{4}{3} \right)^{d/2}$, it becomes large when the dimension
of the embedding space is large. When working in an $\E$-expansion,
$d$ and $D$ are related by $\E=D-d(2-D)/2$ and sending $\E\to0 $
before $D\to2$ imposes that $d\to \infty$. Thus one would like to work
at finite $d$, thus finite $\E$.  The diagrams are UV-convergent for
$d<d_{c}$. They are cut-off in the infrared by a finite membrane
size. Alternatively, we can use a ``soft'' IR cut-off, that is we
introduce a chemical potential $\tau$ into the Hamiltonian and sum over
all manifold sizes. For polymers $(D=1)$ this scheme corresponds to a
``massive scheme'' in an $O(N)$-symmetric, scalar $\phi^{4}$-theory as
can be seen via a de Gennes Laplace transformation on the $N=0$
component limit. (The correlator in momentum-space reads
$\frac{1}{k^{2}+t}$) In the following we keep a hard IR regulator and
calculate the two-loop diagram in fixed dimension. We use the
analytical expression given in (\ref{4.18.1}).  For the numerical
integration in $1<D<2$ we proceed as before. The result for different
embedding dimensions $d$ is shown graphically on figure \ref{f5.1},
where also some explicit values in the limit of $D=2$ are given. Note,
that the exact factorization-property in the limit of $D=1$ is only valid in
the MS-scheme or in a massive scheme with suitable soft cut-off as
mentioned above. However, even the method is not tailored to capture
that feature, the obtained value $\omega=0.6$ in the case of
$(D,d)=(1,1)$ is not too far from the exact value $1/2$. We expect
that higher order terms would again reproduce the exact value of
$\omega =1/2$. Also note that results for
membranes ($D=2$) are close to those  from the $\E$-expansion,
especially for small $d$.
\begin{figure}[hhhh]
\centerline{\hfill \parbox{0.6\textwidth}
        {\fig{0.6\textwidth}{./eps/2-loop-fixDdc.eps}}
\hfill 
\begin{tabular}[hc]{|c|c|}
          \hline
          $d$ & $I_{2|_{D{=}2}}$ \\
          \hline
          0.5 & -0.437 \\
          1 & -0.410\\
          1.5 & -0.386\\
          2 & -0.358\\
          2.5 & -0.331 \\
          3 & -0.299\\
          3.5 & -0.265 \\
          4 &   -0.229 \\
          \hline
\end{tabular} \hfill 
}
\caption{2-loop diagram in fixed dimension for embedding dimensions
       $d{=}0.5$ to $d{=}4$ (from bottom to top). The curves are only shown
       for values of $D$ such that $\E{\geq}0$.}
    \label{f5.1}
\end{figure}
\begin{figure}[htbp]
\centerline{\hfill 
\parbox{0.6\textwidth}{\fig{0.6\textwidth}{./eps/omega-fixDdc.eps}}
\hfill 
\begin{tabular}[hc]{|c|c|}
          \hline
          $d$ & $\omega_{|_{D{=}2}}$ \\
          \hline
          0.5 & 2.82 \\
          1 & 2.80\\
          1.5 & 2.77\\
          2 & 2.75\\
          2.5 & 2.72 \\
          3 & 2.69\\
          3.5 & 2.65 \\
          4 &   2.60 \\
          \hline
\end{tabular} \hfill }
\caption{Correction to scaling exponent $\omega$  for embedding
dimension $d{=}0.5$ to $d{=}4$ 
    (from top to bottom). $\E$ is restricted to $\E{\geq}0$; otherwise
one has to work with an explicit UV-cutoff.} 
  \label{f5.2}
\end{figure}


\subsection{Self-avoiding membranes}
Up to now we were considering a $D$-dimensional polymerized
``phantom'' membrane interacting with a point-like impurity in
$d$-dimensional embedding space modeled through a
$\delta$-potential. The infinite fractal dimension of the free manifold was
causing  problems, and since all operators of the type
$(-\Delta_{r})^{n}\delta(r),\ n{\geq}0$, attain the same relevance in
$D=2$, it is not clear whether the model will break down. Physically,
the problem is much better defined for self-avoiding membranes. From
field-theoretical calculations, which have recently been refined up to
 two-loop  \cite{DavidWiese1996,WieseDavid1997}, we know that two-dimensional
self-avoiding   manifolds embedded in three dimensional
space have an anomalous dimension of $\nu^{*}\approx 0.85$.  We now
try to overcome the problem of an infinite Hausdorff-dimension by a
crude simplification: We approximate the self-avoiding manifold by a
Gaussian theory with the same scaling dimension, that is a Hamiltonian
of the type \cite{DDG1}
\begin{equation}
  \label{5.2}
  \mathcal{H}[r] = \frac{1}{k{-}D}\int_{x}\
  \frac{1}{2}\,r(x)(-\Delta_{x})^{k/2}r(x)\ ,
\end{equation}
where
\begin{displaymath}
  r:\ x \in \R^{D} \longrightarrow  r( x) \in \R^{d}\ 
\end{displaymath}
as before providing the two-point correlator in our normalizations:
\begin{equation}
  \label{5.3}
  C(x,y)= |x{-}y|^{k-D}\ .
\end{equation}
The scaling dimension is
\begin{equation}
  \label{5.4}
  \nu=\frac{k{-}D}{2}
\ .
\end{equation}
We recover (\ref{3.7}) by setting $k=2$, but the model can be 
continued analytically to any real value of $k$, with $k\ge2$.
Setting
\begin{equation}
  \label{5.5}
  k^{*}=2\nu^{*}+D
\end{equation}
we get a Gaussian manifold with identical scaling dimension as obtained for
self-avoiding crumpled membranes at two-loop. The critical embedding dimension
of the interaction then reads
\begin{equation}
  \label{5.6}
  d_{c}=\frac{2D}{k^{*}{-}D}=\frac{D}{\nu^{*}}\ ,
\end{equation}
that is the interacting is relevant in $d{<}d_{c}$. For membranes
$d_{c}{=}\frac{2}{0.85}\approx 2.4$.
All operators of the type
$(-\Delta_{r}^{n}\delta^{d}(r))$ are at least naively irrelevant for $n>0$, as 
\begin{equation}
  \label{5.7}
  [(-\Delta_{r}^{n}\delta^{d}(r))]=-\nu^{*} (d+2n)\ 
\end{equation}
and the corresponding coupling has in inverse length units the dimension
\begin{equation}
  \label{5.8}
  [g^{(n)}_{0}]=D-\nu^{*} (d+2n)<0\ ,\qquad n>0\ .
\end{equation}
We may repeat the calculation in the massive scheme setting the free scaling
dimension $\nu^{*}{=}0.85$ and $(D,d){=}(2,1)$. The result is shown in table
(\ref{t5.1}).\\
Let us stress that this method can not be turned into a systematic expansion,
since perturbation theory neglects all effects of the non-Gaussian nature of
the nontrivial crumpled state. We only know that for $d\to \infty$ the
Gaussian variational approximation for the Hamiltonian, i.e.\ (\ref{5.2})
becomes exact \cite{Goulian1991,LeDoussal1992,GuitterPalmeri1992}. 
\begin{table}[htbp]
  \begin{center}
    \begin{minipage}[h]{3.2cm}
      \begin{tabbing}
        \hspace{2cm}
        \begin{tabular}[h]{|c|c|c|c|}
          \hline
                & I & $\omega(g^{*})$ & $\theta(g^{*})$   \\
          \hline
          $1$-loop & 1.000 & 1.15 & 2.71  \\
          \hline
          $2$-loop & -0.379  & 1.49 & 3.10  \\
          \hline
        \end{tabular} 
      \end{tabbing}
    \end{minipage}
    \label{t5.1}
  \end{center}
  \caption{Two-loop results for self-avoiding membranes $(D{=}2)$ approximated through a
                Gaussian theory having identical scaling dimension
                $\nu^{*}{=}0.85$. Results are shown for the corresponding
                diagram, the correction to scaling exponent $\omega$ and the
                contact exponent $\theta$.}
\end{table}





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\newpage \section{Summation of the perturbation series in the limit $D=2$}
In the last section we introduced a ``massive'' scheme allowing to
perform the renormalization procedure in fixed dimension. As can be
seen in figures \ref{f5.1} and \ref{f5.2}, the limit $D\to 2$ can be
taken in the 2-loop diagram and is smooth.  Surprisingly, this limit
can be calculated analytically: Setting
$\nu=0$ in  (\ref{4.18.1}), the integrand becomes constant, and thus
trivial to calculate. This is a
striking property. Even more, if we slightly modify the regularization
prescription, this property holds to {\em all loop orders} in
perturbation theory. Below, we will give an analytic
expression for the complete perturbation series, which allows for
analyzing the strong coupling limit. Furthermore, we show how
corrections in $(2-D)$ can be incorporated. We will see that only the
latter depend on the explicit cut-off procedure, a point which will be
further clarified below.\\
In the following-- since we perform calculations on a torus -- we slightly
modify the rescaling of the field and the bare coupling:
\begin{equation}
  \label{6.01}
  r\to r(S_{D}(2-D))^{1/2}\ ,\quad g_{0}\to g_{0}(S_{D}(2-D))^{d/2}\ .
\end{equation}
Our aim is to evaluate the perturbation series of the effective coupling,
which reads:
\begin{equation}
  \label{6.02}
  g=\frac{g_{0}L^{\varepsilon}}{\mathcal{V_{\mathcal{M}}}}\sum_{N=0}^{\infty}\frac{(-g_{0})^{N}}{(N{+}1)!}\left<\prod_{i=1}^{N{+}1}\int_{x_{i}}\delta^{d}(r(x_{i}))\right>_{0}\
  .\\
\end{equation}
The two-point function
gets a factor $(S_{D}(2-D))^{d/2}$ due to the rescaling (\ref{6.01}). All other
factors $(S_{D}(2-D))^{d/2}$  will be dropped. We discuss this at the end of this section in \ref{analyticcont}.

\subsection{$N$-loop order}
In order to calculate the $N$--loop contribution to the perturbation
theory we need 
the SDE of $N+1$ contact interactions. The normal ordered product of the
corresponding vertex operators reads
\begin{eqnarray}
  \label{6.1}
  \rme^{ik_{1}r(x_{1})}\rme^{ik_{2}r(x_{2})}\cdot \dots \cdot
  \rme^{ik_{N+1}r(x_{N+1})}\ =\ \bn
\rme^{i(\sum_{n=1}^{N+1}k_{n}r(x_{n}))}\en \ 
  \prod_{i,j=1}^{N+1}\rme^{\frac{1}{2}k_{i}k_{j}C(x_{i}{-}x_{j})}\ .
\end{eqnarray}
We choose $x_{N{+}1}$ as  root in the renormalization procedure (for the
precise definition, see below) and therefore $k_{N{+}1}$
to integrate 
over in order to obtain the $\delta $-distribution (or its
derivative), onto   which to
project. We thus shift 
\begin{displaymath}
  k_{N{+}1}\ \longrightarrow \ k_{N{+}1}-\sum_{j=1}^{N}k_{j}
\end{displaymath}
and rewrite the quadratic form in (\ref{6.1}) as
\begin{eqnarray}
  \frac{1}{2}\sum_{i,j=1}^{N+1}&&k_{i}k_{j}C(x_{i}{-}x_{j})=\sum_{j=1}^{N}k_{N{+}1}k_{j}C(x_{N{+}1}{-}x_{j})+\frac{1}{2}\sum_{i,j=1}^{N}k_{i}k_{j}C(x_{i}{-}x_{j})\nn\\
  &&\longrightarrow \
  \sum_{j=1}^{N}k_{N{+}1}k_{j}C(x_{N{+}1}{-}x_{j})-\sum_{i,j=1}^{N}k_{i}k_{j}C(x_{N{+}1}{-}x_{j})+\frac{1}{2}\sum_{i,j=1}^{N}k_{i}k_{j}C(x_{i}{-}x_{j})\nn \\
  &&=\sum_{j=1}^{N}k_{N{+}1}k_{j}C(x_{N{+}1}{-}x_{j})-\sum_{i,j=1}^{N}k_{i}k_{j}\frac{C(x_{N{+}1}{-}x_{i})+C(x_{N{+}1}{-}x_{j})-C(x_{i}{-}x_{j})}{2}\ .\nn\\
\label{6.2.a}
\end{eqnarray}
(\ref{6.1}) then becomes (up to subdominant terms\footnote{Here we
argue in terms of an OPE. Note 
that in order to make these arguments rigorous, one should ask of
whether these terms affect the effective action at {\em  constant}
background-field. This is not the case, since then $\nabla r=0$.} involving
spatial derivatives of $r$)
\begin{eqnarray}
&&\bn \exp\left[ik_{N+1}r (x_{N+1}) \right]\en \nn \\
&&\quad\times\exp\left[\sum_{j=1}^{N}k_{N{+}1}k_{j}C(x_{N{+}1}{-}x_{j})-\sum_{i,j=1}^{N}k_{i}k_{j}\frac{C(x_{N{+}1}{-}x_{i})+C(x_{N{+}1}{-}x_{j})-C(x_{i}{-}x_{j})}{2}  \right]\nn\\
\end{eqnarray}
and the integration over $k_{N+1}$ produces the $\delta$-interaction
and its derivatives; the latter  come from the first term in
(\ref{6.2.a}). They are subdominant and thus will be dropped. We abbreviate the
quadratic form in (\ref{6.2.a}) as $(D_{ij})$, where the matrix elements are
\begin{eqnarray}
  \label{6.2}
  D_{ij} &=&
  \frac{1}{2}\left[C(x_{N{+}1}{-}x_{i})+C(x_{N{+}1}{-}x_{j})-C(x_{i}{-}x_{j})\right]\ ,\quad
  i\not=j\ ,\nonumber\\
  D_{ii} &=& C(x_{N{+}1}{-}x_{i})\ .
\end{eqnarray}
Integrating out the momenta $k_{1},\dots k_{N}$ yields the
operator product expansion of $N+1$ contact interactions as
\begin{eqnarray}
  \label{6.3}
  \tilde \delta^{d}(r(x_{1}))\, \tilde \delta^{d}(r(x_{2}))\,\dots\,\tilde \delta^{d}(r(x_{N+1})) &=&
  [\det (D_{ij})]^{-d/2}\, \tilde \delta^{d}(r(x_{N{+}1}))\nonumber \\
&& +\,\textrm{subdominant operators}\ .
\end{eqnarray}

\subsection{The limit $D\to 2$ and $(2{-}D)$-expansion on the torus}
Whereas (\ref{6.3}) is still completely general, we now want to study
the limit $D\to2$ as well as perturbations above it.  We start by
remarking that the propagator $C(x)$ may be rewritten as 
\begin{equation}
  \label{6.4.0}
  C(x)=\C^{(0)}(x)+(2-D)\C(x)\ ,
\end{equation}
where $\C^{(0)}(x)$ denotes the limit $D\to 2$ of $C(x)$. Accordingly, we define
matrices $\mathfrak{D}^{(0)}$ and $\mathfrak{D}$, such that
\begin{equation}
  \label{6.4}
  D_{ij}=\mathfrak{D}_{ij}^{(0)}+(2-D)\mathfrak{D}_{ij}\ ,
\end{equation}
where
\begin{eqnarray}
  \label{6.4.1}
  \mathfrak{D}_{ij}^{(0)}&=&\frac{1}{2}\ ,\qquad i\not=j\nonumber\\
  \mathfrak{D}_{ii}^{(0)}&=&1
\end{eqnarray}
and
\begin{eqnarray}
  \label{6.5}
  \mathfrak{D}_{ij}&=&\frac{\mu^{-2\nu}}{2}(\C|(x_{N{+}1}{-}x_{i})\mu|+\C|(x_{N{+}1}{-}x_{j})\mu|-\C|(x_{i}{-}x_{j})\mu|)\
  ,\qquad i\not= j\nonumber\\
  \mathfrak{D}_{ii}&=&\mu^{-2\nu}\C|(x_{N{+}1}{-}x_{i})\mu|\ ,
\end{eqnarray}
$\mu=1/L$. Note, that we obtained (\ref{6.4.1}) from the limit $D\to 2$ in
(\ref{6.2}), and we used for $C(x)$ the infinite $D$-space propagator
$C(x)=|x|^{2{-}D}$. However, the constance in the limit $D\to 2$ for
noncoinciding points stays conserved, even if we are considering closed
manifolds. Thus, (\ref{6.4}) together with (\ref{6.4.1}) always stay valid as
well as the following considerations:\\
Due to (\ref{6.4}),
\begin{eqnarray}
  \label{6.6}
  \det (D)&&=\exp[\mbox{Tr}\ln (D_{ij})]\nonumber\\
  &&=\det(\mathfrak{D}^{(0)})\cdot\exp[\mbox{Tr}\ln[\mathbf{1}+(2-D)[\mathfrak{D}^{(0)}]^{{-1}}\tilde
  D]]\ ,
\end{eqnarray}
where $[\mathfrak{D}^{(0)}]^{{-1}}$ denotes the inverse matrix to $\mathfrak{D}^{(0)}$.
Denoting
\begin{equation}
  \label{6.62}
  M:=[\mathfrak{D}^{(0)}]^{{-1}}\mathfrak{D}\ ,
\end{equation}
let us expand the OPE-coefficient in (\ref{6.3}) up to fourth order in
$2{-}D$. Expanding first the logarithm,
\begin{displaymath}
  \ln[\mathbf{1}+(2{-}D)M]=(2{-}D)M-\frac{(2{-}D)^{2}}{2}M^{2}+\frac{(2{-}D)^{3}}{3}M^{3}-\frac{(2{-}D)^{4}}{4}M^{4}+O((2{-}D)^{5})
\end{displaymath}
and inserting into (\ref{6.6}) one arrives at
\begin{eqnarray}
        \label{6.71}
        &&[\det(D)]^{-d/2}=[\det(\mathfrak{D}^{(0)})]^{-d/2}\left[1-\frac{d}{2}\left[(2-D)\mbox{Tr}M\right.\right.\nonumber\\
        &&\left.-\frac{(2-D)^2}{2}\mbox{Tr}M^2+\frac{(2-D)^3}{2}\mbox{Tr}M^3-\frac{(2-D)^4}{4}\mbox{Tr}M^4\right]\nonumber\\
        &&+\frac{d^2}{8}\left[(2-D)^2\mbox{Tr}^2M-(2-D)^3 \mbox{Tr}M\ \mbox{Tr}M^2+(2-D)^4\left[\frac{1}{4}\mbox{Tr}^2M^2+\frac{2}{3}\mbox{Tr}M\ \mbox{Tr}M^3\right]\right]\nonumber\\
        &&\left.-\frac{d^3}{48}\left[(2-D)^3\mbox{Tr}^3M-(2-D)^4
        \frac{3}{2}\mbox{Tr}^2M\
        \mbox{Tr}M^2\right]+\frac{d^4}{384}(2-D)^4\mbox{Tr}^4M\right]+O((2-D)^5)\nonumber\\
    &&
\end{eqnarray}
To check the above result consider that
\[\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots\]
and
\[(ax+bx^2+cx^3+cx^4)^2=a^2x^2+2abx^3+(b^2+2ac)x^4+\cdots\]
as well as
\[(ax+bx^2+cx^3+cx^4)^3=a^3x^3+3a^2bx^4\ .\]
The zeroth order is $[\det(\mathfrak{D}^{(0)})]^{-d/2}$. In order to proceed we
need the latter and the inverse matrix $[\mathfrak{D}^{(0)}]^{-1}$. Let us first
calculate the determinant. From (\ref{6.4}) we have
\begin{eqnarray}
  \label{6.8}
  \mathfrak{D}^{(0)}&=&\mu^{-2\nu}\left(
    \begin{array}{cccc}
      1 & \frac{1}{2} & \ldots & \frac{1}{2} \\
      \frac{1}{2} & 1 & \ddots & \vdots \\
      \vdots & \ddots & \ddots & \frac{1}{2} \\
      \frac{1}{2} & \ldots & \frac{1}{2} & 1
    \end{array}
\right)=\frac{\mu^{-2\nu}}{2}\left(\left(
    \begin{array}{ccc}
      1 & \ldots & 0 \\
      \vdots & \ddots & \vdots \\
      0 & \vdots & 1
    \end{array}
  \right)+\left(
    \begin{array}{ccc}
      1 & \ldots & 1 \\
      \vdots & \ddots & \vdots \\
      1 & \ldots & 1
    \end{array}
\right)\right) =\frac{\mu^{-2\nu}}{2}(\mathbf{1}+N\mathbf{P})\ ,\nonumber\\
   \mathbf{P}&=&\frac{1}{N}\left(\begin{array}{ccc}
    1 & \ldots & 1 \\
    \vdots & \ddots & \vdots \\
    1 & \ldots & 1
  \end{array}\right)
\end{eqnarray}
being the projector onto $(1,1,\dots,1)$. Denoting by
    $\mbox{dim Im(\bf{P})}$ the dimension of the space onto which $\mathbf{P}$
    projects one has the general formula:
\begin{displaymath}
  \det (\mathbf{1}+a\mathbf{P})=(1+a)^{\mbox{dim Im(\bf{P})}}\ . 
\end{displaymath}
Since here $\mbox{dim Im(\bf{P})}=1$, we find
\begin{equation}
  \label{6.9}
  \det(\mathfrak{D}^{(0)})=\mu^{-2\nu N}\,\frac{1+N}{2^{N}}\ .
\end{equation}
The inverse matrix of $(\mathbf{1}+N\mathbf{P})$ can be inferred from the ansatz
\begin{displaymath}
  (\mathbf{1}+N\mathbf{P})(\mathbf{1}+b\mathbf{P})=\mathbf{1}\ .
\end{displaymath}
Since $\mathbf{P}$ is a projector $\mathbf{P}^{2}=\mathbf{P}$ and this implies
\begin{displaymath}
  (N+b+Nb)\ \mathbf{P}=\mathbf{0}\ .
\end{displaymath}
Therefore, $b=-N/(N+1)$ and
\begin{equation}
  \label{6.10}
  [\mathfrak{D}^{(0)}]^{-1}=2\left(\mathbf{1}-\frac{N}{N+1}\mathbf{P}\right)\mu^{2\nu}\ .
\end{equation}
To obtain the first order contribution in $(2{-}D)$ from eq.\ (\ref{6.6}) we only need
$ M=[\mathfrak{D}^{(0)}]^{-1} \mathfrak{D}$, which reads
\begin{eqnarray}
  \label{6.11}
 (M_{ij})=\left( [\mathfrak{D}^{(0)}]^{-1} \mathfrak{D}\right)_{ij}=\left(2\mathfrak{D}_{ij}
-\frac{2}{1+N}\sum_{k=1}^{N}\mathfrak{D}_{ik}\right)\mu^{2\nu}\ . 
\end{eqnarray}
The trace of (\ref{6.11}) can easily be performed, with the result
\begin{eqnarray}
  \label{6.12}
  \mbox{Tr} M=\left(\frac{2N}{1+N}\sum_{i=1}^{N}\mathfrak{D}_{ii}-
\frac{2}{1+N}\sum_{i=1}^{N}\sum_{k=1}^{N}(1-\delta_{ik})\mathfrak{D}_{ik}\right)
\mu^{2\nu} 
\ .
\end{eqnarray}
In  each order of perturbation theory we have to integrate the
expression  (\ref{6.3}) over internal distances. These integrals
have to be regularized in the infrared through an appropriate IR cut-off. In
the preceding sections we have cut off the integrals such that all relative
distances between internal points were smaller than  $L=1/\mu$.
Here, we are considering a finite manifold of toroidal topology. In order to
be consistent we need modify the regularization prescription (\ref{3.16}): Specializing to a squared piece of membrane all components of internal
$D$-dimensional vectors $\vec x_{i}$ are running from $0$ to $L$. Accordingly,
all components of the difference vectors $\vec x_{i}-\vec x_{N+1}$, where $\vec
x_{N+1}$ is a fixed, but arbitrarily chosen point, the root of the subtraction
prescription, are bounded by $L$. Implicitly, all other distances are
restricted through $L$ as well (see fig.\ \ref{f6.1}). Of course, on the torus the
correlator has to be modified. 
\begin{figure}[t]
\begin{center}
\raisebox{-10mm}{\includegraphics[scale=0.4]{./eps/sdeNtore2.eps}}
\end{center}
\caption{Regularization scheme for the $N$-loop diagrams on manifolds with
  toroidal topology (periodic boundary conditions). Here: $D=2$}
    \label{f6.1}
\end{figure}

To simplify the calculations, we further introduce the following notation:
\begin{equation}
  \label{6.13.1}
  \overline{f(x_{i_{1}},\dots,x_{i_{k}})} := \int_{x_{1}}\cdots\int_{x_{N}}f(x_{i_{1}},\dots,x_{i_{k}})
\end{equation}
with normalization
\begin{equation}
  \label{6.13.2}
  \int_{x}\ :=\ \int\mbox{d}^{D}x\, \prod_{j=1}^{D}\theta(x^{j}<1)\ ,
\end{equation}
which is chosen such that the overbar can be thought of as an
averaging procedure, and especially
\begin{equation}
  \label{6.13.3}
  \overline{1}=1\ 
\end{equation}
Thanks to our regularization prescription the integral of (\ref{6.12})
over internal points can be replaced by $\mu^{-ND}$ (for the
integration measure $L^{ND}$) times 
\begin{eqnarray}
  \label{6.13}
  \overline{\mbox{Tr} M}&=&\frac{2N^{2}}{1+N}\ \overline{\C(x_{N{+}1}-x_{i})}\nonumber\\
  &&-\left(\frac{2N(N-1)}{1+N}\right)\left(\overline{\C(x_{N{+}1}-x_{i})}-\frac{1}{2}\
  \overline{\C(x_{i}-x_{j})}\right)\nonumber\\
  &=&\frac{2N}{1+N}\ \overline{\C(x_{N{+}1}-x_{i})}+\frac{N(N-1)}{1+N}\ \overline{\C(x_{i}-x_{j})}\ ,
\ .
\end{eqnarray}
Due to the internal symmetry of closed manifolds the expression above can be further simplified, since
\begin{equation}
  \label{6.13.1}
  \overline{\C(x_{N{+}1}-x_{i})}=\overline{\C(x_{i}-x_{j})}\equiv \overline{\C(x)}\ .
\end{equation}
Introducing a diagrammatic notation
\begin{equation}
  \label{6.14}
  \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cij.eps}} :=
\overline{\C(x)}\ , 
\end{equation}
the $N$-loop integral reads up to first order in $2{-}D$
\begin{eqnarray}
  \label{6.16}
  \int (\det D)^{-d/2} =\mu^{-N \E}
    \left(\frac{1+N}{2^{N}}\right)^{-d/2}\!\!\!\!\!\!\times
\Bigg[1&-&\frac{d}{2}(2{-}D)\left(N\
      \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cij.eps}}\right)+ O((2{-}D)^{2}) \Bigg]\ .
\end{eqnarray}
For the further analysis we will not only need (\ref{6.13}), but also the
terms appearing to second order in $2-D$ in (\ref{6.71}). We derived
expressions like (\ref{6.13}) for $\mbox{Tr}^{2}M$ and $\mbox{Tr}M^{2}$ with
\mathematica. The \mathematica--program will be presented and discussed in the
appendix. It is based on the fact that all terms to appear in the expansion
(\ref{6.71}) are of the form $\mbox{Tr}^{n}M^{m}$ or products of the latter
and therefore can be written as $\mbox{P}(N)/(N{+}1)^{k}$, where $n,m,k\in
\mathbf{N}$ and $\mbox{P}(N)$ is some polynomial in $N$. It will turn out soon
that it is convenient to expand the polynomial $\mbox{P}(N)$ in terms of the
following base:
\begin{equation}
  \label{6.17}
  \lbrace{1,N,N(N{-}1),N(N{-}1)(N{-}2),\dots,\prod_{j=0}^{k}(N{-}j),\dots\rbrace}\ .
\end{equation}
We obtain:
\begin{eqnarray}
  \label{6.17.1}
  \overline{\mbox{Tr}M^{2}}=\frac{-2N(N{-}1)-N(N{-}1)(N{-}2)}{1{+}N}\
  \overline{\C(x)}^{2}+\frac{2N+3N(N{-}1)+N(N{-}1)(N{-}2)}{1{+}N}\ \overline{\C^{2}(x)}\ ,\nonumber\\
\end{eqnarray}
and
\begin{eqnarray}
  \label{6.17.2}
  \overline{\mbox{Tr}^{2} M}=\frac{4N(N{-}1)+N(N{-}1)(N{-}2)}{1{+}N}\
  \overline{\C(x)}^{2}+\frac{2N}{1{+}N}\ \overline{\C^{2}(x)}\ .
\end{eqnarray}
Diagrammatically, the averages can be rewritten as
\begin{equation}
  \label{6.17.3}
  \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cijk.eps}}
  :=\overline{\C(x)}^{2}\ ,
\end{equation}
and
\begin{equation}
  \label{6.17.4}
  \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cijij.eps}} :=
  \overline{\C^{2}(x)}\ .
\end{equation}
Like in the case of the  first order diagram (\ref{6.17.1}) and (\ref{6.17.2})
are highly simplified as compared to an open manifold. The averages over both
any open chain of two propagators like in (\ref{6.17.3}) and any closed loop
of two propagators like in (\ref{6.17.4}) are independent from the position of
the root $x_{N{+}1}$. Diagrammatically, this symmetry can be written as
\begin{eqnarray}
  \label{6.17.5}
  \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cijk.eps}}\equiv \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/CijkR.eps}}=\raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/CNijR.eps}}=\raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/CiNjR.eps}}=\raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cij_kNR.eps}}=\raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cij_klR.eps}}\ ,
\end{eqnarray}
and
\begin{eqnarray}
  \label{6.17.6}
  \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cijij.eps}}\equiv \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/CijijR.eps}}=\raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/CiNiNR.eps}}\ .
\end{eqnarray}
Combining (\ref{6.71}), (\ref{6.13}), (\ref{6.17.1}) and (\ref{6.17.2}) the
$N$--loop integral reads up to the second order in $2-D$:
\begin{eqnarray}
  \label{6.18}
  &&\int (\det D)^{-d/2}=\mu^{-N \E}
    \left(\frac{1+N}{2^{N}}\right)^{-d/2}\!\!\!\!\!\!\times
\Bigg[1-\frac{d}{2}(2{-}D)\left(N\
      \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cij.eps}}\right)\nonumber\\
    &+&\frac{d}{4}(2{-}D)^{2}\left(\frac{-2N(N{-}1)-N(N{-}1)(N{-}2)}{1{+}N}\
  \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cijk.eps}}+\frac{2N+3N(N{-}1)+N(N{-}1)(N{-}2)}{1{+}N}\ \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cijij.eps}}\right)\nonumber\\
    &+&\frac{d^{2}}{8}(2{-}D)^{2}\left(\frac{4N(N{-}1)+N(N{-}1)(N{-}2)}{1{+}N}\ \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cijk.eps}}
  +\frac{2N}{1{+}N}\ \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cijij.eps}}\right)\nonumber\\
    &&\qquad + O((2{-}D)^{3}) \Bigg]\ .
\end{eqnarray}
Then, we finally obtain for the
renormalized coupling $g$ up to the second order in\footnote{The
  expression in parenthesis is the inverse renormalization
$Z_{g}$-factor which relates 
  the bare coupling $g_{0}$ to the renormalized coupling $g$ according to $g Z_{g}\mu^{\varepsilon}=g_{0}$.} $2-D$:
\begin{eqnarray}
  \label{6.23}
  g&=& g_{0}\mu^{-\varepsilon}\ \left(1+\sum_{N=1}^{\infty}\frac{(-g_{0}\
  2^{d/2}\mu^{-\varepsilon})^{N}}{(N{+}1)!(1{+}N)^{d/2}}\left[1-(2{-}D)\frac{d}{2}\left(N\,\raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cij.eps}}\right)+\right.\right.\nn\\
  &+&\frac{d}{4}(2{-}D)^{2}\left(\frac{-2N(N{-}1)-N(N{-}1)(N{-}2)}{1{+}N}\
  \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cijk.eps}}+\frac{2N+3N(N{-}1)+N(N{-}1)(N{-}2)}{1{+}N}\ \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cijij.eps}}\right)\nn\\
  &+&\frac{d^{2}}{8}(2{-}D)^{2}\left(\frac{4N(N{-}1)+N(N{-}1)(N{-}2)}{1{+}N}\ \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cijk.eps}}
  +\frac{2N}{1{+}N}\ \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cijij.eps}}\right)\nn\\
  &&\qquad \qquad +O(2-D)^{3}\Bigg]\Bigg) 
\end{eqnarray}
This can also be written as
\begin{eqnarray}
g 2^{d/2} &=& - \sum_{N=0}^{\infty } \frac{( -g_{0}2^{d/2}\mu
  ^{-\E})^{N{+}1}}{N!\,(N{+}1)^{d/2{+}1}} + (2{-}D)\frac{d}{2} \sum
_{N=0}^{\infty }  \frac{( -g_{0}2^{d/2}\mu ^{-\E})^{N{+}1}
  N}{N!\,(N{+}1)^{d/2+1}}\ \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cij.eps}}\nn\\
&&  - (2{-}D)^{2}\frac{d}{4} \sum _{N=0}^{\infty }  \frac{( -g_{0}2^{d/2}\mu
  ^{-\E})^{N{+}1}) (-2N(N{-}1)-N(N{-}1)(N{-}2))}{N!\,(N{+}1)^{d/2+2}}\ \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cijk.eps}}\nn\\
&&  -(2{-}D)^{2}\frac{d}{4} \sum _{N=0}^{\infty }  \frac{( -g_{0}2^{d/2}\mu
  ^{-\E})^{N{+}1}) (2N+3N(N{-}1)+N(N{-}1)(N{-}2))}{N!\,(N{+}1)^{d/2+2}}\
  \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cijij.eps}}\nn\\
  && -(2{-}D)^{2}\frac{d^{2}}{8}\sum _{N=0}^{\infty }  \frac{( -g_{0}2^{d/2}\mu
  ^{-\E})^{N{+}1}) (4N(N{-}1)+N(N{-}1)(N{-}2))}{N!\,(N{+}1)^{d/2+2}}\
  \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cijk.eps}}\nn\\
  && -(2{-}D)^{2}\frac{d^{2}}{8}\sum _{N=0}^{\infty }  \frac{( -g_{0}2^{d/2}\mu
  ^{-\E})^{N{+}1}) (2N)}{N!\,(N{+}1)^{d/2+2}}\
  \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cijij.eps}}\qquad +O(2-D)^{3}
\end{eqnarray}
Absorbing a factor of $2^{d/2}$ both in $g$ and $g_{0}$ and defining
the dimensionless coupling $z:=g_{0}\mu ^{-\E}$, we obtain the final result
\begin{eqnarray}
\label{6.26}
g &=& z \sum_{N=0}^{\infty } \frac{( -z)^{N}}{N!\,(N{+}1)^{d/2{+}1}} +
(2{-}D)\frac{d}{2}\ z^{2}\sum _{N=0}^{\infty } \frac{( -z)^{N}
}{N!\,(N{+}2)^{d/2+1}}\
\raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cij.eps}}\nn\\
  &&- (2{-}D)^{2}\frac{d}{4}\left( 2z^{3}\sum _{N=0}^{\infty }
\frac{( -z)^{N}}{N!\,(N{+}3)^{d/2+2}}-z^{4}\sum _{N=0}^{\infty }
\frac{( -z)^{N}}{N!\,(N{+}4)^{d/2+2}}\right)\
\raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cijk.eps}}\nn\\
&&- (2{-}D)^{2}\frac{d}{4}\left(2z^{2}\sum _{N=0}^{\infty }
\frac{( -z)^{N}}{N!\,(N{+}2)^{d/2+2}}{-}3z^{3}\sum _{N=0}^{\infty }
\frac{( -z)^{N}}{N!\,(N{+}3)^{d/2+2}}{+}z^{4}\sum _{N=0}^{\infty }
\frac{( -z)^{N}}{N!\,(N{+}4)^{d/2+2}}\right)\
\raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cijij.eps}}\nn\\
&&-(2{-}D)^{2}\frac{d^{2}}{8}\left(-4z^{3}\sum _{N=0}^{\infty }
\frac{( -z)^{N}}{N!\,(N{+}3)^{d/2+2}}{+}z^{4}\sum _{N=0}^{\infty }
\frac{( -z)^{N}}{N!\,(N{+}4)^{d/2+2}}\right)\ \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cijk.eps}}\nn\\
&&-(2{-}D)^{2}\frac{d^{2}}{8}\left(2z^{2}\sum _{N=0}^{\infty }
\frac{( -z)^{N}}{N!\,(N{+}2)^{d/2+2}}\right)\ \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cijij.eps}}\qquad +O(2-D)^{3}
\ .
\end{eqnarray}
This formula is the starting point for further analysis in the
subsequent sections.

\subsection{Asymptotic behavior of the series}
In the following we    are interested in the limit of large $z$
(strong repulsion), which also is the  scaling behavior of infinitely
large membranes. We therefore need an analytical expression for sums like
(\ref{6.26}) in the limit of large $z$. An example is the leading 
order expression
\begin{equation}
  \label{6.26.1}
  g = z\sum_{N=0}^{\infty}\frac{(-z)^{N}}{N!\ (N{+}1)^{d/2+1}}\ .
\end{equation}
For the following analysis we need an integral representation of the above series.
We claim that for all $k,d>0$
\begin{equation}\label{6.28}
  \sum_{N=0}^{\infty}\frac{(-z)^{N}}{N!(k+N)^{d/2}}=\frac{1}{\Gamma(\frac{d}{2})}\int_{0}^{\infty}\mbox{d}r\
  r^{d/2-1}\exp[-z\
  \rme^{-r }-k r]\ .
\end{equation}
This can be proven as follows:
\begin{eqnarray*}
  \frac{1}{\Gamma(\frac{d}{2})}\int_{0}^{\infty}\mbox{d}r\
  r^{d/2-1}\exp[-z\,
  \rme^{-r}-k r]&=&\frac{1}{\Gamma(\frac{d}{2})}\sum_{N=0}^{\infty}\frac{(-z)^{N}}{N!}\int_{0}^{\infty}\mbox{d}r\
  r^{d/2-1}\rme^{-(N+k)r}\\
  &=&\frac{1}{\Gamma(\frac{d}{2})}\sum_{N=0}^{\infty}\frac{(-z)^{N}}{N!}\frac{\Gamma(\frac{d}{2})}{(N+k)^{d/2}}\ .
\end{eqnarray*}
This integral-representation is not the most practical for our
purpose. It is better to set $r\to s:=\rme^{-r}$ which yields 
\begin{equation}
  \sum_{N=0}^{\infty}\frac{(-z)^{N}}{N!(k+N)^{d/2}}=\frac{1}{\Gamma(\frac{d}{2})}\int_{0}^{1}\mbox{d}s\, s^{k-1} (-\ln s)^{d/2-1} \rme^{-s z}\ .
\end{equation}
This formula is already very useful for some purposes. It is still
advantageous to make a second variable-transformation $s\to y := sz$, yielding
\begin{equation}
  \sum_{N=0}^{\infty}\frac{(-z)^{N}}{N!(k+N)^{d/2}}=\frac{(\ln z)^{d/2-1}}{\Gamma(\frac{d}{2})z^{k}}\int_{0}^{z}\mbox{d}y\, y^{k-1} \left(1-\frac{\ln y}{\ln z} \right)^{d/2-1} \rme^{-y}\ .
\end{equation}
Finally we remark that we usually have the following combination
\begin{equation}
f_{k}^{d} (z):= z^{k}
\sum_{N=0}^{\infty}\frac{(-z)^{N}}{N!(k+N)^{d/2}}=\frac{(\ln
z)^{d/2-1}}{\Gamma(\frac{d}{2})}\int_{0}^{z}\rmd y\, y^{k-1} \left(1-\frac{\ln y}{\ln z} \right)^{d/2-1} \rme^{-y}\ .
\end{equation}
It satisfies the following simple recursion relation, which is
helpful to calculate the $\beta$-function:
\begin{equation}\label{recursion}
z \frac{\rmd }{\rmd z} f_{k}^{d} (z) = f_{k}^{d-2} (z) \ . 
\end{equation}
The derivative above can also be rewritten as
\begin{equation}
  \label{recursion2}
  z \frac{\rmd }{\rmd z} f_{k}^{d} (z) = kf_{k}^{d} (z)-f_{k{+}1}^{d} (z)\ ,
\end{equation}
such that one obtains a useful formula in order to isolate the dominant
behavior for large $z$:
\begin{equation}
  \label{fkdreduce}
  f_{k{+}1}^{d} (z) = kf_{k}^{d} (z)-f_{k}^{d{-}2} (z)
\end{equation}
From (\ref{6.28}) $f_k (z)>0$ for all $k,d>0$ and the behavior for large $z$ is obtained by expanding
$\left(1-\frac{\ln y}{\ln z} \right)^{d/2-1}$ for small $\frac{1}{\ln z}$
\begin{eqnarray}
  \label{fkdexpansion}
f_{k}^{d} (z) &=\ds \frac{(\ln z)^{d/2-1}}{\Gamma(\frac{d}{2})} \Bigg[&
\int_{0}^{\infty } \rmd y\, y^{k-1} \rme^{-y} \nonumber \\
&& -\frac{1}{\ln z}\left(\frac{d}{2}-1 \right)\int_{0}^{\infty }
\rmd y\, y^{k-1}\ln y\, \rme^{-y}\nonumber \\
&& +\, O  \left(\frac{1}{(\ln z)^{2}} \right)
 \Bigg]\nonumber \\
&\ds +\, O (\rme^{-z})\ \ \hfill&
\ .
\end{eqnarray}
The result is 
\begin{equation}
f_{k}^{d} (z) =\frac{(\ln z)^{d/2-1}\Gamma (k)}{\Gamma(\frac{d}{2})}
\left(1- \frac{1}{\ln z} \frac{d{-}2}{2} \frac{\Gamma' (k)}{\Gamma
(k)} + \dots  \right)
\ .
\end{equation}
For later reference, we note  a list of useful formulas ($\gamma
=0.577216\dots $ is the Euler-constant)
\begin{eqnarray}
f_{1}^{d} (z) &=& \frac{(\ln z)^{d/2-1}}{\Gamma
(\frac{d}{2})}+\frac{\gamma (\ln  z) ^{d/2-2}}{\Gamma (\frac{d-2}{2})}
+\frac{( 6 \gamma ^{2}+\pi ^{2}) (\ln  z)^{d/2-3}}{12 \Gamma (\frac{d-4}{2})}+ \dots \\
f_{2}^{d} (z) &=& \frac{(\ln z)^{d/2-1}}{\Gamma
(\frac{d}{2})}+\frac{(\gamma-1) (\ln z)^{d/2-2}}{\Gamma (\frac{d-2}{2})}
+\frac{( 6 \gamma (\gamma -2) +\pi ^{2}) (\ln z)^{d/2-3}}{12 \Gamma (\frac{d-4}{2})}+ \dots\\
f_{3}^{d} (z) &=& \frac{2 (\ln  z)^{d/2-1}}{\Gamma
(\frac{d}{2})}+\frac{(2\gamma-3) (\ln z)^{d/2-2}}{\Gamma (\frac{d-2}{2})}
+\frac{( 6 \gamma^{2} -18 \gamma +6 +\pi ^{2}) (\ln z)^{d/2-3}}{6 \Gamma (\frac{d-4}{2})}+ \dots\ .\nn\\
\end{eqnarray}
Note that with the above notations, the relation (\ref{6.26})
expressing $g$ as a function of $z$ becomes
\begin{eqnarray}\label{star}
g (z) &=& f_{1}^{{d+2}} (z) + (2{-}D)\frac{d}{2} f_{2}^{d+2} (z)\
 \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cij.eps}}\nn\\
 &&-(2{-}D)^{2}\frac{d}{4}\left[\left(2 f_{3}^{d+4}(z)-f_{4}^{d+4}(z)\right)\
 \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cijk.eps}}+\left(2
 f_{2}^{d+4}(z)-3f_{3}^{d+4}(z)\right)\
 \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cijij.eps}} \right]\nn\\
  &&-(2{-}D)\frac{d^{2}}{8}\left[\left(-4 f_{3}^{d+4}(z)+f_{4}^{d+4}(z)\right)\
 \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cijk.eps}}+\left(2
 f_{2}^{d+4}(z)\right)\
 \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cijij.eps}}\right]\quad+
 O (2-D)^{3}\ . \nn\\
\end{eqnarray}
Making several times use of the relation (\ref{fkdreduce}), which can be
rationalized with the help of an algebraic manipulation program (see
\mathematica--program in the appendix), the last equation can be cast into a
form, that makes the leading behavior for large $z$ obvious at each order of
the expansion in $2-D$. Let us return to the notation of (\ref{6.13.1}) and
let us define a "connected" correlation function $\C_{c}(x)$
as
\begin{equation}
  \label{conn1}
  \C_{c}(x):=\C(x)-\overline{\C(x)}\ .
\end{equation}
Skipping the argument of the correlation function to be averaged, the notation
simplifies to
\begin{equation}
  \label{simp}
  \overline{\C(x)}\equiv \overline{\C}\qquad \textrm{a.s.f.,}
\end{equation}
and we have
\begin{eqnarray}
  \label{conn2}
  \overline{\C_{c}^{2}}=\overline{\C^{2}-2\overline{\C}\C+\overline{\C}^{2}}=\overline{\C^{2}}-\overline{\C}^{2}\ ,
\end{eqnarray}
such that (\ref{star}) becomes
\begin{eqnarray}
  \label{star!}
  g(z)&=& f_{1}^{{d+2}} (z) + (2{-}D)\frac{d}{2}\overline{\C}\ f_{2}^{d+2} (z)\nn\\
 &&-(2{-}D)^{2}\frac{d}{4}\left[2\overline{\C_{c}^{2}}\
 f_{1}^{d+4}(z)+(\overline{\C}^{2}{-}4\overline{\C_{c}^{2}})\
 f_{1}^{d+2}(z)+(2\overline{\C_{c}^{2}}{-}\overline{\C}^{2})\
 f_{1}^{d}(z)-\overline{\C_{c}^{2}}\
 f_{1}^{d-2}(z)\right]\nn\\
 &&-(2{-}D)^{2}\frac{d^{2}}{8}\left[2\overline{\C_{c}^{2}}\
 f_{1}^{d+4}(z)-(2\overline{\C_{c}^{2}}{+}\overline{\C}^{2})\
 f_{1}^{d+2}(z)+2\overline{\C}^{2}\
 f_{1}^{d}(z)-\overline{\C}^{2}\ f_{1}^{d-2}(z)\right]\nn\\
 &&\qquad + \ O(2-D)^{3}\ .
\end{eqnarray}
Note, that the leading behavior for large values of $z$ has been absorbed
into the terms ahead, respectively. That is, at second order in $2-D$, the
leading behavior is isolated in $f_{1}^{d+4}(z)$. In (\ref{star!}) it turns
out that the diagrams to be calculated are $\overline{\C}$ and
$\overline{\C_{c}^{2}}$. On the torus this corresponds to the evaluation of
two discrete sums as will be reported in the appendix. We find:
\begin{equation}
  \label{6.29}
  \overline{\C}=S_{D}\sum_{k\not=0}\frac{1}{{\vec
  k}^{2}}-\frac{1}{2\pi(2-D)}=-1.4967(6)+S_{D}\cdot0.3422(1)(2-D)\ +\ O(2-D)^{2}
\end{equation}
and
\begin{equation}
  \label{6.30}
  \overline{\C_{c}^{2}}=S_{D}^{2}\sum_{k\not=0}\frac{1}{{\vec k}^{4}}=0.043(1)\ +\ O(2-D)\ .
\end{equation}
$\vec k$ is $D$-dimensional with components $k_{i}=2\pi/L\ n_{i}$, and the indices $n_{i}$ are running from
$-\infty$ to $\infty$, $n_{i}=0$ being excluded from the summation.\\
\\
In the next two sections we shall present two formalisms which give 
results for $\omega $ at second order. It will turn out that
the results catch qualitatively correctly the cross-over to $D=1$;
however they differ in the numerical values. This problem has already been
reported in \cite{PinnowWiese2001}, where the same procedure has been applied to open
manifolds. There, we suspected this to be a reflection of
the fact that   the  $(2{-}D)$-expansion were not
systematic due to an inconsistent regularization prescription. Now working
with closed manifolds of finite size, regularization problems do not
arise. Although the expansion on a torus considerably simplifies at second
order as compared to an open manifold ((\ref{6.17.5}), (\ref{6.17.6})),
however there remain series problems with these schemes. They can be
attributed to the fact that at second and higher order in $2-D$ the
$\beta$--function does not show up a finite limit for $z\to \infty$, as it
should at the fixed point ($\lim_{z\to \infty}\beta(z)=0$). This problem,
however, is no surprise. We expect the $\beta$-function to decay for large
values of $z$ up to some scaling function like a powerlaw:
\begin{equation}
  \label{powerl}
  \beta(z)\sim z^{-\omega/\E}\ .
\end{equation}
The dominant exponent containing the correction-to-scaling exponent $\omega$
 will have some expansion in $2-D$. Therefore, an expansion of the
 $\beta$-function in powers of $2-D$ will provide some alternating series
 involving at each order powers of logarithms of $z$.
 Since the expansion
of $\beta(z)$ is alternating in $2-D$ as we consider the leading behavior,
there arises a zero at some finite $\ln z^{*}\sim
(2-D)^{-2}$. The scheme to
be presented first is trying to circumvent the problem of a diverging (truncated)
$\beta$--function by treating the zero as a fixed point. Within the second
 scheme to be presented thereafter, we first invert the series for the
 renormalized coupling providing the bare coupling as a function of the
 latter. This also allows for expressing the $\beta$-function in terms of the
 renormalized coupling.\\
Unfortunately, it turns out that within the $2-D$--expansion both schemes do
 not provide a systematic way to gain information about the correction-to-scaling exponent $\omega$ at the fixed point.

\subsection{The RG-functions in the bare coupling}
The renormalization group $\beta$-function is defined as
\begin{equation}
\beta (g):= \mu \frac{\rmd }{\rmd \mu }\lts_0 g\ .
\end{equation}
Since it is hard to invert the function $g (z)$, it is advantageous to
study  $\beta $ as function of the bare coupling $z$
\begin{equation}
  \label{fb.7}
\beta (g)\equiv \beta (z) = -\E z \frac{\rmd }{\rmd z}g\ .
\end{equation}
Taking to each order only those terms into account, which carry the dominant
behavior for large values of $z$, up to second order in $2-D$, we find from (\ref{star}) using
(\ref{recursion}) 
\begin{equation}
  \label{fb.8}
  \beta (z) =  -\varepsilon\left
  [ f_{1}^{d} (z) + (2-D)\frac{d}{2}\ \overline{\C}\
  f_{2}^{d}(z)-(2-D)^{2}\frac{d}{2}\left(1{+}\frac{d}{2}\right)\ \overline{\C_{c}^{2}}\ f_{1}^{d+2}+\dots\right]\ .
\end{equation}
Inserting the asymptotic expansions from the last section, this 
becomes
\begin{equation}\label{fb.8.a}
  \beta (z) =  -\frac{\varepsilon}{\Gamma (\frac{d}{2})}\left[
(\ln z)^{d/2-1}\left(1+(2-D)\frac{d}{2}\ \overline{\C}\right) - (2-D)^{2}\left(1{+}\frac{d}{2}\right)\ \overline{\C_{c}^{2}}\  ( \ln z)^{d/2} +\dots 
\right]
\ ,
\end{equation}
where  omitted terms have either one more power of $(2-D)$ or
$1/\ln z$. 
As shown in section \ref{Model and physical observables}, universal
properties are given by the correction to scaling exponent
$\omega$, reading 
\begin{equation}
  \label{fb.9}
  \omega(z):=\frac{\rmd \beta (g)}{\rmd g}=\frac{\rmd \beta (z)}{\rmd z}\frac{1}{\frac{\rmd g}{\rmd z}}=-\frac{\varepsilon z}{\beta (z)}\frac{\mbox{d}\beta(z)}{\mbox{d}z}\ .
\end{equation}
Let us first study  $D=2$. $\beta (z)$ and $\omega (z)$ are plotted on
figure \ref{f6.2}. We see that for $d<2$ the $\beta$-function becomes
0 for $z\to\infty$. For $d>2$, $\beta (z)$ has no zero, but the flow
of $g$ is still to infinity. 
\begin{figure}[t]
\noindent
\epsfxsize=0.47\textwidth\epsfbox{./eps/betaN.epsi}
\hfill 
\epsfxsize=0.47\textwidth\epsfbox{./eps/omegaN.epsi}
\vspace{-1cm}
    \caption{$\beta$-function (left) and $\omega$ (right) as functions of the dimensionless
      bare coupling $z$ for different dimensions $d$: $d=0.25,\ 0.5,\
      0.75,\dots,\ 3$ (from top to bottom, respectively). Note, that $\beta$
      always has a fixed point at $z{=}0$ with
      $\omega(0){=}-\varepsilon$. Furthermore, there is a fixed point for
      $0{<}d{<}2$ at $z{=}\infty$ with $\omega(\infty){=}0$, the latter
      remaining true for $d{\geq}2$. In $d{=}2$ $\beta$ tends to $-\varepsilon$
      as $z\to \infty$. Above, $\beta$ diverges.}
    \label{f6.2}
\end{figure}%
In both cases $\omega $ is given by $\omega =\lim_{z\to\infty}\omega (z)$.
Taking only the leading term in the $(2-D)$-expansion of $\beta (z) $,
$\omega (z)$ is given by  
\begin{equation}
  \label{fb.9.0}
\omega (z) = -\varepsilon \frac{z\frac{\rmd}{\rmd z}f_{1}^{d}(z)}{f_{1}^{d}(z)}\to \frac{d-2}{2 \ln z} \to  0\ .
\end{equation}
From (\ref{fb.8.a}) it is clear, that to first order in $2-D$ we do not
obtain any correction to the above result. So, let us also  take into account the second order in $2-D$. Then the "fixed
point" (a zero) for small enough $2-D$ of (\ref{fb.8.a}) is at the finite value
\begin{equation}\label{fb.8.b}
\ln z^{*} =  \overline{\C_{c}^{2}}\ \frac{2}{2+d}\frac{1}{(2-D)^{2}}+\cdots\ . 
\end{equation}
Next, when trying to calculate $\omega (z)$, we face the following
problem: Since we truncated the series for $g$ at order two in $2-D$,
$\beta' (z)$ does not vanish at the fixed point, i.e.\ the zero of
$\beta (z)$. This might lead to the conclusion that $\omega (z^{*})$
is always $\infty $, an absurd result. However since this is a consequence
of the truncation of the series, we follow the strategy also to expand
the denominator of $\omega (z)$ in powers of $2-D$. From (\ref{fb.8})
and (\ref{fb.9}) we obtain:
\begin{eqnarray}
  \label{fb.12}
  \omega(z)=-\varepsilon\left[\frac{f_{1}^{d-2}(z)}{f_{1}^{d}(z)}\right.&-&(2-D)^{2}\frac{d}{2}\left(1{+}\frac{d}{2}\right)\ \overline{\C_{c}^{2}}\ \frac{f_{1}^{d}(z)}{f_{1}^{d}(z)}\nn\\
  &+&\left.(2-D)^{2}\frac{d}{2}\left(1+\frac{d}{2}\right)\ \overline{\C_{c}^{2}}\ \frac{f_{1}^{d+2}(z)f_{1}^{d-2}(z)}{(f_{1}^{d}(z))^{2}}+\cdots \right]
\ .
\end{eqnarray}
Inserting the asymptotic series we find
\begin{equation}
  \label{fb.9.1}
\omega (z) = -\E \left[\frac{d-2}{2}\frac{1}{\ln
    z}-(2-D)^{2}\frac{d}{2}\left(1{+}\frac{d}{2}\right)\
  \overline{\C_{c}^{2}}+(2-D)^{2}\frac{d}{2}\left(1+\frac{d}{2}\right)\
  \overline{\C_{c}^{2}}\ \frac{d-2}{d}+\cdots\right]\ .
\end{equation}
Inserting the fixed point (\ref{fb.8.b}), we arrive at
\begin{equation}
  \label{fb.13}
\omega (z^{*}) = \E \left[2+\frac{d}{2}-\frac{d^{2}}{4}\right]\
\overline{\C_{c}^{2}}\ (2-D)^{2}\ .
\end{equation}
This should be checked against the exact result in $D=1$, which 
reads
\begin{equation}
\omega (D{=}1) = \E \ .
\end{equation}
Due to the smallness of $\overline{\C_{c}^{2}}$ our resummation-procedure thus
only reproduces qualitatively the behavior for finite $2-D$.\\
Let us point out that the limit $D=2$ has been analyzed in a completely
consistent way within this scheme. Setting $D=2$ in (\ref{fb.9.1}) the
correction-to-scaling exponent is obtained from the limit $z\to \infty$ in
(\ref{fb.9.0}). Therefore, we always find $\omega=0$ (independently from
$d>0$) in $D=2$.\\
On the other hand, starting from an ansatz like (\ref{powerl}) and expanding
it in powers of $2-D$, the lowest order coefficient of $\omega/\E$ is 
obtained from the constant lowest order term in $2-D$ in the corresponding equation to
(\ref{fb.9.1}). However, the ansatz (\ref{powerl}) is oversimplified as will
be discussed in section \ref{2mDtorus}. In particular it does not cover the limit $D=2$.
Before further discussing the  validity of 
the procedure, let us turn to the second scheme, namely the
calculation in the renormalized coupling.
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\subsection{Calculation in the renormalized coupling}
We start from $g (z)$ given in (\ref{star!}). Taking to each order in $2-D$
only the dominant term for large $z$ into account this can be
approximated by
\begin{equation}
  \label{fr.2}
  g(z)=\frac{(\ln
  z)^{d/2}}{\Gamma(\frac{d}{2}{+}1)}\left[1+(2-D)\frac{d}{2}\
  \overline{\C}-(2-D)^{2}\frac{d}{2}\ \overline{\C_{c}^{2}}\ \ln z\right]\ .
\end{equation}
Using (\ref{fr.2}), we can 
write $\ln z$ as a function of $g$. To second order in $2-D$, this reads
\begin{eqnarray}
\ln z  &=& \tilde
g^{2/d}\left(1-(2{-}D)\overline{\C}+(2{-}D)^{2}\frac{d}{2}\overline{\C}^{2}\right)
+ \tilde g^{4/d}(2{-}D)^{2}\ \overline{\C_{c}^{2}}\ ,\nn\\
   &&\qquad \qquad  \qquad \tx \tilde g:=g\,\Gamma(\frac{d}{2}+1)
\ .
\end{eqnarray} 
\begin{figure}[t]
  \begin{center}
    \vspace{1cm}
    \setlength{\unitlength}{1cm}
    \begin{picture}(7,5)
    \put(0,0){\raisebox{0mm}{\includegraphics[scale=0.8]{./eps/betapa.eps}}}
    \thicklines
    \put(0,4.65){\vector(1,0){9.0}}
    \put(0.88,-0.2){\vector(0,1){5.5}}
    \put(9.0,4.4){$g$}
    \put(-0.0,5.2){$\beta(g)$}
      \end{picture}
    \end{center} 
\caption{$\beta-$function in terms of the
    renormalized coupling $g$  truncated at order 160, Pade-resummed,
and plotting 
only that part for which the truncated series converges. (This can
e.g.\ be tested by taking away the last few terms of the series.) This
is  compared to the  asymptotic behavior (\ref{fr.10})
    (proportional to $1/ g$ for large $g$).  $d$ is set to $1$,
and we used the diagonal (80,80)-Pade approximant, which was find to
converge best. (The non-resummed expression starts to diverge already at
$g\approx 1.8$ at this order.)} 
    \label{betapa}
\end{figure}
We now write the $\beta$-function in terms of $\tilde g$. Starting from
(\ref{fb.8.a})
we obtain
\begin{equation}
  \label{fr.10}
  \beta(\tilde g)=-\frac{\varepsilon}{\Gamma(\frac{d}{2})}\left[{\tilde
  g^{1-2/d}}\left(1+(2{-}D)\ \overline{\C}+(2{-}D)^{2}\ \overline{\C}^{2}\
  \frac{(d{-}4)(d{-}2)}{8}\right)-(2{-}D)^{2}\ 2\overline{\C_{c}^{2}}\ \tilde
  g\right]\ .
\end{equation}
Fixed points are at the zeros of the $\beta$-function. The non-trivial
one is at
\begin{eqnarray}
  \label{fr.12}
  \tilde g^{*-2/d}=(2-D)^{2}\ 2\overline{\C_{c}^{2}}+\cdots\ .
\end{eqnarray}
The correction to scaling exponent $\omega$ is simply obtained by evaluating
the derivative of $\beta$ with respect to $g$ at the fixed point. In terms
of $\tilde g$  this is
\begin{eqnarray}
  \label{fr.13}
  \omega(\tilde g)&=&\frac{\rmd \beta (g)}{\rmd g}=\frac{\rmd \beta(\tilde
  g)}{\mbox{d}\tilde g}\frac{\rmd \tilde g}{\rmd g}=\Gamma({\tx\frac{d}{2}+1})\, \frac{\mbox{d}\beta(\tilde
  g)}{\mbox{d}\tilde g}\nonumber\\
  &=&-\varepsilon\frac{\Gamma(\frac{d}{2}{+}1)}{\Gamma(\frac{d}{2})}\left[\left(1{-}\frac{2}{d}\right)\tilde g^{-2/d}\left(1+(2{-}D)\ \overline{\C}+(2{-}D)^{2}\ \overline{\C}^{2}\
  \frac{(d{-}4)(d{-}2)}{8}\right)-(2{-}D)^{2}\ 2\overline{\C_{c}^{2}}\right]\
  .\nn\\
  &&
\end{eqnarray}
 Inserting the fixed point
$\tilde g^{*}$ from (\ref{fr.12}) we find
\begin{equation}
  \label{fr.14}
  \omega(\tilde
  g^{*})=\varepsilon\ 2\overline{\C_{c}^{2}}\ (2-D)^{2}\ .
\end{equation}
Again, this is only qualitatively correct as compared to the exact result 
 $\omega (D{=}1) =\E$ in $D=1$, and coincides with the result obtained from
 the previous scheme only in $d=0$ and $d=2$.\\
There is a series problem that has to be mentioned and that applies to
the "bare scheme" from the previous subsection as well as to the calculation
in renormalized coupling: In the "bare scheme" $\omega(z)$ is evaluated at the
fixed point $\ln z^{*}\sim (2-D)^{-2}$. However, each higher order power in
$\ln z$ in (\ref{fb.9.1}) is coming up only with an extra power in $2-D$,
therefore providing arbitrarily diverging contributions in $2-D$. The same
applies to the second scheme, where $\omega(g)$ has an expansion in powers of
$(2{-}D)g^{2/d}$ and the fixed point lies at $g^{*-2/d}\sim (2-D)^{2}$.\\ 


Let us finally point out that in the limit $D=2$ the true asymptotic
behavior of the $\beta-$function in terms of the renormalized coupling
$g$ is obtained from the completely summed series (\ref{6.26.1})
leading to (\ref{fr.10}) for large $g$. Conversely, if one tries to
invert (\ref{6.26.1}) and truncates it taking only a finite number of
orders into account, it is at least possible to reach the asymptotic
regime -- however, for large enough $g$ the truncated $\beta-$function
wildly oscillates and thus strongly deviates from the true
behavior. In figure (\ref{betapa}) the Pade-resummed truncated
$\beta-$function up to order $g^{160}$ in $d=1$ is compared with the exact,
asymptotic flow-function. One notices that the truncated
$\beta$-function even though improved through a Pade-Resummation hardly
gets into touch with the asymptotic regime. The same applies to the
slope-function $\omega(g)$, which is not shown in fig. (\ref{betapa}).

Note that the above arguments suggest quite intuitively the behavior
of the exact $\beta $-function in $1<D<2$. Whereas for $D=1$, the 
$\beta $-function is a parabola, and for $D=2$ it decays like a
power-law for large $g$ at least as long as $d<2$, the $\beta$-function
for values of $D$ between these two extremes should cut the axes
$\beta (g)=0$ at a finite value of $g$, which for $D\to 2$ wanders off 
to infinity, thus by continuity forcing the exponent $\omega $ to go 
to 0 for $D\to 2$. 
\begin{figure}[t]
\centerline{\fig{0.5\textwidth}{./eps/guessbeta.eps}}
\caption{Qualitative behavior for the $\beta $-function in $D=1$, $D=2$
and result anticipated for $D\approx 1.5$.}
\label{guessbeta}
\end{figure}

\subsection{The limit $d=0$}
Let us shortly consider the limit of vanishing bulk space dimension $d$, which
is completely solvable for arbitrary internal dimensions $D$. However, the
strong coupling limit of $g(z)$ is not analytic around $d=0$. Starting from
(\ref{6.26.1}) and setting $d=0$ one almost immediately obtains:
\begin{equation}
  \label{d0.0}
  g(z)=1-\mbox{e}^{-z}\ ,
\end{equation}
which can be inverted providing the bare coupling in terms of the renormalized
coupling as
\begin{equation}
  \label{d0.1}
  z=\ln (1-g)^{-1}\ .
\end{equation}
Recalling the definition of the $\beta-$function (\ref{2.4}) we find:
\begin{equation}
  \label{d0.2}
  \beta(g)=\varepsilon (1-g)\ln (1-g)
\end{equation}
with fixed points at 
\begin{equation}
  \label{d0.3}
  g^{*}=0 \qquad \vee \qquad g^{*}=1\ .
\end{equation}
Then, the correction-to-scaling exponent $\omega$ as well can be evaluated in
terms of the renormalized coupling as
\begin{equation}
  \label{d0.4}
  \omega(g)=\left.\frac{\mbox{d}\beta(g)}{\mbox{d}g}\right|_{g^{*}}=-\varepsilon\left.(1+\ln (1-g))\right|_{g^{*}}\ ,
\end{equation}
such that one finds
\begin{equation}
  \label{d0.5}
  \omega(g^{*}{=}0)=-\varepsilon \qquad \textrm{and}\qquad
  \omega(g^{*}{=}1)=+\infty\ .
\end{equation}
Note that (\ref{d0.0}) is obtained from the leading order contribution in
(\ref{star!}). All other terms do not contribute. The same result is found
from the exact solution of the limit $D=1$ setting $d=0$ as is shown in the
appendix (\ref{a.27.2}).

 

\subsection{Laplace-transformed picture}
Let us finally point out that the correction to scaling exponent $\omega$
found at 
leading order remains  unchanged in the Laplace-transformed picture
introduced in section \ref{massive}, that is we sum over all
membrane-sizes weighted by a  chemical potential $\tau$. The sum can
easily be done, leading to an additional factor of 
$\Gamma(1{+}\frac{N\varepsilon}{D})$, which is
$N!$ in the limit of $D\to 2$. (\ref{6.26}) then has to be replaced by
\begin{equation}
  \label{6.52}
  g\ =\ -\sum_{N=1}^{\infty}\frac{(-z)^{N}}{N^{d/2}}\ ,
\end{equation}
which is a special function called polylogarithmic function
\begin{equation}
  \label{6.53}
  g\ =\ -\mbox{Li}_{\frac{d}{2}}(-z)\ ,
\end{equation}
and accordingly from (\ref{fb.7}) the RG $\beta$-function reads
\begin{equation}
  \label{6.54}
  \beta(z)\ =\ \varepsilon\cdot \mbox{Li}_{\frac{d}{2}{-}1}(-z)
\end{equation}
while the correction to scaling exponent becomes (\ref{fb.9})
\begin{equation}
  \label{6.55}
  \omega(z)\ =\ -\varepsilon
  \frac{\mbox{Li}_{\frac{d{-}4}{2}}(-z)}{\mbox{Li}_{\frac{d{-}2}{2}}(-z)}\ .
\end{equation}
Indeed, we find the same fixed point behavior as before and
$\lim_{z\rightarrow \infty}\omega(z)=0$ for arbitrary $d{>}0$. This follows
from the logarithmic divergence of $g(z)$ in the limit of strong
coupling.
\begin{comment} 
Instead of $f_{k}^{d\prime}s$ we will find in the expansion in
powers of $2-D$ the following functions, which we denote by $\tilde
f_{k}^{d\prime}s$:
\begin{eqnarray}
  \label{fr.15}
  \tilde
  f_{k}^{d}(z)&:=&z^{k}\sum_{N=0}^{\infty}\frac{(-z)^{N}}{(k+N)^{d/2}}=\frac{z^{k}}{\Gamma(\frac{d}{2})}\int\limits_{0}^{\infty}\mbox{d}r\ r^{d/2-1}\frac{\mbox{e}^{-kr}}{1+z\mbox{e}^{-r}}\nn\\
  &=&\frac{(\ln z)^{d/2-1}}{\Gamma(\frac{d}{2})}\int\limits_{0}^{z}\mbox{d}y\
  y^{k-1}\left(1-\frac{\ln y}{\ln z}\right)^{d/2-1}\frac{1}{1+y}\ .
\end{eqnarray}
In contrast to (\ref{fkdexpansion}) the upper bound in the integral above
can not be extended to infinity for arbitrary values of $k$. Nevertheless, in the
limit $D=2$ we obtain:
\begin{equation}
  \label{fr.16}
  g(z)=\tilde f_{1}^{d}(z),
\end{equation}
and setting $k=1$ in (\ref{fr.15}), $g(z)$ can be easily estimated to be
logarithmically diverging in the strong coupling limit for $d>0$, from what
one may conclude that then $\omega=0$ (\ref{6.55}).
\end{comment}

\subsection{Discussion of the analytical continuation to
  $D=2$.}\label{analyticcont}
Before we go on with the analysis of the expansion in powers of
$2-D$ let us discuss the analytical continuation to $D=2$ in more detail. In
particular we would like to discuss the proper treatment of the factor
$(2-D)^{d/2}$, which becomes singular in $D=2$ and which appears to the $N$th
power in the $N$th loop integral in the perturbation expansion of the
effective coupling $g(z)$. It is due to the form of the two-point difference
correlator, which reads in dimensions $0<D<2$:
\begin{equation}
  \label{AC1}
  C(x-y):=\frac{1}{2d}\left<|\vec r(\vec x)-\vec r(\vec
  y)|^{2}\right>_{\mathcal{H}_{0}}=\frac{|x-y|^{2-D}}{(2-D)S_{D}}\ ,
\end{equation}
where $S_{D}$ denotes the $D$-dimensional volume of the unitsphere.\\
Let us recall the expression for the perturbation series for the effective
coupling in powers of the scaling variable $z:=g_{0}L^{\varepsilon}$:
\begin{equation}
  \label{AC2}
  g(z)=z\sum_{N=0}^{\infty}\frac{(-z)^{N}}{(N{+}1)!}\int\limits_{x_{1}}\cdots\int\limits_{x_{N}}\left(\det
  D_{ij}\right)^{-d/2}\ , 
\end{equation}
where the internal linear size of the manifold has been scaled out of the
integrations, which are left with the measure (\ref{6.13.2}) on the torus. The
quadratic form $D_{ij}$ is 
\begin{equation}
  \label{AC3}
  D_{ij} =
  \frac{1}{2}\left[C(x_{N{+}1}{-}x_{i})+C(x_{N{+}1}{-}x_{j})-C(x_{i}{-}x_{j})\right]\ .
\end{equation}
The arising of $N$ powers of $(2-D)^{d/2}$ in the $N$th loop order is then
clear.\\
Let us show how this factor can be absorbed into a rescaling of the bare
coupling $z$. Substituting $r^{\prime}=r(S_{D}(2-D))^{1/2}$ in the functional
integration, the free part of the Hamiltonian (\ref{2.1}) gets a prefactor
$(S_{D}(2-D))^{-1}$ and, accordingly, the bare coupling is  replaced with
\begin{equation}
  \label{AC4}
  z\to z(S_{D}(2-D))^{d/2}
\end{equation}
in order to leave the physics invariant. 
Then, the two-point function in the
rescaled variables reads
\begin{equation}
  \label{AC5}
  C(x-y)=|x-y|^{2-D}\ .
\end{equation}
Expressing the perturbation series (\ref{AC2}) still in powers of the old
coupling one has:
\begin{eqnarray}
  \label{AC6}
  g(z)=(S_{D}(2{-}D))^{-d/2}\left(z(S_{D}(2{-}D))^{d/2}\sum_{N=0}^{\infty}\frac{(-z(S_{D}(2{-}D))^{d/2})^{N}}{(N{+}1)!}\int\limits_{x_{1}}\cdots\int\limits_{x_{N}}\left(\det
  D_{ij}\right)^{-d/2}\right)\nn\\
\end{eqnarray}
where, thanks to (\ref{AC5}) the remaining integrand, $(\det D_{ij})^{-d/2}$, 
now has an expansion in powers of $2-D$ as we discussed in the preceding
subsections. The overall factor $(S_{D}(2{-}D))^{-d/2}$ is contributed from
the Jacobian in the integration over the position of the center of
mass. Actually, as compared to (\ref{AC2}) in (\ref{AC6}) nothing has changed. Summing up all loops at each order in $2-D$ of the
integrand, we arrive at (\ref{star!}):
\begin{equation}
  \label{AC7}
  (S_{D}(2{-}D))^{d/2}g(z)= f_{1}^{{d+2}} (z(S_{D}(2{-}D))^{d/2}) +
  (2{-}D)\frac{d}{2}\overline{\C}\ f_{2}^{d+2} (z(S_{D}(2{-}D))^{d/2})+\dots\ .
\end{equation}
Let us  consider a reference membrane of linear internal size
$L$ in $D=2$: The expansion of the (dimensionfull) effective coupling of a manifold of linear internal
size $L/(S_{D}(2{-}D))^{d/(2\varepsilon)}$ in $D<2$ then reads:
\begin{equation}
  \label{AC8}
  L^{\varepsilon}g(z)= L^{\varepsilon}(f_{1}^{{d+2}} (z) +
  (2{-}D)\frac{d}{2}\overline{\C}\ f_{2}^{d+2} (z)+O(2-D)^{2})\ ,
\end{equation}
where the singular scaling factor $(S_{D}(2{-}D))^{d/(2\varepsilon)}$ has canceled out.\\
In the limit $D=2$ we are left with
\begin{equation}
  \label{AC9}
  g(z)= f_{1}^{{d+2}} (z)\ .
\end{equation}
It is important to point out that performing the limit $D\to 2$ in this way corresponds to studying the scaling limit $a/L\ll 1$, where $a$ denotes a microscopic
cutoff. Therefore, in addition to (\ref{AC9}) we expect corrections to
scaling for small membrane sizes. Equivalently, our statement is that after summing
up the complete perturbation series in the limit $D=2$ calculating each
diagram with an explicit ultraviolet cutoff $a$ and the exact
two point function, then, the long-distance behavior is expected to be
independent from the cutoff $a$ in the scaling limit $a/L\ll 1$. It will coincide
with the behavior that we obtained from the resummed, analytically continued diagrams after
having sent $a$ to zero.\\
Let us finally emphasize that there is no alternative to forcing the membrane
size to $\infty$ when approaching the limit $D=2$. This is in contrast to what
is done in the rescaling procedures in (\ref{3.8}) and (\ref{6.01}), which
imply an infinite bare coupling $g_{0}$ in the limit $D=2$. As discussed above
the rescaling of
the field contributes a Jacobian in the perturbation series of $g(z)$ from the integration over the position of the
center of mass and can be only eliminated by forcing the size to $\infty$. The
Jacobian has been dropped in the preceding sections, which is completely safe
as long as one is only interested in the exponent $\omega$.\\ 
There is a simple way to anticipate the divergence of the fixed point in the
limit $D\to 2$ and to justify that the analytical continuation of the exponent
$\omega$ to $D=2$ must become zero as we found it: The $\beta$-function at two
loops reads (\ref{4.32})
\begin{equation}
  \label{AC10}
  \beta(g)=-\varepsilon g+\frac{g^{2}}{2}I_{1}(D)-\frac{g^{3}}{3}I_{2}(D)+O(g^{4})\ ,
\end{equation}
where in contrast to (\ref{4.32}) we have not made the value of the one loop
integral, $I_{1}(D)$, explicit. Within the $\varepsilon$-expansion one finds a
nontrivial fixed point at $g^{*}$ (\ref{4.35}):
\begin{equation}
  \label{AC11}
  g^{*}=\frac{2\varepsilon}{I_{1}(D)}+\frac{8}{3}\frac{I_{2}(D)}{I^{3}_{1}(D)}\varepsilon^{2}+O(\varepsilon^{3})\ .
\end{equation}
As we already discussed above each loop integral $I_{N}$ is coming up with $N$
powers of $(2-D)^{d/2}$, as long as we do not perform the rescaling procedure,
such that the shift to infinity in the limit $D=2$ of the fixed point $g^{*}$
can already be anticipated from (\ref{AC11}). However calculating the
correction -to-scaling exponent $\omega$, the $\varepsilon$-expansion reads
(\ref{4.36}):
\begin{equation}
  \label{AC12}
  \omega(g^{*})=\varepsilon-\frac{4
  }{3}\frac{I_{2}(D)}{I^{2}_{1}(D)}\varepsilon^{2}+O(\varepsilon^{3})\ .
\end{equation}
It turns out that all loop integrals appear only in a form, where they are
normed with respect to the one loop integral. This way any singular factor from
the one loop integral drops out. The reasoning above is not restricted to the
$\varepsilon$-expansion. Even more, it has turned out that only within the
calculations in fixed dimensions $(D,d)$ the remaining combinations
$I_{2}(D)/I_{1}^{2}(D)$ have smooth expansions in $2-D$, from what we may
conclude that as long as one is interested in the analytical continuation of
the exponent $\omega$ to $D=2$ it is completely justified to norm all loop integrals with
respect to the one loop integral. This is what the rescaling procedure does. 




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\newpage \section{Higher order terms in the $(2-D)$- expansion on the torus
  }\label{2mDtorus}
In this section we will render the analysis of the strong coupling behavior of
the theory in $D$ slightly below $2$ more systematic. For this purpose we will
  carry out the expansion in
powers of $2-D$ up to the fourth order. This will allow for an insight into
the general structure of the $2-D$-expansion. Next, we will give two different
ans\"atze for the exact  renormalized coupling $g(D,z)$ as a function of the
internal dimension $D$ and show, how their free parameters get matched with
the obtained expansion in $2-D$.\\
From the second order on, it is quite a tedious work to collect all terms to appear
in the expansion. We accomplished this with the help of a
\mathematica-program, which we discuss in the appendix. However, we have only
calculated the diagrams to appear up to the second order. To get an idea of
the general form of the expansion we also needed the resummed terms to the
third and fourth order, even if we do not have the explicit values of the
corresponding diagrams available. Of course, our main purpose is to get
information about the strong coupling behavior for finite $2-D$ close to
zero.\\
First, let us list the diagrams to appear up to the fourth order:
\begin{eqnarray}
  \label{D1}
  \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cij.eps}} \ =\ 
\overline{\C(x_{i}{-}x_{j})}\ ,
\end{eqnarray}
which contributes to first order in $2-D$. To second order one needs in
addition
\begin{eqnarray}
  \label{D2}
  \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cijij.eps}} \ =\
  \overline{\C^{2}(x_{i}{-}x_{j})}\ .
\end{eqnarray}
To third order diagrams with new topology are
\begin{eqnarray}
  \label{D3}
  \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/C3.eps}} \ =\
  \overline{\C^{3}(x_{i}{-}x_{j})}\ ,\nn\\
  \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Ctri.eps}} \ =\
  \overline{\C(x_{i}{-}x_{j})\C(x_{j}{-}x_{k})\C(x_{i}{-}x_{k})}\ .
\end{eqnarray}
Finally, to fourth order we need:
\begin{eqnarray}
  \label{D4}
  \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/C4.eps}} \ =\
  \overline{\C^{4}(x_{i}{-}x_{j})}\ ,\nn\\
  \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cblobtr.eps}} \ =\
  \overline{\C^{2}(x_{i}{-}x_{j})\C(x_{k}{-}x_{j})\C(x_{k}{-}x_{i})}\ ,\nn\\
  \raisebox{-4.5mm}{\includegraphics[scale=0.09]{./eps/Cdiamond.eps}} \ =\
  \overline{\C(x_{i}{-}x_{j})\C(x_{k}{-}x_{j})\C(x_{l}{-}x_{k})\C(x_{i}{-}x_{l})}\ .
\end{eqnarray}
We will express the resummed series in terms of averages over both the
correlator $\C(x)$ (\ref{6.4.0}) and the connected correlator $\C_{c}(x)$
(\ref{conn1}). One can switch to one or the other representation depending on
which might be the more convenient. Before stating the results let us remind
the definition of the correlator $\C(x)$, which was
\begin{eqnarray}
  \label{Dij}
C (x) = \C^{(0)}(x)+(2-D)\C(x)\ ,
\end{eqnarray}
where $\C^{(0)}(x)$ is the limit $D\to 2$ of the two-point correlator $C(x)$. In (\ref{conn1}) we have already introduced a connected correlation function
\begin{equation}
  \label{conn3}
  \C_{c}(x)=\C(x)-\overline{\C}\ ,
\end{equation}
such that
\begin{equation}
  \label{conn4}
  \overline{\C_{c}^{2}}=\overline{\C^{2}}-\overline{\C}^{2}\ ,
\end{equation}
and the notation for the leading behavior of the renormalized coupling $g$
could be shortened (\ref{star!}). From the definition (\ref{conn3}), we obtain
other useful averages, which we will need at higher orders in $2-D$. To third
order we will need:
\begin{eqnarray}
  \label{conn5}
  \overline{\C_{c}^{3}}=\overline{\C^{3}}-3\overline{\C}\overline{\C^{2}}+2\overline{\C}^{3} \ 
\end{eqnarray}
and
\begin{eqnarray}
  \label{conn6}
  \overline{\C_{c}^{\triangle}}&=&\overline{\C_{c}(x_{i}{-}x_{j})\C_{c}(x_{j}{-}x_{k})\C_{c}(x_{k}{-}x_{i})}\nn\\
  &=&\overline{\C(x_{i}{-}x_{j})\C(x_{j}{-}x_{k})\C(x_{k}{-}x_{i})}+3\overline{\C}^{2}\overline{\C(x_{i}{-}x_{j})}-3\overline{\C}\overline{\C(x_{i}{-}x_{j})\C(x_{j}{-}x_{k})}-\overline{\C}^{3}\nn\\
  &=&\overline{\C^{\triangle}}-\overline{\C}^{3}\ ,
\end{eqnarray}
where $x_{i},x_{j},x_{k}$ are distinct points, and the average is over their
positions. In (\ref{conn6}) we exploited the symmetry of the closed manifold
as it is diagrammatically expressed in (\ref{6.17.5}), and the definition of
$\overline{\C^{\triangle}}$ is self-evident. Furthermore, we will need to fourth
order in $2-D$:
\begin{eqnarray}
  \label{conn7}
  \Ccbisq=\overline{\C^{4}}+12\overline{\C^{2}}\overline{\C}^{2}-4\overline{\C^{3}}\overline{\C}-3\overline{\C}^{4}\ ,
\end{eqnarray}
\begin{eqnarray}
  \label{conn8}
  \Ccdia&=&\overline{\C_{c}(x_{i}{-}x_{j})\C_{c}(x_{j}{-}x_{k})\C_{c}(x_{k}{-}x_{l})\C_{c}(x_{l}{-}x_{i})}\nn\\
  &=&\overline{\C^{\diamond}}+5\overline{\C}^{4}
\end{eqnarray}
and
\begin{eqnarray}
  \label{conn9}
  \Ccblobt&=&\overline{\C^{2}_{c}(x_{i}{-}x_{j})\C_{c}(x_{i}{-}x_{k})\C_{c}(x_{k}{-}x_{j})}\nn\\
  &=&\overline{\C^{\BT}}-2\overline{\C^{\triangle}}\overline{\C}-\overline{\C^{2}}\overline{\C}^{2}+2\overline{\C}^{4}\ .
\end{eqnarray}

\newpage
\subsection{Resummed contributions to the expansion in $2-D$ up to the fourth
  order}\label{2mDresults}
In the following we will state all terms, which appear in
the expansion of the renormalized coupling $g(z)$ up to the fourth order in
$2-D$ according to (\ref{6.71}). 
We have to calculate at order $N$ of perturbation theory:
\begin{eqnarray}\label{2mD3}
\overline{\Tr M} &=& N\, \overline {{\C
}}
\end{eqnarray}
Inserting this into the perturbation series and summing up the resulting terms
to all orders in $N$ generates the following contributions in the
$2-D$-expansion of the renormalized coupling:
\begin{eqnarray}
  \label{2mD3.01}
  \sum_{\mbox{\tiny{loops}}}(\det\mathfrak{D}^{(0)})^{-d/2}\frac{\overline{\Tr M}(-z)^{N{+}1}}{N!}=\overline {{\C}} f_1^{d+2}(z)-\overline {{\C}} f_1^{d}(z)\ ,
\end{eqnarray}
which contributes to first order in $2-D$. To second
order in $2-D$, we have:
\begin{eqnarray}
  \label{2mD3.0} 
 \overline{\Tr M^{2}} &=&\frac{-2N(N{-}1)-N(N{-}1)(N{-}2)}{1{+}N}\
  \overline{\C}^{2}+\frac{2N+3N(N{-}1)+N(N{-}1)(N{-}2)}{1{+}N}\
  \overline{\C^{2}}\ ,\nn\\
\end{eqnarray}
providing
\begin{eqnarray}
  \label{2mD3.00}
  \sum_{\mbox{\tiny{loops}}}(\det\mathfrak{D}^{(0)})^{-d/2}\frac{\overline{\Tr
   M^{2}}(-z)^{N{+}1}}{N!}&=&(-2\overline{\C}^2+2\overline{\C^2})f_1^{d+4}(z)+(5\overline{\C}^2-4\overline{\C^2})f_1^{d+2}+(-4\overline{\C}^2+3\overline{\C^2})f_1^d(z)\nonumber\\
   &&-(-\overline{\C}^2+\overline{\C^2})f_1^{d-2}(z)\ ,
\end{eqnarray}
and
\begin{eqnarray}
  \label{2mD3.01}
  \overline{\Tr^{2}M} &=& \frac{4N(N{-}1)+N(N{-}1)(N{-}2)}{1{+}N}\
  \overline{\C}^{2}+\frac{2N}{1{+}N}\ \overline{\C^{2}}\ ,\nn\\
\end{eqnarray}
providing
\begin{eqnarray}
  \label{2mD3.02}
  \sum_{\mbox{\tiny{loops}}}(\det\mathfrak{D}^{(0)})^{-d/2}\frac{\overline{\Tr^{2}M}(-z)^{N{+}1}}{N!}&=&(-2\overline{\C}^2+2\overline{\C^2})f_1^{d+4}(z)+(\overline{\C}^2-2\overline{\C^2})f_1^{d+2}(z)\nonumber\\
        &&+2\overline{\C}^{2}f_1^d(z)-\overline{\C}^2f_1^{d-2}(z)\ .
\end{eqnarray}
Let us rewrite this in terms of the connected correlator (\ref{conn4}):
\begin{eqnarray}
  \label{2mD3.1}
  \sum_{\mbox{\tiny{loops}}}(\det\mathfrak{D}^{(0)})^{-d/2}\frac{\overline{\Tr M^{2}}(-z)^{N{+}1}}{N!} &=&2\overline{\C_{c}^2}\
        f_1^{d+4}(z)+(-4\overline{\C_{c}^2}+\overline{\C}^2)f_1^{d+2}(z)+(-\overline{\C}^{2}+3\overline{\C_{c}^2})\ f_1^d(z)\nonumber\\
        &&-\overline{\C_{c}^2}\ f_1^{d-2}(z)\ ,
\end{eqnarray}
\begin{eqnarray}
  \label{2mD3.2}
  \sum_{\mbox{\tiny{loops}}}(\det\mathfrak{D}^{(0)})^{-d/2}\frac{\overline{\Tr^{2}M}(-z)^{N{+}1}}{N!} &=&2\overline{\C_{c}^2}\ f_1^{d+4}(z)-(2\overline{\C_{c}^2}+\overline{\C}^2)f_1^{d+2}(z)\nonumber\\
        &&+2\overline{\C}^2\ f_1^d(z)-\overline{\C}^2\ f_1^{d-2}(z)\
\end{eqnarray}

\begin{comment}
Therefore, to second order in $2-D$ we find for the renormalized coupling $g$
in terms of the $f_{k}^{d}$'s (\ref{star}):
\begin{eqnarray}\label{2mD5}
g (z) &=& f_{1}^{{d+2}(z)} (z) +\frac{ (2-D)d}{2} f_{2}^{d+2}(z)
\overline{\C}+   \frac{(2-D)^{2} d}{4} f_{2}^{d+2}(z)
\overline{\C^2}\nonumber \\
&&+ \frac{(2-D)^{2}d}{4} \left[f_{3}^{d+4}(z)-f_{3}^{d+2}(z)
\right]\overline{\C}^{2} + \frac{(2-D)^{2}d}{4} \left[f_{3}^{d+2} (z)-2f_{2}^{d+4}(z)
\right]\overline{\C^{2}}\nonumber \\
&&+ \frac{(2-D)^{2}d^{2}}{8}\left[f_{3}^{d+2} (z)+f_{3}^{d+4}(z)
\right]\overline{\C}^{2}- \frac{(2-D)^{2}d^{2}}{4}f_{2}^{d+4}(z) \overline{\C^{2}}
+\dots \ , 
\end{eqnarray}
or in terms of the connected correlator (\ref{star!}):
\begin{eqnarray}
  \label{2mD5.1}
  g(z)&=& f_{1}^{{d+2}} (z) + (2{-}D)\frac{d}{2}\overline{\C}\ f_{2}^{d+2} (z)\nn\\
 &&-(2{-}D)^{2}\frac{d}{4}\left[2\overline{\C_{c}^{2}}\
 f_{1}^{d+4}(z)+(\overline{\C}^{2}{-}4\overline{\C_{c}^{2}})\
 f_{1}^{d+2}(z)+\overline{\C_{c}^{2}}\
 f_{2}^{d}(z)+(2\overline{\C_{c}^{2}}{-}\overline{\C}^{2})\
 f_{1}^{d}(z)\right]\nn\\
 &&-(2{-}D)^{2}\frac{d^{2}}{8}\left[2\overline{\C_{c}^{2}}\
 f_{1}^{d+4}(z)-(2\overline{\C_{c}^{2}}{+}\overline{\C}^{2})\
 f_{1}^{d+2}(z)+\overline{\C}^{2}\ f_{2}^{d}(z)+\overline{\C}^{2}\
 f_{1}^{d}(z)\right]\nn\\
 &&\qquad + \ O(2-D)^{3}\ .
\end{eqnarray}
\end{comment}

Let us now state the terms to appear at third order in $2-D$, which we derived
with the help of a \mathematica- program ($N$ denotes the loop order.):
\begin{eqnarray}
  \label{2mD6}
        \sum_{\mbox{\tiny{loops}}}(\det\mathfrak{D}^{(0)})^{-d/2}\frac{\overline{\Tr
        M^{3}}(-z)^{N{+}1}}{N!}
        &=&4(6\overline{\C}^3-3\overline{\C}\overline{\C^2}+\overline{\C^3}-4\overline{\C^{\triangle}})f_1^{d+4}(z)\nonumber\\
      &+&(-62\overline{\C}^3+36\overline{\C}\overline{\C^2}-10\overline{\C^3}+36\overline{\C^{\triangle}})f_1^{d+2}(z)\nonumber\\
      &+&(63\overline{\C}^3-39\overline{\C}\overline{\C^2}+9\overline{\C^3}-32\overline{\C^{\triangle}})f_1^{d}(z)\nonumber\\
      &+&(-33\overline{\C}^3+18\overline{\C}\overline{\C^2}-3\overline{\C^3}+17\overline{\C^{\triangle}})f_1^{d-2}(z)\nonumber\\
      &+&(9\overline{\C}^{3}-3\overline{\C}\overline{\C^2}-6\overline{\C^{\triangle}})f_1^{d-4}(z)+(-\overline{\C}^{3}+\overline{\C^{\triangle}})f_1^{d-6}(z)\nn\\
\end{eqnarray}
\begin{eqnarray}
  \label{2mD7}
  \sum_{\mbox{\tiny{loops}}}(\det\mathfrak{D}^{(0)})^{-d/2}\frac{\overline{\Tr M\ \Tr
  M^{2}}(-z)^{N{+}1}}{N!}
  &=&4(6\overline{\C}^3-3\overline{\C}\overline{\C^2}+\overline{\C^3}-4\overline{\C^{\triangle}})f_1^{d+6}(z)\nonumber\\
  &+&(-50\overline{\C}^3+26\overline{\C}\overline{\C^2}-8\overline{\C^3}+32\overline{\C^{\triangle}})f_1^{d+4}(z)\nonumber\\
  &+&(29\overline{\C}^3-16\overline{\C}\overline{\C^2}+6\overline{\C^3}-20\overline{\C^{\triangle}})f_1^{d+2}(z)\nonumber\\
  &+&(\overline{\C}^3-\overline{\C}\overline{\C^2}-2\overline{\C^3}+4\overline{\C^{\triangle}})f_1^{d}(z)\nonumber\\
  &+&(-5\overline{\C}^3+4\overline{\C}\overline{\C^2})f_1^{d-2}(z)+(\overline{\C}^3-\overline{\C}\overline{\C^2})f_1^{d-4}(z)\nn\\
\end{eqnarray}
\begin{eqnarray}
  \label{2mD8}
  \sum_{\mbox{\tiny{loops}}}(\det\mathfrak{D}^{(0)})^{-d/2}\frac{\overline{\Tr^3 M}(-z)^{N{+}1}}{N!}
  &=&4(6\overline{\C}^3-3\overline{\C}\overline{\C^2}+\overline{\C^3}-4\overline{\C^{\triangle}})f_1^{d+6}(z)\nonumber\\
  &+&(-26\overline{\C}^3+6\overline{\C}\overline{\C^2}-4\overline{\C^3}+24\overline{\C^{\triangle}})f_1^{d+4}(z)\nonumber\\
  &+&(-3\overline{\C}^3+12\overline{\C}\overline{\C^2}-8\overline{\C^{\triangle}})f_1^{d+2}(z)\nonumber\\
  &+&3(\overline{\C}^3-2\overline{\C}\overline{\C^2})f_1^{d}(z)\nn\\
  &+&3\overline{\C}^3f_1^{d-2}(z)-\overline{\C}^3f_1^{d-4}(z)\nn\\
\end{eqnarray}
Again, let us rewrite this in terms of connected correlators:
\begin{eqnarray}
  \label{2mD8.1}
  \sum_{\mbox{\tiny{loops}}}(\det\mathfrak{D}^{(0)})^{-d/2}\frac{\overline{\Tr
  M^{3}}(-z)^{N{+}1}}{N!}
  &=&4(\overline{\C_{c}^{3}}-4\overline{\C_{c}^{\triangle}})f_{1}^{d+4}(z)\nn\\
  &+&(-10\overline{\C_{c}^{3}}+36\overline{\C_{c}^{\triangle}}+6\overline{\C}\overline{\C_{c}^{2}})f_{1}^{d+2}(z)\nn\\
  &+&(9\overline{\C_{c}^{3}}-32\overline{\C_{c}^{\triangle}}-12\overline{\C}\overline{\C_{c}^{2}}+\overline{\C}^{3})f_{1}^{d}(z)\nn\\
  &+&(3\overline{\C_{c}^{3}}-17\overline{\C_{c}^{\triangle}}-9\overline{\C}\overline{\C_{c}^{2}}+\overline{\C}^{3})f_{1}^{d-2}(z)\nn\\
  &-&3(2\overline{\C_{c}^{\triangle}}+\overline{\C}\overline{\C_{c}^{2}})f_{1}^{d-4}(z)+\overline{\C_{c}^{\triangle}}\ f_{1}^{d-6}(z)\ ,\nn\\
\end{eqnarray}
\begin{eqnarray}
  \label{2mD8.2}
  \sum_{\mbox{\tiny{loops}}}(\det\mathfrak{D}^{(0)})^{-d/2}\frac{\overline{\Tr M\ \Tr
  M^{2}}(-z)^{N{+}1}}{N!}   
  &=&4(\overline{\C_{c}^{3}}-4\overline{\C_{c}^{\triangle}})f_{1}^{d+6}(z)\nn\\
  &+&(-8\overline{\C_{c}^{3}}+32\overline{\C_{c}^{\triangle}}+2\overline{\C}\overline{\C_{c}^{2}})f_{1}^{d+4}(z)\nn\\
  &+&(6\overline{\C_{c}^{3}}-20\overline{\C_{c}^{\triangle}}+2\overline{\C}\overline{\C_{c}^{2}}-6\overline{\C}^{3})f_{1}^{d+2}(z)\nn\\
  &+&(-2\overline{\C_{c}^{3}}+4\overline{\C_{c}^{\triangle}}-7\overline{\C}\overline{\C_{c}^{2}}+2\overline{\C}^{3})f_{1}^{d}(z)\nn\\
  &-&(\overline{\C}^{3}-4\overline{\C}\overline{\C_{c}^{2}})f_{1}^{d-2}(z)-\overline{\C}\overline{\C_{c}^{2}}\ f_{1}^{d-4}(z)\ ,\nn\\
\end{eqnarray}
\begin{eqnarray}
  \label{2mD8.3}
  \sum_{\mbox{\tiny{loops}}}(\det\mathfrak{D}^{(0)})^{-d/2}\frac{\overline{\Tr^3
  M}(-z)^{N{+}1}}{N!}
  &=&4(\overline{\C_{c}^{3}}-4\overline{\C_{c}^{\triangle}})f_{1}^{d+6}(z)\nn\\
  &+&(-4\overline{\C_{c}^{3}}+24\overline{\C_{c}^{\triangle}}-6\overline{\C}\overline{\C_{c}^{2}})f_{1}^{d+4}(z)\nn\\
  &+&(-8\overline{\C_{c}^{\triangle}}+12\overline{\C}\overline{\C_{c}^{2}}+\overline{\C}^{3})f_{1}^{d+2}(z)\nn\\
  &-&3(\overline{\C}^{3}+2\overline{\C}\overline{\C_{c}^{2}})f_{1}^{d}(z)\nn\\
  &+&3\overline{\C}^{3}f_{1}^{d-2}(z)-\overline{\C}^{3}f_{1}^{d-4}(z)\ .\nn\\
\end{eqnarray}
To fourth order in $2-D$ we obtain:
\begin{eqnarray}
  \label{2mD9}
  &&\sum_{\mbox{\tiny{loops}}}(\det\mathfrak{D}^{(0)})^{-d/2}\frac{\overline{\Tr M^4}(-z)^{N{+}1}}{N!}\nn\\
  &=&8(90\overline{\C}^4-36\overline{\C}^2\overline{\C^2}+3\overline{\C^2}^2+4\overline{\C}\overline{\C^3}-\overline{\C^4}+24\overline{\C^{\BT}}-36\overline{\C^{\diamond}}-48\overline{\C}\overline{\C^{\triangle}})f_1^{d+8}(z)\nonumber\\
  &+&4(501\overline{\C}^4-222\overline{\C}^2\overline{\C^2}+23\overline{\C^2}^2+28\overline{\C}\overline{\C^3}-6\overline{\C^4}+128\overline{\C^{\BT}}-180\overline{\C^{\diamond}}-272\overline{\C}\overline{\C^{\triangle}})f_1^{d+6}(z)\nonumber\\
  &-&4(561\overline{\C}^4-273\overline{\C}^2\overline{\C^2}+31\overline{\C^2}^2+38\overline{\C}\overline{\C^3}-7\overline{\C^4}+136\overline{\C^{\BT}}-178\overline{\C^{\diamond}}-308\overline{\C}\overline{\C^{\triangle}})f_1^{d+4}(z)\nonumber\\
  &+&(1359\overline{\C}^4-696\overline{\C}^2\overline{\C^2}+82\overline{\C^2}^2+100\overline{\C}\overline{\C^3}-16\overline{\C^4}+304\overline{\C^{\BT}}-396\overline{\C^{\diamond}}-736\overline{\C}\overline{\C^{\triangle}})f_1^{d+2}(z)\nonumber\\
  &+&(-500\overline{\C}^4+248\overline{\C}^2\overline{\C^2}-36\overline{\C^2}^2-32\overline{\C}\overline{\C^3}+5\overline{\C^4}-92\overline{\C^{\BT}}+154\overline{\C^{\diamond}}+252\overline{\C}\overline{\C^{\triangle}})f_1^{d}(z)\nonumber\\
  &+&(116\overline{\C}^4-48\overline{\C}^2\overline{\C^2}+12\overline{\C^2}^2+4\overline{\C}\overline{\C^3}-\overline{\C^4}+12\overline{\C^{\BT}}-47\overline{\C^{\diamond}}-48\overline{\C}\overline{\C^{\triangle}})f_1^{d-2}(z)\nonumber\\
  &+&(-16\overline{\C}^4+4\overline{\C}^2\overline{\C^2}-2\overline{\C^2}^2+10\overline{\C^{\BT}}+4\overline{\C}\overline{\C^{\triangle}})f_1^{d-4}(z)+(\overline{\C}^4-\overline{\C^{\diamond}})f_1^{d-6}(z)\nn\\
\end{eqnarray}
\newpage
\begin{eqnarray}
  \label{2mD10}
  &&\sum_{\mbox{\tiny{loops}}}(\det\mathfrak{D}^{(0)})^{-d/2}\frac{\overline{\Tr^2
  M^{2}}(-z)^{N{+}1}}{N!}\nn\\
  &=&8(90\overline{\C}^4-36\overline{\C}^2\overline{\C^2}+3\overline{\C^2}^2+4\overline{\C}\overline{\C^3}-\overline{\C^4}+24\overline{\C^{\BT}}-36\overline{\C^{\diamond}}-48\overline{\C}\overline{\C^{\triangle}})f_1^{d+8}(z)\nonumber\\
   &+&4(501\overline{\C}^4-222\overline{\C}^2\overline{\C^2}+23\overline{\C^2}^2+28\overline{\C}\overline{\C^3}-6\overline{\C^4}+128\overline{\C^{\BT}}-180\overline{\C^{\diamond}}-272\overline{\C}\overline{\C^{\triangle}})f_1^{d+6}(z)\nonumber\\
  &-&4(537\overline{\C}^4-261\overline{\C}^2\overline{\C^2}+28\overline{\C^2}^2+40\overline{\C}\overline{\C^3}-8\overline{\C^4}+132\overline{\C^{\BT}}-172\overline{\C^{\diamond}}-296\overline{\C}\overline{\C^{\triangle}})f_1^{d+4}(z)\nonumber\\
  &+&(1111\overline{\C}^4-572\overline{\C}^2\overline{\C^2}+44\overline{\C^2}^2+120\overline{\C}\overline{\C^3}-24\overline{\C^4}+272\overline{\C^{\BT}}-624\overline{\C}\overline{\C^{\triangle}})f_1^{d+2}(z)\nonumber\\
  &+&(260\overline{\C}^4-122\overline{\C}^2\overline{\C^2}-10\overline{\C^{2}}^{2}+48\overline{\C}\overline{\C^3}-10\overline{\C^4}+72\overline{\C^{\BT}}-80\overline{\C^{\diamond}}-160\overline{\C}\overline{\C^{\triangle}})f_1^{d}(z)\nonumber\\
  &+&(6\overline{\C}^4+18\overline{\C}^2\overline{\C^2}-15\overline{\C^2}^2+8\overline{\C}\overline{\C^3}-2\overline{\C^4}+8\overline{\C^{\BT}}-8\overline{\C^{\diamond}}-16\overline{\C}\overline{\C^{\triangle}})f_1^{d-2}(z)\nonumber\\
  &+&(8\overline{\C}^4-14\overline{\C}^2\overline{\C^2}+6\overline{\C^2}^2)f_1^{d-4}(z)\nonumber\\
  &+&(-\overline{\C}^4+2\overline{\C}^2\overline{\C^2}-\overline{\C^2}^2)f_1^{d-6}(z)\nn\\
\end{eqnarray}
\begin{eqnarray}
  \label{2mD11}
  &&\sum_{\mbox{\tiny{loops}}}(\det\mathfrak{D}^{(0)})^{-d/2}\frac{\overline{\Tr M\ \Tr
  M^3}(-z)^{N{+}1}}{N!}\nn\\
  &=&8(90\overline{\C}^4-36\overline{\C}^2\overline{\C^2}+3\overline{\C^2}^2+4\overline{\C}\overline{\C^3}-\overline{\C^4}+24\overline{\C^{\BT}}-36\overline{\C^{\diamond}}-48\overline{\C}\overline{\C^{\triangle}})f_1^{d+8}(z)\nonumber\\
  &+&4(441\overline{\C}^4-186\overline{\C}^2\overline{\C^2}+18\overline{\C^2}^2+22\overline{\C}\overline{\C^3}-5\overline{\C^4}+114\overline{\C^{\BT}}-168\overline{\C^{\diamond}}-236\overline{\C}\overline{\C^{\triangle}})f_1^{d+6}(z)\nonumber\\
  &-&2(800\overline{\C}^4-351\overline{\C}^2\overline{\C^2}+39\overline{\C^2}^2+41\overline{\C}\overline{\C^3}-9\overline{\C^4}+204\overline{\C^{\BT}}-294\overline{\C^{\diamond}}-430\overline{\C}\overline{\C^{\triangle}})f_1^{d+4}(z)\nonumber\\
  &+&(661\overline{\C}^4-279\overline{\C}^2\overline{\C^2}+36\overline{\C^2}^2+23\overline{\C}\overline{\C^3}-6\overline{\C^4}+186\overline{\C^{\BT}}-258\overline{\C^{\diamond}}-364\overline{\C}\overline{\C^{\triangle}})f_1^{d+2}(z)\nonumber\\
  &+&(-90\overline{\C}^4+21\overline{\C}^2\overline{\C^2}-6\overline{\C^2}^2+6\overline{\C}\overline{\C^3}-48\overline{\C^{\BT}}+60\overline{\C^{\diamond}}+59\overline{\C}\overline{\C^{\triangle}})f_1^{d}(z)\nonumber\\
  &+&(-24\overline{\C}^4+15\overline{\C}^2\overline{\C^2}-3\overline{\C}\overline{\C^3}+6\overline{\C^{\BT}}-6\overline{\C^{\diamond}}+11\overline{\C}\overline{\C^{\triangle}})f_1^{d-2}(z)\nonumber\\
  &+&(10\overline{\C}^4-3\overline{\C}^2\overline{\C^2}-7\overline{\C}\overline{\C^{\triangle}})f_1^{d-4}(z)+(-\overline{\C}^4+\overline{\C}\overline{\C^{\triangle}})f_1^{d-6}(z)\nn\\
\end{eqnarray}
\begin{eqnarray}
  \label{2mD12}
  &&\sum_{\mbox{\tiny{loops}}}(\det\mathfrak{D}^{(0)})^{-d/2}\frac{\overline{\Tr^2M\ \Tr
  M^2}(-z)^{N{+}1}}{N!}\nn\\
  &=&8(90\overline{\C}^4-36\overline{\C}^2\overline{\C^2}+3\overline{\C^2}^2+4\overline{\C}\overline{\C^3}-\overline{\C^4}+24\overline{\C^{\BT}}-36\overline{\C^{\diamond}}-48\overline{\C}\overline{\C^{\triangle}})f_1^{d+8}(z)\nonumber\\
  &+&4(381\overline{\C}^4-150\overline{\C}^2\overline{\C^2}+13\overline{\C^2}^2+16\overline{\C}\overline{\C^3}+4\overline{\C^4}-100\overline{\C^{\BT}}+156\overline{\C^{\diamond}}-200\overline{\C}\overline{\C^{\triangle}})f_1^{d+6}(z)\nonumber\\
  &+&4(251\overline{\C}^4-90\overline{\C}^2\overline{\C^2}+8\overline{\C^2}^2+8\overline{\C}\overline{\C^3}-3\overline{\C^4}+70\overline{\C^{\BT}}-116\overline{\C^{\diamond}}-128\overline{\C}\overline{\C^{\triangle}})f_1^{d+4}(z)\nonumber\\
  &+&(159\overline{\C}^4-4\overline{\C}^2\overline{\C^2}-4\overline{\C^2}^2-12\overline{\C}\overline{\C^3}+2\overline{\C^4}-40\overline{\C^{\BT}}+72\overline{\C^{\diamond}}-72\overline{\C}\overline{\C^{\triangle}})f_1^{d+2}(z)\nonumber\\
  &+&(50\overline{\C}^4-53\overline{\C}^2\overline{\C^2}+8\overline{\C^2}^2+16\overline{\C}\overline{\C^3}-8\overline{\C^{\BT}}+16\overline{\C^{\diamond}}-32\overline{\C}\overline{\C^{\triangle}})f_1^{d}(z)\nonumber\\
  &+&(-4\overline{\C}^4+5\overline{\C}^2\overline{\C^2}-2\overline{\C^2}^2-4\overline{\C}\overline{\C^3}+8\overline{\C}\overline{\C^{\triangle}})f_1^{d-2}(z)\nonumber\\
  &+&(-6\overline{\C}^4+5\overline{\C}^2\overline{\C^2})f_1^{d-4}(z)+(\overline{\C}^4-\overline{\C}^2\overline{\C^2})f_1^{d-6}(z)\nn\\
\end{eqnarray}
\begin{eqnarray}
  \label{2mD13}
  &&\sum_{\mbox{\tiny{loops}}}(\det\mathfrak{D}^{(0)})^{-d/2}\frac{\overline{\Tr^{4} M}(-z)^{N{+}1}}{N!}\nn\\
  &=&8(90\overline{\C}^4-36\overline{\C}^2\overline{\C^2}+3\overline{\C^2}^2+4\overline{\C}\overline{\C^3}-\overline{\C^4}+24\overline{\C^{\BT}}-36\overline{\C^{\diamond}}-48\overline{\C}\overline{\C^{\triangle}})f_1^{d+8}(z)\nonumber\\
  &+&4(261\overline{\C}^4-78\overline{\C}^2\overline{\C^2}+3\overline{\C^2}^2+4\overline{\C}\overline{\C^3}-2\overline{\C^4}+72\overline{\C^{\BT}}-132\overline{\C^{\diamond}}-128\overline{\C}\overline{\C^{\triangle}})f_1^{d+6}(z)\nonumber\\
  &+&4(61\overline{\C}^4+9\overline{\C}^2\overline{\C^2}-6\overline{\C^2}^2-8\overline{\C}\overline{\C^3}+24\overline{\C^{\BT}}-72\overline{\C^{\diamond}}-8\overline{\C}\overline{\C^{\triangle}})f_1^{d+4}(z)\nonumber\\
  &+&(-89\overline{\C}^4+36\overline{\C}^2\overline{\C^2}-12\overline{\C^2}^2-16\overline{\C}\overline{\C^3}-48\overline{\C^{\diamond}}+128\overline{\C}\overline{\C^{\triangle}})f_1^{d+2}(z)\nonumber\\
  &+&(36\overline{\C}^2\overline{\C^2}-32\overline{\C}\overline{\C^{\triangle}})f_1^{d}(z)+(6\overline{\C}^4-12\overline{\C}^2\overline{\C^2})f_1^{d-2}(z)\nonumber\\
  &+&4\overline{\C}^4f_1^{d-4}(z)-\overline{\C}^4f_1^{d-6}(z)\nn\\
\end{eqnarray}
Rewriting everything in terms of connected correlators we find:
\begin{eqnarray}
  \label{2mD13.1}
  &&\sum_{\mbox{\tiny{loops}}}(\det\mathfrak{D}^{(0)})^{-d/2}\frac{\overline{\Tr^4 M}(-z)^{N{+}1}}{N!}\nn\\
  &=&8(222\overline{\C}^{4}+6\Csq\Ccsq+3\Ccsq^{2}-\Ccbisq+24\Ccblobt-36\Ccdia)f_1^{d+8}(z)\nonumber\\
  &+&4(804\overline{\C}^4+12\Csq\Ccsq+3\Ccsq^{2}-2\Ccbisq+72\Ccblobt-132\Ccdia-4\ovC\Cccub+16\ovC\Cctri)f_1^{d+6}(z)\nonumber\\
  &-&4(432\ovC^{4}-3\ovC^{2}\Ccsq-6\Ccsq^{2}+24\Ccblobt-72\Ccdia-8\ovC\Cccub+40\ovC\Cctri)f_1^{d+4}(z)\nonumber\\
  &+&(287\ovC^{4}-36\ovC^{2}\Ccsq-12\Ccsq^{2}-48\Ccdia-16\ovC\Cccub+128\ovC\Cctri)f_1^{d+2}(z)\nn\\
  &+&(4\ovC^{4}+36\ovC^{2}\Ccsq-32\ovC\Cctri)f_1^{d}(z)\nn\\
  &+&(-6\ovC^{4}-2\ovC^{2}\Ccsq)f_1^{d-2}(z)+4\ovC^{4}f_1^{d-4}(z)-\ovC^{4}f_1^{d-6}(z)\nn\\
\end{eqnarray}
\begin{eqnarray}
  \label{2mD13.2}
  &&\sum_{\mbox{\tiny{loops}}}(\det\mathfrak{D}^{(0)})^{-d/2}\frac{\overline{\Tr M^{2}\
  \Tr^{2}M}(-z)^{N{+}1}}{N!}\nn\\
  &=&8(222\overline{\C}^{4}+6\Csq\Ccsq+3\Ccsq^{2}-\Ccbisq+24\Ccblobt-36\Ccdia)f_1^{d+8}(z)\nonumber\\
   &+&4(960\ovC^{4}+24\ovC^{2}\Ccsq+\Ccsq^{2}-4\Ccbisq+100\Ccblobt-156\Ccdia)f_1^{d+6}(z)\nonumber\\
  &-&4(714\ovC^{4}+20\ovC^{2}\Ccsq+8\Ccsq^{2}-3\Ccbisq+70\Ccblobt-116\Ccdia-4\ovC\Cccub+12\ovC\Cctri)f_1^{d+4}(z)\nonumber\\
  &+&(889\ovC^{4}+36\ovC^{2}\Ccsq-2\Ccsq^{2}-28\ovC\Cccub-4\Ccbisq+80\Ccblobt-144\Ccdia+88\ovC\Cctri)f_1^{d+2}(z)\nn\\
  &+&(-99\ovC^{4}+3\ovC^{2}\Ccsq+8\Ccsq^{2}+16\ovC\Cccub-8\Ccblobt+16\Ccdia-48\ovC\Cctri)f_1^{d}(z)\nn\\
  &+&(3\ovC^{4}-11\ovC^{2}\Ccsq-2\Ccsq^{2}-4\ovC\Cccub+8\ovC\Cctri)f_1^{d-2}(z)\nn\\
  &+&(-\ovC^{4}-5\ovC^{2}\Ccsq)f_1^{d-4}(z)-\ovC^{2}\Ccsq f_1^{d-6}(z)\nn\\
\end{eqnarray}
\newpage
\begin{eqnarray}
  \label{2mD13.3}
  &&\sum_{\mbox{\tiny{loops}}}(\det\mathfrak{D}^{(0)})^{-d/2}\frac{\overline{\Tr^2
  M^{2}}(-z)^{N{+}1}}{N!}\nn\\
  &=&8(222\overline{\C}^{4}+6\Csq\Ccsq+3\Ccsq^{2}-\Ccbisq+24\Ccblobt-36\Ccdia)f_1^{d+8}(z)\nonumber\\
   &+&4(1116\ovC^{4}+36\ovC^{2}\Ccsq+23\Ccsq^{2}-6\Ccbisq+128\Ccblobt-180\Ccdia+4\ovC\Cccub-16\ovC\Cctri)f_1^{d+6}(z)\nonumber\\
  &-&4(1080\ovC^{4}+47\ovC^{2}\Ccsq+28\Ccsq^{2}-8\Ccbisq+132\Ccblobt-172\Ccdia+8\ovC\Cccub-32\ovC\Cctri)f_1^{d+4}(z)\nonumber\\
  &+&(2111\ovC^{4}+148\ovC^{2}\Ccsq+44\Ccsq^{2}+24\ovC\Cccub-24\Ccbisq+272\Ccblobt-328\Ccdia-80\ovC\Cctri)f_1^{d+2}(z)\nn\\
  &+&(-538\ovC^{4}-74\ovC^{2}\Ccsq+10\Ccsq^{2}-8\ovC\Cccub+10\Ccbisq-72\Ccblobt+80\Ccdia+8\ovC\Cctri)f_1^{d}(z)\nn\\
  &+&(59\ovC^{4}+20\ovC^{2}\Ccsq-15\Ccsq^{2}-2\Ccbisq+8\Ccblobt-8\Ccdia)f_1^{d-2}(z)\nn\\
  &+&(-2\ovC^{2}\Ccsq+6\Ccsq^{2})f_1^{d-4}(z)-\Ccsq^{2}f_1^{d-6}(z)\nn\\
\end{eqnarray}
\begin{eqnarray}
  \label{2mD13.4}
  &&\sum_{\mbox{\tiny{loops}}}(\det\mathfrak{D}^{(0)})^{-d/2}\frac{\overline{\Tr M\ \Tr
  M^3}(-z)^{N{+}1}}{N!}\nn\\
  &=&8(222\overline{\C}^{4}+6\Csq\Ccsq+3\Ccsq^{2}-\Ccbisq+24\Ccblobt-36\Ccdia)f_1^{d+8}(z)\nonumber\\
  &+&4(1038\ovC^{4}+30\ovC^{2}\Ccsq+18\Ccsq^{2}-5\Ccbisq+114\Ccblobt-168\Ccdia+2\ovC\Cccub-8\ovC\Cctri)f_1^{d+6}(z)\nonumber\\
  &-&2(1818\ovC^{4}+54\ovC^{2}\Ccsq+39\Ccsq^{2}+\ovC\Cccub-9\Ccbisq+204\Ccblobt-294\Ccdia-22\ovC\Cctri)f_1^{d+4}(z)\nonumber\\
  &+&(1583\ovC^{4}+48\ovC^{2}\Ccsq+36\Ccsq^{2}-\ovC\Cccub-6\Ccbisq+186\Ccblobt-258\Ccdia+8\ovC\Cctri)f_1^{d+2}(z)\nn\\
  &+&(-358\ovC^{4}-21\ovC^{2}\Ccsq-6\Ccsq^{2}+6\ovC\Cccub-48\Ccblobt+60\Ccdia-37\ovC\Cctri)f_1^{d}(z)\nn\\
  &+&(35\ovC^{4}+12\ovC^{2}\Ccsq-3\ovC\Cccub+6\Ccblobt-6\Ccdia+23\ovC\Cctri)f_1^{d-2}(z)\nn\\
  &+&(-3\ovC^{2}-7\ovC\Cctri)f_1^{d-4}(z)+\ovC\Cctri f_1^{d-6}(z)\nn\\
\end{eqnarray}
\begin{eqnarray}
  \label{2mD13.5}
  &&\sum_{\mbox{\tiny{loops}}}(\det\mathfrak{D}^{(0)})^{-d/2}\frac{\overline{\Tr M^4}(-z)^{N{+}1}}{N!}\nn\\
  &=&8(222\overline{\C}^{4}+6\Csq\Ccsq+3\Ccsq^{2}-\Ccbisq+24\Ccblobt-36\Ccdia)f_1^{d+8}(z)\nonumber\\
  &+&4(1116\ovC^{4}+36\ovC^{2}\Ccsq+23\Ccsq^{2}-6\Ccbisq+128\Ccblobt-180\Ccdia+4\ovC\Cccub-16\ovC\Cctri)f_1^{d+6}(z)\nonumber\\
  &-&4(1110\ovC^{4}+39\ovC^{2}\Ccsq+31\Ccsq^{2}+10\ovC\Cccub-7\Ccbisq+136\Ccblobt-178\Ccdia-36\ovC\Cctri)f_1^{d+4}(z)\nonumber\\
  &+&(2473\ovC^{4}+72\ovC^{2}\Ccsq+82\Ccsq^{2}+36\ovC\Cccub-16\Ccbisq+304\Ccblobt-396\Ccdia-128\ovC\Cctri)f_1^{d+2}(z)\nn\\
  &+&(-955\ovC^{4}-12\ovC^{2}\Ccsq-36\Ccsq^{2}-12\ovC\Cccub+5\Ccbisq-92\Ccblobt+154\Ccdia+68\ovC\Cctri)f_1^{d}(z)\nn\\
  &+&(288\ovC^{4}+12\Ccsq^{2}-\Ccbisq+12\Ccblobt-47\Ccdia-24\ovC\Cctri)f_1^{d-2}(z)\nn\\
  &+&(-60\ovC^{4}-2\Ccsq^{2}+10\Ccdia+4\ovC\Cctri)f_1^{d-4}(z)+(6\ovC^{4}-\Ccdia)f_1^{d-6}(z)\nn\\
\end{eqnarray}

\subsection{Renormalized coupling}
Our purpose is to guess the exact form of the renormalized coupling $g(z)$ as
a function of $2-D$ on a manifold of toroidal topology. The
information that is of our disposal are the exact results in both $D=2$
(\ref{6.26.1}) and $D=1$ (\ref{a.26}) as well as the exact resummed series up
to the fourth order in $2-D$.\\
From the structure of the expansion of $g(z)$ in powers of $2-D$ it is clear
(see also discussion of the \mathematica-program in the appendix) that to each
order in $2-D$ $g(z)$ can be expressed in terms of $f_{1}^{d}s$
(\ref{fkdreduce}).  Since
\begin{eqnarray}
  \label{2mD13.6}
  f_{1}^{d}(z)=\frac{z}{\Gamma(\frac{d}{2})}\int_{0}^{\infty}\mbox{d}r\
  r^{d/2-1}\exp[-z\mbox{e}^{-r}-r]\ ,
\end{eqnarray}
the exact $g(z)$ as a function of $2-D$ will be of the form:
\begin{eqnarray}
  \label{2mD13.7}
  g(z)\equiv g(D,z)=z\int_{0}^{\infty}\mbox{d}r\
  \tilde g(D,r)\exp[-z\mbox{e}^{-r}-r]\ ,
\end{eqnarray}
where $\tilde g(D,r)$ has some expansion in powers of $2-D$. To zeroth
order corresponding to $D=2$ one has:
\begin{equation}
  \label{2mD13.8}
  \tilde g(2,r)=\frac{r^{d/2}}{\Gamma(\frac{d}{2}{+}1)}\ .
\end{equation}
Analyzing the series data from the preceding subsection reveals the following
structure of the expansion of $\tilde g(D,z)$:
\begin{eqnarray}
  \label{2mD13.9}
  \tilde
  g(D,r)=r^{d/2}\left(\frac{1}{\Gamma(\frac{d}{2}{+}1)}+(2-D)\mathcal{P}(2{-}D,r)\right)\ ,
\end{eqnarray}
where $\mathcal{P}(2{-}D,r)$ is analytic in $D=2$, such that
\begin{equation}
  \label{2mD13.10}
  \mathcal{P}(2{-}D,r)=\sum_{n=0}^{\infty}\mathcal{P}_{n}(r)(2-D)^{n}\ ,
\end{equation}
and the $n$th order coefficient $\mathcal{P}_{n}(r)$ has a Laurent series in
$r$:
\begin{eqnarray}
  \label{2mD13.11}
  \mathcal{P}_{n}(r)=\sum_{j=-n_{\mbox{\tiny max}}}^{n}p_{n_{j}}r^{j}\ .
\end{eqnarray}
Note that powers $\alpha$ of $r$ on the level of $\tilde g$ (\ref{2mD13.9}) translate
after integrating over $r$ (\ref{2mD13.7}) into powers $\alpha$ of $\log z$
(clarify \ref{fkdexpansion}). Therefore, positive powers of $r$ dominate the
behavior for large $z$ (strong coupling).\\
Let us write down the terms of the expansion of $\tilde g(D,r)$ up to the
third order in $2-D$ as one
obtains them from the generated series contributions of the preceding
subsection:
\begin{eqnarray}
  \label{2mD13.12}
  &&\tilde g(D,r)=r^{d/2}\left[\frac{1}{\Gamma(\frac{d}{2}{+}1)}+(2-D)\frac{d}{2}\ovC\left(\frac{1}{\Gamma(\frac{d}{2}{+}1)}-\frac{r^{-1}}{\Gamma(\frac{d}{2})}\right)\right.\nn\\
  &&-(2-D)^{2}\left(\frac{d}{2}\left(1{+}\frac{d}{2}\right)\Ccsq\frac{r}{\Gamma(\frac{d}{2}{+}2)}+\frac{\frac{d}{4}\left(\Csq-4\Ccsq\right)-\frac{d^{2}}{8}\left(2\Ccsq+\Csq\right)}{\Gamma(\frac{d}{2}{+}1)}\right.\nn\\
  &&+\left.\frac{\left(\frac{d}{4}\left(-\Csq+3\Ccsq\right)-\frac{d^{2}}{8}\left(2\Csq\right)\right)r^{-1}}{\Gamma(\frac{d}{2})}+\frac{\left(-\frac{d}{4}\Ccsq+\frac{d^{2}}{8}\Csq\right)r^{-2}}{\Gamma(\frac{d}{2}{-}1)}\right)\nn\\
  &&+(2-D)^{3}\left(\frac{\left(\frac{d^{2}}{2}+\frac{d^{3}}{12}\right)\left(\Cccub-4\ \Cctri\right)r^{2}}{\Gamma(\frac{d}{2}{+}3)}\right.\nn\\
  &&+\left.\frac{\left(d\left(\Cccub-4\ \Cctri\right)+\frac{d^{2}}{8}\left(-8\
  \Cccub+32\ \Cctri+2\ \overline{\C}\Ccsq\right)+\frac{d^{3}}{48}\left(-4\
  \Cccub+24\ \Cctri-6\
  \overline{\C}\Ccsq\right)\right)r}{\Gamma(\frac{d}{2}{+}2)}\right.\nn\\
  &&\!\!\!\!\!\!+\frac{\left(\frac{d}{4}\left(-10\ \Cccub+36\
  \Cctri{+}6\overline{\C}\Ccsq\right){+}\frac{d^{2}}{8}\left(6\ \Cccub{-}20\ \Cctri+2\ \overline{\C}\Ccsq{-}6\
  \overline{\C}^{3}\right)+\frac{d^{3}}{48}\left(-8\ \Cctri{+}12\
  \overline{\C}\Ccsq{+}\overline{\C}^{3}\right)\right)}{\Gamma(\frac{d}{2}{+}1)}\nn\\
  &&\!\!\!\!\!\!+\frac{\left(\frac{d}{4}\left(9\overline{\C_{c}^{3}}-32\overline{\C_{c}^{\triangle}}-12\overline{\C}\overline{\C_{c}^{2}}+\overline{\C}^{3}\right){+}\frac{d^{2}}{8}\left(-2\overline{\C_{c}^{3}}+4\overline{\C_{c}^{\triangle}}-7\overline{\C}\overline{\C_{c}^{2}}+2\overline{\C}^{3}\right)-\frac{d^{3}}{16}\left(\overline{\C}^{3}+2\overline{\C}\overline{\C_{c}^{2}}\right)\right)\
  r^{-1}}{\Gamma(\frac{d}{2})}\nn\\
  &&+\frac{\left(\frac{d}{4}\left(3\overline{\C_{c}^{3}}-17\overline{\C_{c}^{\triangle}}-9\overline{\C}\overline{\C_{c}^{2}}+\overline{\C}^{3}\right){-}\frac{d^{2}}{8}\left(\overline{\C}^{3}-4\overline{\C}\overline{\C_{c}^{2}}\right)+\frac{d^{3}}{16}\
  \overline{\C}^{3}\right)\
  r^{-2}}{\Gamma(\frac{d}{2}{-}1)}\nn\\
  &&+\left.\left.\frac{\left(-\frac{3d}{4}\left(2\overline{\C_{c}^{\triangle}}+\overline{\C}\overline{\C_{c}^{2}}\right){-}\frac{d^{2}}{8}\
  \overline{\C}\overline{\C_{c}^{2}}-\frac{d^{3}}{48}\ \overline{\C}^{3}\right)\
  r^{-3}}{\Gamma(\frac{d}{2}{-}2)}+\frac{\frac{d}{4}\ \overline{\C_{c}^{\triangle}}\ r^{-4}}{\Gamma(\frac{d}{2}{-}3)}\right)\right]\nn\\
  &&
\end{eqnarray}
Clearly, one discovers the form (\ref{2mD13.9}). Let us also state $\tilde
g(D,r)$ up to the fourth order, but taking to each order only the dominant
contribution into account:
\begin{eqnarray}
  \label{2mD14}
  \tilde g(D,r)&=&r^{d/2}\
   \left[\frac{1}{\Gamma(\frac{d}{2}{+}1)}\right.+(2-D)\frac{\left(\frac{d}{2}\right)}{\Gamma(\frac{d}{2}{+}1)}\overline{\C}-(2-D)^{2}\frac{\frac{d}{2}\left(1+\frac{d}{2}\right)}{\Gamma(\frac{d}{2}{+}2)}\ \Ccsq \ r\nonumber\\
   &+&(2-D)^{3}\frac{\left(\frac{d^{2}}{2}+\frac{d^{3}}{12}\right)\left(\Cccub-4\Cctri\right)}{\Gamma(\frac{d}{2}{+}3)}r^{2}\nonumber\\
   &-&\left.(2-D)^{4}\frac{\left(d+\frac{3}{4}d^2+\frac{d^{3}}{4}+\frac{d^{4}}{48}\right)\left(222\overline{\C}^{4}+6\Csq\Ccsq+3\Ccsq^{2}-\Ccbisq+24\Ccblobt-36\Ccdia\right)}{\Gamma(\frac{d}{2}{+}4)}r^{3}+\cdots\right]\nn\\
\end{eqnarray}
For completeness let us state the result of the $2-D$-expansion of the
renormalized coupling up to the fourth order. This time performing the
integration in (\ref{2mD13.7}), such that we obtain $g$ as a function of the
bare coupling in terms of the functions $f_{k}^{d}(z)$, whose asymptotic
behavior has been analyzed in (\ref{fkdexpansion}). For this purpose and for the sake of more compactness we need
introduce a new notation: Since all series contributions are of the form as
stated in (\ref{pr.0}), we introduce vectors $\mathbb{M}$ such that
\begin{eqnarray}
  \label{2mD14.00}
  \sum_{\mbox{\tiny{loops}}}(\det\mathfrak{D}^{(0)})^{-d/2}\frac{\overline{\prod_{i=1}^{l}\left(\Tr M^{n_{i}}\right)^{m_{i}}}(-z)^{N}}{N!}&\equiv&\sum_{j=\mbox{\tiny{min}}}^{\mbox{\tiny{max}}}\mathbb{M}_{
    \tiny{\left(\begin{array}[]{cccc}
        \!\!m_{1} & \!\!m_{2} & \!\!\cdots & \!\!m_{l}\\
        \!\!n_{1} & \!\!n_{2} & \!\!\cdots & \!\!n_{l}
    \end{array}\!\!\right)}
}f_{1}^{d+2j}(z)\nn\\
  &\equiv&\mathbb{M}^{j}_{
    \tiny{\left(\begin{array}[]{cccc}
        \!\!m_{1} & \!\!m_{2} & \!\!\cdots & \!\!m_{l}\\
        \!\!n_{1} & \!\!n_{2} & \!\!\cdots & \!\!n_{l}
    \end{array}\!\!\right|\left.\begin{array}[]{c}
      \!\! \mbox{max}\\
      \!\! \mbox{min}
    \end{array}\right)}
}f_{1}^{d+2j}(z)\ ,
\end{eqnarray}
where $\mbox{\small{max}}$ and $\mbox{\small{min}}$ are some integers, and
summation over the double index $j$ is implicit. Inserting the results for the
resummed series contributions into (\ref{6.71}) we find for the renormalized
coupling to the fourth order in $2-D$:
%\newpage
\begin{eqnarray}
  \label{star!!!}
  g(z)&=&f_{1}^{d+2}(z)-(2-D)\ \frac{d}{2}\mathbb{M}^{j}_{
    \tiny{\left(\begin{array}[]{c}   %%%% Tr M %%%%
          1\ \\
          1\ 
    \end{array}\!\!\right|\left.\begin{array}[]{c} 
      \!1\\
      \!0
    \end{array}\right)}
}f_{1}^{d+2j}(z)+(2-D)^{2}\left(\frac{d}{4}\mathbb{M}^{j}_{
    \tiny{\left(\begin{array}[]{c}  %%%% Tr M^{2} %%%%
          1\ \\
          2\ 
    \end{array}\!\!\right|\left.\begin{array}[]{c}
      \!\!\!2\\
      \!\!\!-1
    \end{array}\right)}
}f_{1}^{d+2j}(z)+\frac{d^{2}}{8}\mathbb{M}^{j}_{
    \tiny{\left(\begin{array}[]{c}   %%%% Tr^{2} M %%%%
          2\ \\
          1\ 
    \end{array}\!\!\right|\left.\begin{array}[]{c}
      \!\!\!2\\
      \!\!\!-1
    \end{array}\right)}
}f_{1}^{d+2j}(z)\right)\nn\\
  &-&(2-D)^{3}\left(\frac{d}{4}\mathbb{M}^{j}_{
    \tiny{\left(\begin{array}[]{c}   %%%% Tr M^{3} %%%%
          1\ \\
          3\ 
    \end{array}\!\!\right|\left.\begin{array}[]{c}
      \!\!\!2\\
      \!\!\!-3
    \end{array}\right)}
}f_{1}^{d+2j}(z)+\frac{d^{2}}{8}\mathbb{M}^{j}_{
    \tiny{\left(\begin{array}[]{cc}   %%%% Tr M Tr M^{2} %%%%
          1 & 1\ \\
          1 & 2\ 
    \end{array}\!\!\right|\left.\begin{array}[]{c}
      \!\!\!3\\
      \!\!\!-2
    \end{array}\right)}
}f_{1}^{d+2j}(z)+\frac{d^{3}}{48}\mathbb{M}^{j}_{
    \tiny{\left(\begin{array}[]{c}   %%%% Tr^{3} M %%%%
          3\ \\
          1\ 
    \end{array}\!\!\right|\left.\begin{array}[]{c}
      \!\!\!3\\
      \!\!\!-2
    \end{array}\right)}
}f_{1}^{d+2j}(z)\right)\nn\\
  &+&(2-D)^{4}\left(\frac{d}{8}\mathbb{M}^{j}_{
    \tiny{\left(\begin{array}[]{c}   %%%% Tr M^{4} %%%%
          4\ \\
          1\ 
    \end{array}\!\!\right|\left.\begin{array}[]{c}
      \!\!\!4\\
      \!\!\!-3
    \end{array}\right)}
}f_{1}^{d+2j}(z)+\frac{d^{2}}{8}\left(\frac{1}{4}\mathbb{M}^{j}_{
    \tiny{\left(\begin{array}[]{c}   %%%%  r^{2} M^{2} %%%%
          2\ \\
          2\ 
    \end{array}\!\!\right|\left.\begin{array}[]{c}
      \!\!\!4\\
      \!\!\!-3
    \end{array}\right)}
}f_{1}^{d+2j}(z)+\frac{2}{3}\mathbb{M}^{j}_{
    \tiny{\left(\begin{array}[]{cc}   %%%% Tr M Tr M^{3} %%%%
          1 & 1\ \\
          1 & 3\ 
    \end{array}\!\!\right|\left.\begin{array}[]{c}
      \!\!\!4\\
      \!\!\!-3
    \end{array}\right)}
}f_{1}^{d+2j}(z)\right)\right.\nn\\
  &&\left.+\frac{d^{3}}{32}\mathbb{M}^{j}_{
    \tiny{\left(\begin{array}[]{cc}   %%%% Tr^{2} M Tr M^{2} %%%%
          2 & 1\ \\
          1 & 2\ 
    \end{array}\!\!\right|\left.\begin{array}[]{c}
      \!\!\!4\\
      \!\!\!-3
    \end{array}\right)}
}f_{1}^{d+2j}(z)+\frac{d^{4}}{384}\mathbb{M}^{j}_{
    \tiny{\left(\begin{array}[]{c}   %%%% Tr^{4} M %%%%
          4\ \\
          1\ 
    \end{array}\!\!\right|\left.\begin{array}[]{c}
      \!\!\!4\\
      \!\!\!-3
    \end{array}\right)}
}f_{1}^{d+2j}(z)\right)+O(2-D)^{5}
\end{eqnarray}
The vector entries $\mathbb{M}^{j}$ are to be be taken from subsection \ref{2mDresults}.

%\subparagraph{}
%\begin{equation}\label{3.3.7}
%\qquad \fbox{\rule[-6mm]{0mm}{15mm}$\displaystyle \quad \kappa^{\prime}_0\ =\ \kappa_0\cdot (1-5\cdot \frac{\langle \triangle A \rangle}{A_{base}})\ . \quad $}
%\end{equation}



\subsection{Discussion of possible forms of $\tilde g(D,r)$}
Before making a guess for the exact $\tilde g(r)$ and thus implicitly for
$g(z)$, let us summarize the properties, to which $g(z)$ as a function of the
internal dimension $D$ has to obey.
\begin{itemize}
\item[(I)] In the limit $D=2$ $g(z)$ has to tend to the exact form as it is given
  by (\ref{6.26.1}). Note, that in this limit $g(z)$  diverges
  logarithmically. There is no strong coupling expansion in $D=2$.
\item[(II)] A phantom polymer closed to form a ring corresponds to the limit $D=1$
  of the toroidal manifold. This case has also been
  solved exactly. There are both weak- and strong coupling expansions
  available (\ref{a.25.1},\ref{a.33}). In particular, $g(z)$ tends to a finite
  fixed point value as $z$ tends to $\infty$. In view of the logarithmic
  divergence in $D=2$ and the renormalizability of the theory for internal
  dimensions $0<D<2$ \cite{DDG1,DDG2}, we expect finiteness of $\lim_{z\to
    \infty}=g^{*}$ as soon as $2-D$ becomes positive. Then, we expect
  the existence of a strong coupling expansion starting with the fixed point
  value and a term, which encodes the correction to scaling behavior of the theory:
\end{itemize}
\begin{eqnarray}
  \label{2mD14.1}
  g(D,z)=g^{*}(D)+C_{1}(D,\ln z)\ z^{-\omega_{1}/\varepsilon}+C_{2}(D, \ln z)\
  z^{-\omega_{2}/\varepsilon}+\cdots\ ,
\end{eqnarray}
$\omega_{1}, \omega_{2}, \omega_{2}{-}\omega_{1}>0$ a.s.f.., and the scaling
functions $C_{i}$ diverge at most subexponentially for large arguments.
Powerlaw behavior in the strong coupling limit necessitates the resummation
of the series expansion (\ref{2mD13.12}) to turn into some exponentially growing
function $\tilde g(D,r)$. This can be seen as follows:
\begin{eqnarray}
  \label{2mD14.2}
  f^{2}_{1{+}\frac{\omega}{\varepsilon}}(z)&=&z^{1{+}\frac{\omega}{\varepsilon}}\int_{0}^{\infty}\mbox{d}r\ \exp[-z\
  \rme^{-r}-(1{+}\omega/\E)r]=\Gamma(1{+}\frac{\omega}{\varepsilon})+O(\mathrm{e}^{-z})\nn\\
  \Rightarrow z^{-\omega/\E}&=&\frac{z}{\Gamma(1{+}\frac{\omega}{\varepsilon})}\int_{0}^{\infty}\mbox{d}r\ \exp[-z\
  \rme^{-r}-(1{+}\omega/\E)r]+O(\mathrm{e}^{-z})\nn\\
  &=&\frac{z}{\Gamma(1{+}\frac{\omega}{\varepsilon})}\sum_{n=0}^{\infty}\int_{0}^{\infty}\mbox{d}r\
  \frac{(-\omega/\E\ r)^{n}}{n!}\exp[-z\ \rme^{-r}-r]+O(\mathrm{e}^{-z})\nn\\
  &=&\frac{1}{\Gamma(1{+}\frac{\omega}{\varepsilon})}\sum_{n=0}^{\infty}\frac{(-\omega/\E)^{n}}{n!}f_{1}^{2(n{+}1)}(z)+O(\mathrm{e}^{-z})\ ,
\end{eqnarray}
where it is to be understood that $\omega$ is expanded in powers of $2-D$.\\ 
Let us now postulate a possible form of $\tilde g(D,r)$ obeying to the
demanded properties (I) and (II):
\begin{eqnarray}
  \label{2mD14.3}
  \tilde
  g(D,r)=\mathcal{C}\left(\frac{1-\mbox{e}^{-\frac{\omega}{\E}r\mathcal{S}(2{-}D,r)}}{\omega/\E}\right)^{d/2}\ ,
\end{eqnarray}
where $\mathcal{S}$ is analytic in $D=2$ such that
\begin{equation}
  \label{2mD14.4}
  \mathcal{S}(2{-}D,r)=\sum_{n=0}^{\infty}\mathcal{S}_{n}(r)(2-D)^{n}
\end{equation}
and to each order in $2-D$, there is a Laurent series of the
$\mathcal{S}_{n}(r)$ of the form
\begin{eqnarray}
  \label{2mD14.5}
  \mathcal{S}_{n}(r)=\sum_{j=-\infty}^{0}s_{n_{j}}\ r^{j}\ .
\end{eqnarray}
Furthermore, $\omega/\E$ has some expansion in powers of $2-D$:
\begin{equation}
  \label{2mD14.6}
  \omega/\E=\omega_{n}(2-D)^{n}+O(2-D)^{n{+}1}\ .
\end{equation}
Let us once again stress that powers in $r$ at the level of $\tilde g(r)$
translate into powers in $\log z$, while exponential growth in $r$ is
providing powerlaw behavior for large $z$. Therefore, (\ref{2mD14.3}) obeys to
the demanded properties (I) and (II). Note in particular, that setting
$\mathcal{S}=1$ and integrating over $r$ according to (\ref{2mD13.6}) the expression in parentheses resembles to
\begin{equation}
  \label{2mD14.7}
  \log z=\lim_{D\to 2}\frac{1-z^{-(2{-}D)}}{2-D}\ .
\end{equation}
The r.h.s. expression above grows logarithmically in the limit $D=2$ and
shows up a fixed point together with a strong coupling expansion as soon as $2-D>0$.\\
Let us expand (\ref{2mD13.3}) in powers of $2-D$ taking only the leading term
of $\omega/\E$ into account:
\begin{eqnarray}
  \label{2mD15}
  \tilde g(D,r)=\mathcal{C}\left(r\ \mathcal{S}(2{-}D,r)-\frac{\omega_{n}(2{-}D)^{n}r^{2}}{2!}\mathcal{S}^{2}(2{-}D,r)+\cdots\right)^{d/2}
\end{eqnarray}
Inserting the expansion of $\mathcal{S}(2{-}D,r)$,
\begin{equation}
  \label{2mD16}
  \mathcal{S}(2{-}D,r)=\mathcal{S}_{0}(r)+(2-D)\mathcal{S}_{1}(r)+O(2-D)^{2}\ ,
\end{equation}
and expanding everything provides:
\begin{eqnarray}
  \label{2mD17}
  \tilde g(D,r)&=&\mathcal{C}\ r^{d/2}\ \mathcal{S}_{0}(r)\times\nn\\
  &\times&\left(1+(2{-}D)\frac{d}{2}\frac{\mathcal{S}_{1}(r)}{\mathcal{S}_{0}(r)}+(2{-}D)^{2}\frac{d}{2}\frac{\mathcal{S}_{2}(r)}{\mathcal{S}_{0}(r)}-\frac{\omega_{n}(2{-}D)^{n}d}{4}r\right.\nn\\
  &&\qquad\left.+\frac{(2{-}D)^{2}}{2!}\frac{d}{2}\left(\frac{d}{2}{-}1\right)\left(\frac{\mathcal{S}_{1}(r)}{\mathcal{S}_{0}(r)}\right)^{2}+O(2-D)^{3}\right)
\end{eqnarray}
Matching the coefficients of the expansion above with the result in
(\ref{2mD13.12}) we set:
\begin{eqnarray}
  \label{2mD18}
  \mathcal{C}&=&\frac{1}{\Gamma(\frac{d}{2}{+}1)}\nn\\
  \mathcal{S}_{0}(r)&=&1\nn\\
  \mathcal{S}_{1}(r)&=&\ovC\
  \left(1-\frac{d}{2}\frac{1}{r}+O(1/r^{2})\right)\nn\\
  \omega_{n}&\equiv&\omega_{2}=2\ \Ccsq\ ,
\end{eqnarray}
where we have skipped all higher order poles in the Laurent series of $\mathcal{S}_{1}(r)$\\
From the above considerations it follows that the correction to scaling
exponent $\omega$ reads to lowest order in $2-D$:
\begin{equation}
  \label{2mD19}
  \omega=\E\ 2\Ccsq\ (2-D)^{2}+O(2-D)^{3}\ .
\end{equation}
The result coincides with what we have found in the preceding section from
the calculation within the scheme in the renormalized coupling.\\  
Let us point out, however, that the scheme above is not unambiguous. Going to
the next order it will turn out that the sum in (\ref{2mD14.5}) actually has
to be extended to $j=1$ in order to provide consistency with the expansion
$(\ref{2mD13.12})$. But then, it is not unambiguous, whether the term
proportional to $r^{d/2+1}$ has to be attributed to $\omega_{2}$ or rather to
$\mathcal{S}_{2}(r)$. From some simple general considerations one can easily
see that it is in principle impossible to extract any information about the
correction to scaling exponent from an expansion like (\ref{2mD13.12}) as long
as one has no further information about the behavior of scaling functions like
$\mathcal{S}(r)$ available. The only thing that we can conclude for sure is
that the correction to scaling exponent in an ensemble of a manifold of
toroidal topology and internal dimension close to $D=2$, which 
interacts with a single short-range defect, is proportional to some power of
$2-D$ and that this power is larger
than $1$:
\begin{equation}
  \label{2mD20}
  \omega \sim (2-D)^{n}\ ,\qquad n\geq 2\ .
\end{equation}
Actually, higher than linear powers of $r$ in the exponent (\ref{2mD14.3}) are
fatal, because they destroy the scaling behavior for $z\to \infty$, which was
intended to be encoded in the exponent $\omega/\E$.\\
Therefore, let us discuss another ansatz for $\tilde g(D,r)$ that does not show higher
than linear powers of $r$ in the exponent as in (\ref{2mD14.3}):
\begin{equation}
  \label{2mD21}
  \tilde g(D,r)=\mathcal{C}\left(\frac{1-\mathcal{S}(2{-}D,r)\
  \mbox{e}^{-\frac{\omega}{\E}r}}{\omega/\E}\right)^{d/2}\ ,
\end{equation}
where $\mathcal{S}$ can be written as
\begin{equation}
  \label{2mD21.1}
  \mathcal{S}(2{-}D,r)=1+\frac{\omega}{\E}r\ \tilde \mathcal{S}(2{-}D,r)
\end{equation}
$\tilde \mathcal{S}$ being analytic in $D=2$ such that
\begin{equation}
  \label{2mD22}
  \tilde \mathcal{S}(2{-}D,r)=\sum_{n=1}^{\infty}\tilde\mathcal{S}_{n}(r)(2-D)^{n}
\end{equation}
and to each order in $2-D$, there is a Laurent series of the
$\tilde \mathcal{S}_{n}(r)$ of the form
\begin{eqnarray}
  \label{2mD23}
  \tilde \mathcal{S}_{n}(r)=\sum_{j=-n_{\tiny{\mbox{min}}}}^{n_{\tiny{\mbox{max}}}}\tilde s_{n_{j}}\ r^{j}\ .
\end{eqnarray}
In contrast to (\ref{2mD14.5}) there is no restriction for positive powers in
$r$, but we have to set 
\begin{equation}
  \label{2mD23.1}
  \mathcal{S}_{0}(r)\equiv 1
\end{equation}
from the beginning in order to obey to the properties (I) and (II).\\
Furthermore, $\omega/\E$ has some expansion in powers of $2-D$:
\begin{equation}
  \label{2mD24}
  \omega/\E=\omega_{n}(2-D)^{n}+O(2-D)^{n{+}1}\ .
\end{equation}
Expanding first the expression in parenthesis of (\ref{2mD21}) taking only
the lowest order of $\omega/\E$
into account leads to
\begin{eqnarray}
  \label{2mD24.1}
  \frac{1-\mathcal{S}(2{-}D,r)\ \mbox{e}^{-\frac{\omega}{\E}r}}{\omega/\E}&=&-r\
  \left(\tilde \mathcal{S}_{1}(r)(2-D)+\tilde
  \mathcal{S}_{2}(r)(2-D)^{2}+\tilde S_{3}(r)(2-D)^{3}\right)\nn\\
   &+&r\
  \left(1+\frac{\omega}{\E}r\ \tilde
  \mathcal{S}_{1}(r)(2-D)\right)-\frac{\omega}{\E}\frac{r^{2}}{2!}+\cdots\ ,
\end{eqnarray}
where we have expanded $\mathcal{S}$ up to the third order in $2-D$. $\omega$
is at least of second order in $2-D$. Skipping all higher than third order
terms and expanding everything in (\ref{2mD15}), we obtain:
\begin{eqnarray}
  \label{2mD25}
  \tilde
  g(D,r)&=&\mathcal{C}\
  r^{d/2}\left(1-\tilde\mathcal{S}_{1}(r)(2-D)-\left(\frac{\omega_{2}}{2!}r+\tilde\mathcal{S}_{2}(r)\right)(2-D)^{2}\right.\nn\\
  &&\qquad\left.-\left(-\frac{\omega_{2}}{2!}r\
  \tilde
  S_{1}(r)+\frac{\omega_{3}}{2!}r+\tilde S_{3}(r)\right)(2-D)^{3}\right)^{d/2}\nn\\
  &=&\mathcal{C}\
  r^{d/2}\left(1-\frac{d}{2}\left(\tilde\mathcal{S}_{1}(r)(2-D)+\left(\frac{\omega_{2}}{2!}r+\tilde\mathcal{S}_{2}(r)\right)(2-D)^{2}\right.\right.\nn\\
  &&+\left.\left(-\frac{\omega_{2}}{2!}r\
  \tilde
  S_{1}(r)+\frac{\omega_{3}}{2!}r+\tilde S_{3}(r)\right)(2-D)^{3}\right)\nn\\
  &&+\frac{d}{4}\left(\frac{d}{2}{-}1\right)\left(\tilde
  \mathcal{S}_{1}^{2}(r)(2-D)^{2}+2\tilde S_{1}(r)\left(\tilde
  S_{2}(r)+\frac{\omega_{2}}{2!}\right)(2-D)^{3}\right)\nn\\
  &&+\left.\frac{d}{12}\left(\frac{d}{2}{-}1\right)\left(\frac{d}{2}{-}2\right)\left(\tilde
  S_{1}^{3}(r)(2-D)^{3}\right)\right)
\end{eqnarray}
This time one may set
\begin{eqnarray}
  \label{2mD26}
  \mathcal{C}&=&\frac{1}{\Gamma(\frac{d}{2}{+}1)}\nn\\
  \mathcal{S}_{0}(r)&=&1\nn\\
  \tilde \mathcal{S}_{1}(r)&=&-\ovC\
  \left(1-\frac{d}{2}\frac{1}{r}+O(1/r^{2})\right)\nn\\
  \omega_{2}&=&2\ \Ccsq\ ,
\end{eqnarray}
in order to match with the expansion in (\ref{2mD13.12}). Again, we recover
the same result for $\omega$ as from the previous ansatz and the calculation
within the renormalized coupling:
\begin{equation}
  \label{2mD26.1}
  \omega=\E\ 2\Ccsq\ (2-D)^{2}+O(2-D)^{3}\ .
\end{equation}
However, there still remains the ambiguity in attributing the second order
terms either to $\omega_{2}$ or to $\tilde\mathcal{S}_{2}$ as it is obvious
from (\ref{2mD25}). Nevertheless, making any choice for $\omega_{2}$ it is
clear, that the third order term $\tilde \mathcal{S}_{3}(r)$ has sufficiently many
parameters free to get the third order reconciliated with the second order
a.s.f.. Of course, the behavior of $\mathcal{S}(r)$ for large values of $r$ is supposed
to stay subexponential. This corresponds to the same demand on the scaling
functions $C_{i}$ in (\ref{2mD14.1}). There is some additional information
about the scaling function $\mathcal{S}(r)$ necessary, in order to enable an
unambiguous analysis of the scaling behavior for dimensions D close to $2$
. From the structure of the expansion (\ref{star!!!}), however, it is not
possible to make a guess about the precise properties of $\mathcal{S}(r)$. A
way out might be a complete solution of the problem in $D=1$ within the $2-D$ expansion.

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\newpage \section{Discussion and conclusion}\label{conclusion}
In this work, we have discussed the perturbation expansion of 
a $D$-dimensional elastic manifold interacting via a $\delta
$-interaction with a  fixed point in embedding space. This calculation
can be done exactly for $D=1$, but becomes non-trivial for $D\neq 1$. 
Interestingly and quite surprisingly, it simplifies considerably in
the limit of $D\to2$, if one decides to work at finite $\E $. In that limit
we were able to obtain an explicit expression for the renormalized
coupling as function of the bare one, in terms of a non-trivial
series. Analysis of this series shows that in the limit of large values of the
bare coupling $z$, corresponding to strong repulsion or equivalently to large
membrane sizes, the renormalized coupling $g$ diverges logarithmically as a
function of $z$. This yields a vanishing correction-to-scaling exponent
$\omega$ in the limit $D=2$. Furthermore, dependent on the dimensionality $d$
of the embedding space the $\beta$-function shows up a zero at infinite bare
coupling as $0\leq d<2$. Then, the renormalization flow tends to a fixed
point, and the theory becomes scale invariant in this limit. Due to the
logarithmic divergence of the renormalized coupling, however, the
corresponding zero of the $\beta$-function in terms of the latter is, too,
shifted to infinity. This is quite a remarkable result, because it reveals
that the strong coupling behavior of a "phantom", polymerized membrane in a
crumpled phase interacting with a $\delta$-like defect in embedding space is
only accessible through a non-perturbative treatment and always largely
deviates qualitatively from any finite loop expansion, be it within a minimal
subtraction scheme or at finite $\E$.\\
Also, it is important to note that this result is completely independent of
the regularization procedure. This does no longer hold true beyond the leading
order, which should be accessible to an expansion in $2-D$. We constructed its
first order in a specific regularization scheme in
\cite{PinnowWiese2001}. While this reproduces qualitatively correctly the
known result in $D=1$ (and even exactly for $d=0$), it suffers from its
inconsistent renormalization scheme, which neglects the boundaries of a finite
manifold. We used a hard cutoff in position space, while working with the
infinite $D$-space correlator. It seems that only in an $\E$-expansion this
procedure is systematic. Now, we have overcome this problem by constructing
the $2-D$-expansion on a manifold of toroidal shape of finite size, thus
imposing periodic boundary conditions on the field. There is no further
infrared cutoff necessary. We have carried out the expansion of the renormalized
coupling up to the fourth order in $2-D$. This reveals part of the general
structure of the expansion. It is important to point out that in considering
$g$ as a function of the bare coupling the limits $D\to 2$ and strong
coupling ($z\to \infty$) can not be interchanged. While $g$ tends to infinity
as $z$ does in $D=2$, we expect finiteness of this limit as soon as $2-D>0$
and the existence of a strong coupling expansion as found for polymers
($D=1$). We were able to guess an exact $g(D,z)$ as a function of $z$ and the
internal dimension $D$, which satisfies these properties and which can be
reconciliated with the available expansion in $2-D$ by an appropriate matching
of its free parameters. However, this matching is not unambiguous and
therefore does not provide a decent determination of the prefactor in the
relation
\begin{equation}
  \label{con1}
  \omega\sim (2-D)^{2}\ .
\end{equation}



\begin{comment}
Analysis of this series shows that the fixed point lies at
infinity in the bare coupling, a limit in which we were able to derive an
asymptotic expansion for the renormalized coupling as a function of
the bare one. This yields a vanishing exponent $\omega $ in the
limit of 
$D\to 2$. Also it is important to note that this result is completely
independent of the regularization procedure.
  This does no longer hold
true  beyond the leading order, which should be accessible to 
an expansion in $(2-D)$. We here constructed its first order in a
specific regularization scheme. While this reproduces qualitatively
correctly the known results in $D=1$ (and even exactly for $d=0$), it
also  shows by its dependence 
on the renormalization scheme (here working with the $\beta
$-function either in terms of the renormalized or bare coupling) that
this expansion in $(2-D)$ is  not systematic. Even though
experience with this new kind of expansion is still lacking, this is
likely to be caused by the use of  a hard cutoff in position space while
working with the correlator of an infinite membrane. It seems that
only in an $\E$-expansion this procedure is systematic. Work is in
progress \cite{PinnowWieseProgress} to study the model on a sphere or
torus of finite size, such 
that no further infrared cutoff is necessary. 
This should lead to exact results beyond the leading order and provide
for the crossover between polymers and membranes.
\end{comment}
The correction-to-scaling exponent $\omega$ is closely related to observables,
which are accessible to Monte-Carlo simulations. This is for instance the
monomer density at the potential of a membrane, which is pinned by one of its
points close to the defect.\\

While results for the pinning problem are interesting in its own, the
main motivation is certainly to obtain a better understanding of 
self-avoiding polymerized membranes. This model is well-behaved
physically, since  its fractal dimension $d_{f}$ is bounded by the
dimension of the embedding space $d_{f}<d$.  Preliminary studies
\cite{PinnowWieseProgress} indicate that  this problem can also
be attacked by the methods developed in this work. This would be
very welcome to check the 2-loop calculations
\cite{WieseDavid1997,DavidWiese1996} and the large-order behavior
\cite{DavidWiese1998} on one side and numerical results (e.g.\
\cite{BowickTravesset2001})  on the other.  






 

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\begin{comment}
  \newpage \section{Higher order terms in the $(2-D)$- expansion on the torus }\label{2mDtorus}

Therefore, to second order in $2-D$ we find for the renormalized coupling $g$
in terms of the $f_{k}^{d}$'s (\ref{star}):
\begin{eqnarray}\label{2mD5}
g (z) &=& f_{1}^{{d+2}} (z) +\frac{ (2-D)d}{2} f_{2}^{d+2} (z)
\overline{\C}+   \frac{(2-D)^{2} d}{4} f_{2}^{d+2} (z)
\overline{\C^2}\nonumber \\
&&+ \frac{(2-D)^{2}d}{4} \left[f_{3}^{d+4}(z)-f_{3}^{d+2} (z)
\right]\overline{\C}^{2} + \frac{(2-D)^{2}d}{4} \left[f_{3}^{d+2} (z)-2f_{2}^{d+4}(z)
\right]\overline{\C^{2}}\nonumber \\
&&+ \frac{(2-D)^{2}d^{2}}{8}\left[f_{3}^{d+2} (z)+f_{3}^{d+4}(z)
\right]\overline{\C}^{2}- \frac{(2-D)^{2}d^{2}}{4}f_{2}^{d+4}(z) \overline{\C^{2}}
+\dots \ , 
\end{eqnarray}
or in terms of the connected correlator (\ref{star!}):
\begin{eqnarray}
  \label{2mD5.1}
  g(z)&=& f_{1}^{{d+2}} (z) + (2{-}D)\frac{d}{2}\overline{\C}\ f_{2}^{d+2} (z)\nn\\
 &&-(2{-}D)^{2}\frac{d}{4}\left[2\overline{\C_{c}^{2}}\
 f_{1}^{d+4}(z)+(\overline{\C}^{2}{-}4\overline{\C_{c}^{2}})\
 f_{1}^{d+2}(z)+\overline{\C_{c}^{2}}\
 f_{2}^{d}(z)+(2\overline{\C_{c}^{2}}{-}\overline{\C}^{2})\
 f_{1}^{d}(z)\right]\nn\\
 &&-(2{-}D)^{2}\frac{d^{2}}{8}\left[2\overline{\C_{c}^{2}}\
 f_{1}^{d+4}(z)-(2\overline{\C_{c}^{2}}{+}\overline{\C}^{2})\
 f_{1}^{d+2}(z)+\overline{\C}^{2}\ f_{2}^{d}(z)+\overline{\C}^{2}\
 f_{1}^{d}(z)\right]\nn\\
 &&\qquad + \ O(2-D)^{3}\ .
\end{eqnarray}

Let us give the expansion of the renormalized coupling $g(z)$ in powers of
$2{-}D$ up to the fourth order taking only the dominant terms into account:
\begin{eqnarray}
  \label{2mD14}
  g(z)=(\ln z)^{d/2}\
   \left[\frac{1}{\Gamma(\frac{d{+}2}{2})}\right.&+&(2-D)\frac{\left(\frac{d}{2}\right)}{\Gamma(\frac{d{+}2}{2})}\overline{\C}+(2-D)^{2}\frac{\left(-\frac{d}{2}-\frac{d^{2}}{4}\right)}{\Gamma(\frac{d{+}4}{2})}\ \Ccsq \ (\ln
  z)\nonumber\\
     &+&(2-D)^{3}\frac{\left(\frac{d^{2}}{2}+\frac{d^{3}}{12}\right)}{\Gamma(\frac{d{+}6}{2})}(\Cccub-4\Cctri)(\ln z)^{2}\nonumber\\
   &-&(2-D)^{4}\frac{\left(d+\frac{3}{4}d^2+\frac{d^{3}}{4}+\frac{d^{4}}{48}\right)}{\Gamma(\frac{d{+}8}{2})}\times\nonumber\\
   &&\hspace{-1cm}\times(222\overline{\C}^{4}+6\Csq\Ccsq+3\Ccsq^{2}-\Ccbisq+24\Ccblobt-36\Ccdia)(\ln z)^{3}\nonumber\\
 &+&\left.\cdots\right]\nonumber\\
 \qquad &&\qquad
\end{eqnarray}
Expecting powerlaw behavior of the $\beta$- function as soon as $2-D$ becomes
finite, what we need is an expansion of $g(z)$ in powers of $\ln z$ with
coefficients being expanded in powers of $2-D$. Therefore, let as state $g(z)$
up to the third order in $2-D$ taking all divergent terms into account:
\begin{eqnarray}
  \label{2mD15}
  g(z)=f_{1}^{d+2}(z)&+&(2-D)\frac{d}{2}\ \overline{\C}\ f_{2}^{d+2}(z)\nn\\
  &-&(2-D)^{2}\frac{d}{4}\left[2\ \Ccsq\
  f_{1}^{d+4}(z)+(\Csq-4\Ccsq)f_{1}^{d+2}(z)\right]\nn\\
  &-&(2-D)^{2}\frac{d^{2}}{8}\left[2\ \Ccsq\ f_{1}^{d+4}(z)-(2\
  \Ccsq+\Csq)f_{1}^{d+2}(z)\right]\nn\\
  &+&(2-D)^{3}\frac{d}{4}\left[4(\Cccub-4\ \Cctri)f_{1}^{d+4}(z)+(-10\ \Cccub+36\
  \Cctri+6\overline{\C}\Ccsq)f_{1}^{d+2}(z)\right]\nn\\
  &+&(2-D)^{3}\frac{d^{2}}{8}\left[4(\Cccub-4\ \Cctri)f_{1}^{d+6}(z)+(-8\
  \Cccub+32\ \Cctri+2\ \overline{\C}\Ccsq)f_{1}^{d+4}(z)\right.\nn\\
  &&\qquad \qquad \qquad \left.+(6\ \Cccub-20\ \Cctri+2\ \overline{\C}\Ccsq-6\
  \overline{\C}^{3})f_{1}^{d+2}(z)\right]\nn\\
  &+&(2-D)^{3}\frac{d^{3}}{48}\left[4(\Cccub-4\ \Cctri)f_{1}^{d+6}(z)+(-4\
  \Cccub+24\ \Cctri-6\ \overline{\C}\Ccsq)f_{1}^{d+4}(z)\right.\nn\\
  &&\qquad \qquad \qquad \left.+(-8\ \Cctri+12\
  \overline{\C}\Ccsq+\overline{\C}^{3})f_{1}^{d+2}(z)\right]\nn\\
  &&
\end{eqnarray}
From the last expression we easily obtain the $\beta$- function taking
advantage of (\ref{recursion}). Collecting all terms, which contribute to the
same power of $\ln z$, respectively, and taking only divergent terms into account, we find
for $\beta(z)$:
\begin{eqnarray}
  \label{2mD16}
  \beta(z)&=&-\varepsilon\left[\frac{(\ln
  z)^{d/2}}{\Gamma(\frac{d}{2}{+}1)}\left(-(2-D)^{2}\left[\frac{d}{2}\Ccsq+\frac{d^{2}}{4}\Ccsq\right]+(2-D)^{3}\left[\left(d+\frac{d^{2} \gamma}{2}\right)(\Cccub-4\ \Cctri)\right.\right.\right.\nn\\
  &&\qquad \qquad \qquad \left.\left.+\frac{d^{3}}{48}(4(\Cccub-4\
  \Cctri)\gamma+(-4\ \Cccub+24\ \Cctri-6\
  \overline{\C}\Ccsq))\right]\right)\nn\\
  &+&\frac{(\ln
  z)^{d/2+1}}{\Gamma(\frac{d}{2}{+}2)}\left((2-D)^{3}\left(\frac{d^{2}}{2}+\frac{d^{3}}{12}\right)(\Cccub-4\
  \Cctri)\right)\nn\\
  &-&\left.\frac{(\ln
  z)^{d/2+2}}{\Gamma(\frac{d}{2}{+}3)}\left((2-D)^{4}\left(d+\frac{3}{4}d^2+\frac{d^{3}}{4}+\frac{d^{4}}{48}\right)(222\overline{\C}^{4}+6\Csq\Ccsq+3\Ccsq^{2}-\Ccbisq+24\Ccblobt-36\Ccdia)\right)\right]\
  ,\nn\\
\end{eqnarray}
where to fourth order in $2-D$ we have only taken the most divergent terms
into account. Of course, we might have given an expansion like (\ref{2mD16}) already at the
level of the renormalized coupling (\ref{2mD15}).



\subsection{Discussion}
Let us start our analysis with a guess for the renormalized coupling $g(z)$
for small $2-D$ obeying to the following properties:
We know the behavior of $g(z)$ exactly in the limit $D=2$. In that case $g$ diverges logarithmically in the strong coupling
limit for all embedding dimensions $d$. From what follows that the correction
to scaling exponent $\omega$ tends to zero then. On the other hand we know of
an other completely solvable limit, which is $D=1$ corresponding to
fluctuating flexible lines or polymers. Having a look at the strong coupling
expansion of the renormalized coupling $g$ in this case (\ref{a.26}), we find
a finite limit $g^{*}$ and the dominant behavior for large values of the bare
coupling is isolated in the first term of the expansion ($\sim
z^{-1}$). Furthermore, we may conclude from renormalizability in dimensions
$0<D<2$ that $g(z)$ has to stay finite in the limit $z{\to}\infty$ as soon as
$2-D>0$. We thus expect a pole in $2-D$ for the strong coupling expansion of
$g(z)$.\\
Let us now make a simple guess for $g(z)$ with $2-D>0$, obeying to all
demanded properties and having an expansion like (\ref{2mD15}). Be $S$ any
function diverging only subexponentially for large arguments. Then
\begin{eqnarray}
  \label{2mD17}
  g(z):=\left(\frac{S(\ln z)\ z^{\omega/\varepsilon}}{1+(2{-}D)^{2}\ S(\ln z)\
  z^{\omega/\varepsilon
  }}\right)^{d/2}=\frac{1}{(2-D)^{d}}\left(1+\frac{(S(\ln
  z))^{-1}z^{-\omega/\varepsilon}}{(2{-}D)^{2}}\right)^{-d/2}\nn\\
\end{eqnarray}
This leads to the following strong coupling expansion:
\begin{eqnarray}
  \label{2mD18}
  g(z)=\frac{1}{(2-D)^{d}}\left(1-\frac{d}{2}\frac{(S(\ln
  z))^{-1}z^{-\omega/\varepsilon}}{(2{-}D)^{2}}+\cdots\right)\ ,
\end{eqnarray}
where $\omega/\varepsilon$ has an expansion in $2-D$ as well as the scaling function
$S(\ln z)$. Let us now look upon how $g$ is expanded in powers of $2-D$. Since
\begin{displaymath}
  z^{\omega/\varepsilon}=1+\omega/\varepsilon\ \ln z+\cdots\ ,
\end{displaymath}
we find:
\begin{eqnarray}
  \label{2mD19}
  g(z)&=&\left(\frac{S(\ln z)(1+\omega/\varepsilon\ \ln
  z+\cdots)}{1+(2{-}D)^{2}\ S(\ln z)(1+\omega/\varepsilon \ \ln
  z+\cdots)}\right)^{d/2}\nn\\
   &=&(S(\ln z))^{d/2}\ \left(1+\frac{d}{2}(\omega/\varepsilon-(2{-}D)S(\ln
  z))\ln z+\cdots\right)\ .
\end{eqnarray}
Note that $(S(\ln z))^{d/2}$ may as well be expanded in powers of $2-D$ such that in
the limit $D=2$ one may recover the form (\ref{star}).\\
From the strong coupling expansion (\ref{2mD18}) it is clear that
$\omega$ is equaling the correction to scaling exponent.\\
\\
Let us now turn to a discussion about how to extract information about the correction to
scaling exponent $\omega$ starting from a generalized form of the strong
coupling expansion as given in (\ref{2mD18}). Be
$S$ again any scaling function diverging only subexponentially for large
arguments and $\omega$ the correction to scaling exponent having some
expansion in powers of $2-D$. Then, the strong coupling expansion of the
renormalized coupling $g(z)$ might
be of the following general form:
\begin{eqnarray}
  \label{2mD20}
  g(z)=g^{*}+ S(\ln z)\ z^{-\omega/\varepsilon}+\cdots\ ,
\end{eqnarray}
where both the fixed point $g^{*}$ and the scaling function $S$ might have any
Laurent-expansion in powers of $2-D$. 
From the definition of $\beta(z)$ and $\omega$ it follows immediately that
\begin{eqnarray}
  \label{2mD21}
  \beta(z)=-\varepsilon z^{-\omega/\varepsilon}\left[\frac{\partial S(\ln
  z)}{\partial (\ln z)}-\frac{\omega}{\varepsilon}\ S(\ln z)\right]
\end{eqnarray}
and
\begin{equation}
  \label{2mD22}
  \omega(z)=-\varepsilon z\ \frac{\partial \beta(z)}{\partial z}\Bigg/
  \beta(z)=-\varepsilon\left[\frac{\frac{\partial^{2}S(\ln z)}{\partial (\ln
  z)^{2}}-\frac{\omega}{\varepsilon}\frac{\partial S(\ln z)}{\partial (\ln
  z)}}{\frac{\partial S(\ln z)}{\partial (\ln
  z)}-\frac{\omega}{\varepsilon}S(\ln z)}-\frac{\omega}{\varepsilon}\right]\ . 
\end{equation}
From the properties of $S$ it clearly follows that $\lim_{z\to
  \infty}\omega(z)=\omega$. Furthermore, the bracketed expression in
  (\ref{2mD22}) has a Taylor-expansion in powers of $2-D$. Assuming
\begin{eqnarray}
  \label{2mD23}
  \omega(z)=\omega_{n}(2-D)^{n}+O(2-D)^{n{+}1}
\end{eqnarray}
and
\begin{eqnarray}
  \label{2mD24}
  S(\ln z)=S_{0}(\ln z)+(2-D)S_{1}(\ln z)+\cdots\ ,
\end{eqnarray}
which is legitimate, since any divergent prefactor (in $2-D$) drops out, let
us expand the term being contributed from the scaling function in
(\ref{2mD22}):
\begin{eqnarray}
  \label{2mD23}
  \frac{\frac{\partial^{2}S(\ln z)}{\partial (\ln
  z)^{2}}-\frac{\omega}{\varepsilon}\frac{\partial S(\ln z)}{\partial (\ln
  z)}}{\frac{\partial S(\ln z)}{\partial (\ln
  z)}-\frac{\omega}{\varepsilon}S(\ln z)}&\simeq & \frac{1}{\frac{\partial
  S_{0}(\ln z)}{\partial (\ln z)}}\frac{\frac{\partial^{2}S_{0}(\ln
  z)}{\partial (\ln z)^{2}}+(2{-}D)\frac{\partial^{2}S_{1}(\ln z)}{\partial
  (\ln z)^{2}}-\frac{\omega}{\varepsilon}\left(\frac{\partial S_{0}(\ln
  z)}{\partial (\ln z)}\right)}{1+(2{-}D)\frac{\partial S_{1}(\ln z)}{\partial
  (\ln z)}\bigg/\frac{\partial S_{0}(\ln z)}{\partial (\ln
  z)}-\frac{\omega}{\varepsilon}S_{0}(\ln z)\bigg/\frac{\partial S_{0}(\ln
  z)}{\partial (\ln z)}}\nn\\
  &\simeq & \frac{\frac{\partial^{2}S_{0}(\ln
  z)}{\partial (\ln z)^{2}}}{\frac{\partial
  S_{0}(\ln z)}{\partial (\ln z)}}\left(1-(2{-}D)\frac{\frac{\partial
  S_{1}(\ln z)}{\partial (\ln z)}}{\frac{\partial S_{0}(\ln z)}{\partial (\ln z)}}+\frac{\omega}{\varepsilon}\frac{S_{0}(\ln z)}{\frac{\partial
  S_{0}(\ln z)}{\partial (\ln z)}}\right)+(2{-}D)\frac{\frac{\partial^{2}S_{1}(\ln z)}{\partial
  (\ln z)^{2}}}{\frac{\partial
  S_{0}(\ln z)}{\partial (\ln z)}}-\frac{\omega}{\varepsilon}\nn\\
\end{eqnarray}
Therefore, in addition to $(\ln z)$-dependent terms there has been generated
another $-\omega/\varepsilon$. This is due to the expansion of the denominator
in powers of $2-D$. Actually, the leading term is less dominant for large
$(\ln z)$, which would suppress this contribution in the limit $z\to
\infty$.\\
Note that the factor $-\varepsilon$ ahead is always exact, but only
$\omega/\varepsilon$ is expanded in $2-D$.\\
\\
We are now going to generalize the discussion above, expanding
$z^{-\omega/\varepsilon}$ from the beginning and taking also subleading terms
into account. Once again, be $S$ as well as $\tilde S$ any scaling functions
diverging at most subexponentially for large arguments. Then, we start our
analysis from the strong coupling expansion of the renormalized coupling being
of the following form:
\begin{eqnarray}
  \label{2mD24}
  g(z)&=&g^{*}+S(\ln z)\ z^{-\omega/\E}+\tilde S(\ln z)\
  z^{(\omega+\delta\omega)/\E}+\cdots\nn\\
  &\simeq&g^{*}+(S(\ln z)+\tilde S(\ln z))(1-\frac{\omega}{\E}\ln
  z)-\frac{\delta \omega}{\E}\tilde
  S(\ln z)\ln z\ .
\end{eqnarray}
The expansion of the $\beta$-function reads:
\begin{eqnarray}
  \label{2mD25}
  \beta(z)&=&-\E\left[\left(\frac{\partial S(\ln z)}{\partial (\ln
  z)}+\frac{\partial \tilde S(\ln z)}{\partial (\ln
  z)}\right)\left(1-\frac{\omega}{\E}\ln z\right)-\frac{\omega}{\E}(S(\ln
  z)+\tilde S(\ln z))\right.\nn\\
   &&\hspace{5cm}\left.-\frac{\delta \omega}{\E}\left(\frac{\partial \tilde
  S(\ln z)}{\partial (\ln z)}\ln z+\tilde S(\ln z)\right)\right]
\end{eqnarray}
In order to calculate $\omega(z)$ we need
\begin{eqnarray}
  \label{2mD26}
  z\frac{\partial \beta(z)}{\partial z}&=&-\E\left[\left(\frac{\partial^{2}
  S(\ln z)}{\partial (\ln z)^{2}}+\frac{\partial^{2} \tilde S(\ln z)}{\partial
  (\ln z)^{2}}\right)\left(1-\frac{\omega}{\E}\ln
  z\right)-2\frac{\omega}{\E}\left(\frac{\partial S(\ln z)}{\partial (\ln
  z)}+\frac{\partial \tilde S(\ln z)}{\partial (\ln z)}\right)\right.\nn\\
  &&\hspace{5cm}\left.-\frac{\delta \omega}{\E}\left(\frac{\partial^{2} \tilde S(\ln
  z)}{\partial (\ln z)^{2}}\ln z+2\frac{\partial \tilde S(\ln z)}{\partial
  (\ln z)}\right)\right]
\end{eqnarray}
and
\begin{eqnarray}
  \label{2mD27}
  \frac{1}{\beta(z)}&=&\frac{1}{-\E\left(\frac{\partial
  S(\ln z)}{\partial (\ln z)}+\frac{\partial \tilde S(\ln z)}{\partial
  (\ln z)}\right)}\left(1+\frac{\omega}{\E}\ln
  z+\frac{\omega}{\E}\frac{\left(S(\ln
  z)+\tilde S(\ln z)\right)}{\left(\frac{\partial
  S(\ln z)}{\partial (\ln z)}+\frac{\partial \tilde S(\ln z)}{\partial
  (\ln z)}\right)}\right.\nn\\
   &&\hspace{5cm}\left.+\frac{\delta \omega}{\E}\frac{\left(\frac{\partial \tilde
  S(\ln z)}{\partial (\ln z)}\ln z+\tilde S(\ln z)\right)}{\left(\frac{\partial
  S(\ln z)}{\partial (\ln z)}+\frac{\partial \tilde S(\ln z)}{\partial
  (\ln z)}\right)}\right)
\end{eqnarray}
Combining the last two equations one arrives at
\begin{eqnarray}
  \label{2mD28}
  \omega(z)&=&-\E\left[\frac{\left(\frac{\partial^{2}
  S(\ln z)}{\partial (\ln z)^{2}}+\frac{\partial^{2} \tilde S(\ln z)}{\partial
  (\ln z)^{2}}\right)}{\left(\frac{\partial
  S(\ln z)}{\partial (\ln z)}+\frac{\partial \tilde S(\ln z)}{\partial
  (\ln z)}\right)}\left(1-\frac{\omega}{\E}\ln
  z\right)-2\frac{\omega}{\E}+\frac{\omega}{\E}\frac{\left(\frac{\partial^{2}
  S(\ln z)}{\partial (\ln z)^{2}}+\frac{\partial^{2} \tilde S(\ln z)}{\partial
  (\ln z)^{2}}\right)\ln z}{\left(\frac{\partial
  S(\ln z)}{\partial (\ln z)}+\frac{\partial \tilde S(\ln z)}{\partial
  (\ln z)}\right)}\right.\nn\\
  &+&\frac{\omega}{\E}\frac{\left(S(\ln
  z)+\tilde S(\ln z)\right)\left(\frac{\partial^{2}
  S(\ln z)}{\partial (\ln z)^{2}}+\frac{\partial^{2} \tilde S(\ln z)}{\partial
  (\ln z)^{2}}\right)}{\left(\frac{\partial
  S(\ln z)}{\partial (\ln z)}+\frac{\partial \tilde S(\ln z)}{\partial
  (\ln z)}\right)^{2}}\nn\\
  &-&\left.\frac{\delta \omega}{\E}\frac{\left(\frac{\partial^{2} \tilde S(\ln
  z)}{\partial (\ln z)^{2}}\ln z+2\frac{\partial \tilde S(\ln z)}{\partial
  (\ln z)}\right)}{\left(\frac{\partial
  S(\ln z)}{\partial (\ln z)}+\frac{\partial \tilde S(\ln z)}{\partial
  (\ln z)}\right)}+\frac{\delta \omega}{\E}\frac{\left(\frac{\partial^{2}
  S(\ln z)}{\partial (\ln z)^{2}}+\frac{\partial^{2} \tilde S(\ln z)}{\partial
  (\ln z)^{2}}\right)\left(\frac{\partial \tilde
  S(\ln z)}{\partial (\ln z)}\ln z+\tilde S(\ln z)\right)}{\left(\frac{\partial
  S(\ln z)}{\partial (\ln z)}+\frac{\partial \tilde S(\ln z)}{\partial
  (\ln z)}\right)^{2}}\right]\nn\\
\end{eqnarray}
From the above formula it is clear that it is impossible to analyze, which
terms belong to the expansion of $\omega$. There is not only the constant term
$-2\omega/\E$, which contributes, but also others, which have constant parts
due to the properties of the scaling functions. \\
The same analysis as above can be repeated starting from a formula like
(\ref{2mD17}) resulting in similar problems. Of course, the situation
simplifies as soon as we enter some additional information about the scaling
functions. Starting from the ansatz (\ref{2mD17}), the precise form of $S$
becomes clear, since we know the limit $D=2$. Unfortunately, (\ref{2mD17}) can
not be correct.\\
\\
Let us point out that it is easier to obtain a guess for the exact relation
between the renormalized and the bare coupling as a function of $2-D$ as
one starts from the integral representation (\ref{6.28}):
\begin{displaymath}
  \sum_{N=0}^{\infty}\frac{(-z)^{N}}{N!(k+N)^{d/2}}=\frac{1}{\Gamma(\frac{d}{2})}\int_{0}^{\infty}\mbox{d}r\
  r^{d/2-1}\exp[-z\
  \rme^{-r }-k r]\ .
\end{displaymath}
Then, power-law behavior in the bare coupling $z$ can be found from an
expansion in $2-D$ in terms of the functions $f_{k}^{d}$ in the following
way:
\begin{eqnarray}
  \label{2mD28}
  z^{-\omega}&=&z\int_{0}^{\infty}\mbox{d}r\ \exp[-z\
  \rme^{-r}-(1{+}\omega)r]=\sum_{n=0}^{\infty}\int_{0}^{\infty}\mbox{d}r\
  \frac{(-\omega r)^{n}}{n!}\exp[-z\ \rme^{-r}-r]\nn\\
  &=&\sum_{n=0}^{\infty}\frac{(-\omega)^{n}}{n!}f_{1}^{2(n{+}1)}(z)\ ,
\end{eqnarray}
where it is to be understood that $\omega$ is expanded in powers of $2-D$.\\
Let us write the exact $g(z)$ in terms of $f_{1}^{d\prime}$s,which is always
possible thanks to the relation (\ref{fkdreduce}):
\begin{equation}
  \label{2mD29}
  g(z)=z\int_{0}^{\infty}\mbox{d}r\ \tilde g(r)\ \exp[-z\
  \rme^{-r}-r]
\end{equation}
and let us then give an appropriate guess for $\tilde g(r)$:
\begin{eqnarray}
  \label{2mD30}
  \tilde g(r)=A(2-D)\left(\frac{1-P(2-D,r)\rme^{-\omega
  r}}{\omega}\right)^{d/2}\ ,
\end{eqnarray}
where both the prefactor $A$ and the function $P(r)$ are expanded in
$2-D$. Furthermore, $P(r)$ has some Laurent-expansion in $r$ to each order in
$2-D$. The expansion of $\omega$ starts from some power $n$ as
\begin{equation}
  \label{2mD31}
  \omega=\omega_{n}(2-D)^{n}+\cdots\ ,\qquad n\geq 1\ .
\end{equation}
Clearly, for finite $2-D$ (\ref{2mD30}) leads to a strong coupling expansion
in $z$. On the other hand, fixing $P(0,r)=1$, the expression can be expanded in
powers of $2-D$:
\begin{eqnarray}
  \label{2mD32}
  \tilde
  g(r)=(A(0)+(2-D)A^{\prime}(0)+\cdots)r^{d/2}\left(1-\frac{d}{2}\left(\omega
  r+(2-D)P^{\prime}(0,r)+\cdots\right)\right)\ .
\end{eqnarray}
The exact $\tilde g(r)$ in powers of $2-D$ reads (\ref{star!}):
\begin{eqnarray}
  \label{2mD33}
  \tilde
  g(r)=\frac{r^{d/2}}{\Gamma(\frac{d}{2}{+}1)}+(2-D)\frac{d}{2}\ovC\left(\frac{r^{d/2}}{\Gamma(\frac{d}{2}{+}1)}-\frac{r^{d/2-1}}{\Gamma(\frac{d}{2})}\right)-(2-D)^{2}\left(\frac{d}{2}\left(1+\frac{d}{2}\right)\Ccsq\right)\frac{r^{d/2+1}}{\Gamma(\frac{d}{2}{+}2)}\ ,\nn\\
\end{eqnarray}
where to order $(2-D)^{2}$ we have only taken the dominant contribution into
account. Comparing with the expansion of the ansatz (\ref{2mD32}), we set:
\begin{eqnarray}
  \label{2mD34}
  A(0)&=&\frac{1}{\Gamma(\frac{d}{2}+1)}\nn\\
  P^{\prime}(0,r)=\ovC\left(1-\frac{d}{2}\frac{1}{r}\right)\nn\\
  \omega=(2-D)^{2}\Ccsq
\end{eqnarray}

\end{comment}

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\begin{appendix}
\newpage \section{Derivation of the kernel}
Starting from the free theory with Hamiltonian
\begin{equation}
  \label{c.0}
  \mathcal{H}_{0}=\frac{1}{2-D}\int_{x}\frac{1}{2}(\nabla r(x))^{2}
\end{equation}
yields the difference correlator
\begin{equation}
  \label{c.1}
  C^{kl}(x_{1}{-}x_{2}):=\left<\frac{1}{2}[r_{k}(x_{1}){-}r_{l}(x_{2})]^{2}\right>_{0}=\delta^{kl}C(x_{1}{-}x_{2})
\end{equation}
with
\begin{equation}
  \label{c.2}
  C(x_{1}{-}x_{2})=|x_{1}{-}x_{2}|^{2-D}\ .
\end{equation}
It can be shown as in \cite{WieseHabil} to be a solution in the sense of distributions
of the Laplacian equation
\begin{equation}
  \label{c.3}
  \Delta C(x)=(2-D)S_{D}\delta^{D}(x)\ .
\end{equation}
The difference correlator may be considered as the analytical continuation to
values of $D<2$ of
the usual two-point correlator. Let us shortly derive (\ref{c.2}) starting
from the well known form of the kernel in Fourier space
\begin{equation}
  \label{c.4}
  G(\vec k_{1},\vec k_{2})=\frac{\delta^{D}(\vec k_{1}{+}\vec k_{2})}{(\vec k_{1})^{2}}\ ,
\end{equation}
such that
\begin{equation}
  \label{c.5}
  G(x)=\frac{1}{(2\pi)^{D}}\int \mbox{d}^{D}k\ \frac{\exp[i\vec k \vec x]}{k^{2}}
\end{equation}
Introducing a Schwinger parametrization and splitting the integration over the
momenta into a part integrating over the component, which is parallel to $\vec
x$, and its complementary part leads to
\begin{eqnarray}
  \label{c.6}
  G(x)=\frac{1}{(2\pi)^{D}}\int\limits_{0}^{\infty}\mbox{d}s\int\limits_{-\infty}^{\infty}\mbox{d}k_{\parallel}\int
  \mbox{d}^{D-1}k_{\perp}\
  \mbox{e}^{ik_{\parallel}x-s(k_{\parallel}^{2}+k_{\perp}^{2})}\nonumber\\
  =\frac{1}{(2\pi)^{D}}\int\limits_{0}^{\infty}\mbox{d}s\int\limits_{-\infty}^{\infty}\mbox{d}k_{\parallel}\
  \mbox{e}^{-sk_{\parallel}^{2}+ik_{\parallel}x}\left(\int\limits_{-\infty}^{\infty}\mbox{d}k_{\perp}\
  \mbox{e}^{-sk_{\perp}^{2}}\right)^{D-1}\nonumber\\
  =\frac{1}{(2\pi)^{D}}\int\limits_{0}^{\infty}\mbox{d}s\
  \left(\frac{\pi}{s}\right)^{D/2}\mbox{e}^{-\frac{x^{2}}{4s}}=\frac{1}{(2\sqrt{\pi})^{D}}\int\limits_{0}^{\infty}\mbox{d}s^{\prime}\
  s^{\prime D/2-2}\mbox{e}^{-s^{\prime}\frac{x^{2}}{4}}\nonumber\\
  =\frac{1}{(2\sqrt{\pi})^{D}}\Gamma\left(\frac{D}{2}-1\right)\left(\frac{x^{2}}{4}\right)^{-(D/2-1)}=\frac{1}{S_{D}(2-D)}|x|^{2-D}
\end{eqnarray}

\newpage \section{\mathematica--programs for the generation of the series contributions
  in the $2-D$-expansion}
All terms to appear in the expansion in $2-D$ are of the form 
\begin{equation}
  \label{pr.0}
  \prod_{i=1}^{k}(\mbox{Tr}M^{m_{i}})^{n_{i}}\ ,m_{i}, n_{i}, k\in \mathbf{N}
\end{equation}
as follows from (\ref{6.71}). To $N$-loop order, $M$ is an $N\times N$-matrix
as defined in (\ref{6.11}). The diagrams to be evaluated at $n$th order in
$2-D$ are positional averages of an ensemble of internal points on the
manifold, which are connected through $n$ correlators as shown in the example of
figure (\ref{fpr.0}). The numerical value of the diagrams is determined through the topology
of the graph, whose knots are given by the internal points and whose links
correspond to the correlators joining them. It furthermore depends on the
number of knots making up closed loops in the graph, but it does not depend on
the loop order.\\
Then, the $N$-loop contribution at $n$th order in $2-D$ depends on the
numerical value of the diagrams each coming up with a combinatorial factor involving the
number of ways to form the corresponding graph from $N{+}1$ independent
internal points. The combinatorial factor is a polynomial in $N$. Together
with some power of the factor $(N+1)^{-1}$ from (\ref{6.10}) the general form
of the term (\ref{pr.0}) can be written as
\begin{eqnarray}
  \label{pr.1}
  \prod_{i=1}^{k}(\mbox{Tr}M^{m_{i}})^{n_{i}}=\frac{P(N)}{(N+1)^{n_{\mbox{\tiny{max}}}}}\
  ,n_{\mbox{\tiny{max}}}\leq \sum_{i=1}^{k}m_{i}n_{i}\ ,
\end{eqnarray}
and $P(N)$ is some polynomial in $N$. The degree of the polynomial is less than
$2n$, if $n$ denotes the order in $2-D$.\\
\begin{figure}[t]
\begin{center}
    \leavevmode
    { \setlength{\unitlength}{1cm}
    \begin{picture}(10,5)
    \put(-1.5,1){\raisebox{-1.7cm}{\includegraphics[scale=0.3]{./eps/Cijijikjl.eps}}$\qquad\equiv\qquad
    \overline{\C^{2}(x_{i}{-}x_{j})\C(x_{i}{-}x_{k})\C(x_{j}{-}x_{k})\C(x_{l}{-}x_{k})}$} 
      \end{picture} }
\end{center}
    \caption{Example of a fifth order diagram in the $2-D$-expansion.}
    \label{fpr.0}
\end{figure}
In order to write down the  term (\ref{pr.0}) as a function of $N$ we
need know $P(N)$. Since its degree is bounded through $2n$, one can
determine its coefficients through a fit-procedure to be applied on the values
of (\ref{pr.0}) at the first $2n{+}1$ loop orders. This is done by the
\mathematica--modules, which are presented on the following pages. The module
"Dreset" initializes the matrices $\mathfrak{D}^{(0)\mathbf{-1}}$ and
$\mathfrak{D}_{ik}$ (\ref{6.11}) to obtain $M$. The "For" loop, which is
individual for each term (\ref{pr.0}) to be analyzed, recognizes the diagrams
to appear and performs the evaluation to $N=1,\dots , N_{\mbox{\tiny{max}}}\leq
2n+1$ loops. Of course, the "values" are expressions involving finitely many
diagrams.\\
The module "AnalyseListfkd" fits the $N_{\mbox{\tiny{max}}}$ points through a
polynomial being expanded in terms of the base given in (\ref{6.17}).\\
Summing up all loop orders each term of the form (\ref{pr.0}) translates into
some expression, which can be written in terms of the functions $f_{k}^{d}(z)$
as introduced in (\ref{6.28}). This procedure is done by the module
"Recognizefkd". Making several times use of the relation (\ref{fkdreduce}) all
$f_{k}^{d\prime}s$ can be transformed into some $f_{1}^{d\prime}s$ making the
leading behavior in the strong coupling limit obvious, which is then
completely encoded in the upper index. This is done by "Reducefkd".\\
Finally, it might be convenient to express the diagrams in terms of
positional averages of internal points, which are rather joint by the
connected correlator as introduced in (\ref{conn3}). The transcription of all
expressions is performed by "Trans2Conn". The application of the discussed
modules on the second order terms in the $2-D$-expansion is also included in
the sequel.
%\vspace{5cm}

\vfill
\begin{center}
\begin{minipage}[t]{17cm}
  \noindent\includegraphics[scale=1.0]{./eps/math1.epsi}
\end{minipage}
\end{center}
\vfill
\begin{center}
\begin{minipage}[t]{17cm}
  \noindent\includegraphics[scale=1.0]{./eps/math2.epsi}
\end{minipage}
\end{center}
\vfill
\begin{center}
\begin{minipage}[t]{17cm}
  \noindent\includegraphics[scale=1.0]{./eps/math3.epsi}
\end{minipage}
\end{center}
   
     




\newpage \section{Calculation of the diagrams in the $2-D$-expansion}
In this section we will calculate the diagrams to appear in the
$2-D$ expansion on the torus. It turns out that to obtain
$\overline{\C}$ and $\overline{\C_{c}^{2}}$ we need evaluate two sums over
discrete wavevectors obeying to periodic boundary conditions on the torus. Let us first derive the latter before turning to the
explicit evaluation. Starting from the definition of the difference correlator $C(x)$,
\begin{equation}
  \label{d.0.1}
  C(x):=G(x)-G(0)\ ,
\end{equation}
where $G(x)$ is the usual two--point correlator, we obtain $C(x)$ through an
inverse discrete 
Fourier--transformation from $G(k)=1/{\vec k}^{2}$, which reads:
\begin{equation}
  \label{d.0.2}
  C(x)=\frac{1}{L^{D}}\sum_{\vec k\not=0}\frac{1}{{\vec k}^{2}}\left(1-\mbox{e}^{i\vec k \vec
  x}\right)\ ,\qquad \vec k =\frac{2\pi}{L}\vec n\ , \ \vec n\in
  \mathbf{Z}\times\mathbf{Z}\backslash\{\{0\}\times\mathbf{N}\}\ .
\end{equation}
Performing the averaging procedure
\begin{equation}
  \label{d.0.3}
  \overline{C(x)}=\int_{x}C(x)\ ,
\end{equation}
where $1/L^{D} \int_{x}\mbox{e}^{i\vec k \vec x}=\delta^{D}_{\vec k}$ is to be taken
into account, the calculation of $\overline{C(x)}$ reduces to
\begin{equation}
  \label{d.1}
  \overline{C(x)}=I_{1}:=\sum_{k\not=0}\frac{1}{{\vec k}^{2}}\ ,\qquad
  \vec k=\frac{2\pi}{L}\vec n\ ,
\end{equation}
where $\vec k$ is $D$-dimensional, and the indices $n_{i}$ are running from
$-\infty$ to $\infty$, $n_{i}=0$ being excluded from the summation. Of course, in
the expansion in powers of $2-D$ we need an analytic continuation to real
values of D. Finally, to obtain $\overline{\C(x)}$ we have to subtract
$\overline{\C^{(0)}(x)}$ from $\overline{C(x)}$. Due to our normalizations:
\begin{equation}
  \label{d.1.1}
  \overline{\C(x)}=S_{D}\left(\overline{C(x)}-\frac{\overline{\C^{(0)}(x)}}{2\pi(2-D)}\right)\ ,
\end{equation}
where $S_{D}$ denotes the volume of the unit sphere and $\overline{\C^{(0)}(x)}=1$.\\
Turning to $\overline{\C_{c}^{2}(x)}$, we first note that within our
normalizations we have
\begin{eqnarray}
  \label{d.1.2}
  S_{D}^{-2}\overline{\C^{2}(x)}=\overline{(C(x)-\C^{(0)}(x)/(2\pi(2{-}D)))^{2}}=\overline{C(x)^{2}}-2\frac{\overline{C(x)}}{2\pi(2{-}D)}+\frac{1}{(2\pi(2{-}D))^{2}}
\end{eqnarray}
and
\begin{eqnarray}
  \label{d.1.3}
  S_{D}^{-2}\overline{\C(x)}^{2}=\overline{C(x)}^{2}-2\frac{\overline{C(x)}}{2\pi(2{-}D)}+\frac{1}{(2\pi(2{-}D))^{2}}
\end{eqnarray}
according to (\ref{d.1.1}), such that
\begin{equation}
  \label{d.1.4}
  \overline{\C_{c}^{2}(x)}\equiv
  \overline{\C^{2}(x)}-\overline{\C(x)}^{2}=S_{D}^{2}\left(\overline{C^{2}(x)}-\overline{C(x)}^{2}\right)\ .
\end{equation}
Knowing already the sum to be evaluated to obtain $\overline{C}$, (\ref{d.1}),
what is left to obtain is:
\begin{eqnarray}
  \label{d.1.5}
  \overline{C^{2}(x)}&=&\int_{y}\int_{x}C^{2}(x)=\frac{1}{L^{2D}}\int_{y}\int_{x}\sum_{k_{i}\not=0}\sum_{p_{j}\not=0}\frac{1}{{\vec k}^{2}}\frac{1}{\vec p^{2}}\Big(\mbox{e}^{i\vec k \vec x}-1\Big)\Big(\mbox{e}^{i\vec p \vec x}-1\Big)\nn\\
  &&=\sum_{k_{i}\not=0}\sum_{p_{j}\not=0}\frac{1}{{\vec k}^{2}}\frac{1}{\vec
  p^{2}}\left(\delta^{D}_{\vec k{+}\vec p}-\delta^{D}_{\vec
  k}-\delta^{D}_{\vec p}+1\right)=\sum_{k_{1}\not=0}\frac{1}{\vec
  k^{4}}+\left[\sum_{k_{1}\not=0}\frac{1}{{\vec k}^{2}}\right]^{2}\ .
\end{eqnarray}
Therefore,
\begin{equation}
  \label{d.2}
  S_{D}^{-2}\overline{\C_{c}^{2}(x)}=I_{2}:=\sum_{k\not=0}\frac{1}{{\vec k}^{4}}\ ,\qquad
  k_{i}=\frac{2\pi}{L}n_{i}\ .
\end{equation}
Let us first calculate $I_{1}$: Setting $L=1$, renaming the indices $n_{i}$
into $k_{i}$ and introducing a Schwinger parametrization we have:
\begin{eqnarray}
  \label{d.3}
  I_{1}=\frac{1}{(2\pi)^{2}}\sum_{k_{i}=-\infty \atop
  k_{i}\not=0}^{\infty}\frac{1}{\vec
  k^{2}}=\frac{1}{(2\pi)^{2}}=\sum_{k_{i}=-\infty \atop
  k_{i}\not=0}^{\infty}\int\limits_{0}^{\infty}\mbox{d}s\ \mbox{e}^{-s{\vec
  k}^{2}}\nonumber\\
   =\frac{1}{(2\pi)^{2}}\int\limits_{0}^{\infty}\mbox{d}s\ \left(\sum_{k=-\infty \atop
  k\not=0}^{\infty}\mbox{e}^{-sk^{2}}\right)^{D}=\frac{1}{(2\pi)^{2}}\int\limits_{0}^{\infty}\frac{\mbox{d}s}{s^{2}}\ \left(\sum_{k=-\infty \atop
  k\not=0}^{\infty}\mbox{e}^{-k^{2}/s}\right)^{D}\ ,
\end{eqnarray}
where it is to be noted that the sum in the last line is only
one-dimensional. Furthermore, from now on it is clear, how $I_{1}$ is
analytically continued to real values of $D$.\\
In order to evaluate this sum, we will make use of a Poisson-transformation,
which reads:
\begin{equation}
  \label{d.4}
  \sum_{n=-\infty}^{\infty}\mbox{e}^{-A(n-z/2)^{2}}=\sqrt{\frac{\pi}{A}}\sum_{l=-\infty}^{\infty}\mbox{e}^{-\frac{\pi^{2}l^{2}}{A}+i\pi lz}\ .
\end{equation}
The contribution from $l=0$ is the approximation of the l.h.s.
through a gaussian integral. Our aim is to calculate the coefficients of the
$2-D$ expansion of $I_{1}$ numerically using some algebraic manipulation
program. Then, the integration interval in (\ref{d.3}) has to be made finite.
This is done as follows: For any $s_{0}>0$ we have
\begin{equation}
  \label{d.5}
  I_{1}=\frac{1}{(2\pi)^{2}}\int\limits_{0}^{s_{0}}\frac{\mbox{d}s}{s^{2}}\ \left(\sum_{k=-\infty \atop
  k\not=0}^{\infty}\mbox{e}^{-k^{2}/s}\right)^{D}+\frac{1}{(2\pi)^{2}}\int\limits_{s_{0}}^{\infty}\frac{\mbox{d}s}{s^{2}}\ \left(\sum_{k=-\infty \atop
  k\not=0}^{\infty}\mbox{e}^{-k^{2}/s}\right)^{D}\ .
\end{equation}
For any finite $s_{0}>0$, the sum in the first integral can be
truncated at some finite $k_{\mbox{\tiny{max}}}$ for all $s\in [0,s_{0}]$.
For the second integral (corresponding to large values of $s$) we make use of
the poissonian formula (\ref{d.4}) with $z=0$ and choose $s_{0}$ in a way that
terms $l\not=0$ can be neglected:
\begin{eqnarray}
  \label{d.6}
  \left(\sum_{k=-\infty \atop
  k\not=0}^{\infty}\mbox{e}^{-k^{2}/s}\right)^{D}=\left(\sqrt{\pi
  s}-1+2\sum_{l=1}^{\infty}\mbox{e}^{-\pi^{2}l^{2}s}\right)^{D}\approx
  (\sqrt{\pi s})^{D}(1-(\sqrt{\pi s})^{-1})^{D}\ .
\end{eqnarray}
Then, the second integral can be evaluated approximately through
\begin{eqnarray}
  \label{d.7}
  \frac{1}{(2\pi)^{2}}\int\limits_{s_{0}}^{\infty}\frac{\mbox{d}s}{s^{2}}\ \left(\sum_{k=-\infty \atop
  k\not=0}^{\infty}\mbox{e}^{-k^{2}/s}\right)^{D}\approx
  \frac{1}{(2\pi)^{2}}\sum_{n=0}^{\infty}\frac{\Gamma(n-D)\pi^{(D-n)/2}}{\Gamma(n+1)\Gamma(-D)}\int\limits_{s_{0}}^{\infty}\mbox{d}s\ s^{\frac{D-n-4}{2}}\nonumber\\
  =\frac{1}{(2\pi)^{2}}\sum_{n=1}^{\infty}\frac{2\Gamma(n-D)\pi^{(D-n)/2}}{\Gamma(n+1)\Gamma(-D)(2-D+n)}s_{0}^{\frac{2-D+n}{2}}+2\frac{s_{0}^{\frac{2-D}{2}}}{2\pi^{(2-D/2)}(2-D)}\ .
\end{eqnarray}
From (\ref{d.5}) and (\ref{d.7}) we then have:
\begin{eqnarray}
  \label{d.8}
  I_{1}\approx\frac{1}{(2\pi)^{2}}\left[\int\limits_{0}^{1}\frac{\mbox{d}s\
  s_{0}}{s^{2}}\
  \left(2\sum_{k=1}^{\infty}\mbox{e}^{-k^{2}\frac{s_{0}}{s}}\right)^{D}+\sum_{n=1}^{\infty}\frac{2\Gamma(n-D)\pi^{(D-n)/2}}{\Gamma(n+1)\Gamma(-D)(2-D+n)}s_{0}^{\frac{2-D+n}{2}}+\frac{2\pi^{D/2}}{2-D}s_{0}^{\frac{2-D}{2}}\right]\ .\nonumber\\
\end{eqnarray}
The third term above shows up a pole in $2-D$, which can be easily subtracted
expanding the expression in powers of
$2-D$. The pole is
\begin{equation}
  \label{d.9}
  I_{1}=\frac{1}{2\pi(2-D)}+O((2-D)^{0}) .
\end{equation}
The precision of the machine that we used to evaluate (\ref{d.8}) was
sufficient in a way
that we could select $s_{0}$ from an interval, such that the sum appearing in
the integrand could
be truncated at some finite $k_{\mbox{\tiny max}}$ and the result was independent
from the precise value of $s_{0}$ within the desired order of
accuracy, therefore, justifying the approximation in (\ref{d.7}). Of course,
the other series can be safely truncated as well. Setting for instance $s_{0}=10$ and $k_{\mbox{\tiny max}}=n_{\mbox{\tiny max}}=20$ we obtain
with {\em Mathematica$^{\bigcirc\hspace{-2.3mm}\mbox{\tiny R}}$\/}\ :
\begin{equation}
  \label{d.10}
  I_{1}=\frac{1}{2\pi(2-D)}-0.2382(1)+0.3422(1)(2-D)+O((2-D)^{2})\ .
\end{equation}
On the torus we scaled the square root of the volume of the $D$-dimensional
unitsphere into the field. Accordingly, comparing with (\ref{d.1}) and (\ref{d.1.1}) we
then find:
  \begin{equation}
    \label{d.10.1}
    \overline{\C}=-1.4967(6)+S_{D}\cdot 0.3422(1)(2-D)+O((2-D)^{2})\ .
  \end{equation}
Let us turn to the evaluation of $I_{2}$ following the same strategy as above.
Again, setting $L=1$ and introducing a Schwinger parametrization leads to:
\begin{eqnarray}
  \label{d.11}
  I_{2}=\frac{1}{(2\pi)^{4}}\int\limits_{0}^{\infty}\mbox{d}s\ s^{-3}\left(\sum_{k=-\infty \atop
  k\not=0}^{\infty}\mbox{e}^{-k^{2}/s}\right)^{D}\nonumber\\
  =\frac{1}{(2\pi)^{4}}\left[\int\limits_{0}^{1}\frac{\mbox{d}s}{s_{0}^{2}s^{3}}\left(2\sum_{k=1}^{\infty}\mbox{e}^{-k^{2}/(s_{0}s)}\right)^{D}+\int\limits_{s_{0}}^{\infty}\mbox{d}s\ s^{-3}\left(\sum_{k=-\infty \atop
  k\not=0}^{\infty}\mbox{e}^{-k^{2}/s}\right)^{D}\right]
\end{eqnarray}
Once again applying the Poisson-transformation (\ref{d.4}) with $z=0$ on the
second integral involving large values of $s$, we approximate
\begin{eqnarray}
  \label{d.12}
  \int\limits_{s_{0}}^{\infty}\mbox{d}s\ s^{-3}\left(\sum_{k=-\infty \atop
  k\not=0}^{\infty}\mbox{e}^{-k^{2}/s}\right)^{D}\approx
  \pi^{D/2}\int\limits_{s_{0}}^{\infty}\mbox{d}s\ s^{D/2-3}\left(1-(\pi
  s)^{(-1/2)}\right)\nonumber\\
  \sum_{n=0}^{\infty}\frac{\Gamma(n-D)\pi^{(D-n)/2}}{\Gamma(n+1)\Gamma(-D)}\int\limits_{s_{0}}^{\infty}\mbox{d}s\ s^{\frac{D-n}{2}-3}=\sum_{n=0}^{\infty}\frac{2\Gamma(n-D)\pi^{(D-n)/2}}{\Gamma(n+1)\Gamma(-D)(4+n-D)}s_{0}^{\frac{D-n-4}{2}}\ ,\nonumber\\
\end{eqnarray}
such that we arrive at
\begin{eqnarray}
  \label{d.13}
  I_{2}\approx\frac{1}{(2\pi)^{4}}\left[\int\limits_{0}^{1}\frac{\mbox{d}s}{s_{0}^{2}s^{3}}\left(2\sum_{k=1}^{\infty}\mbox{e}^{-k^{2}/(s_{0}s)}\right)^{D}+\sum_{n=0}^{\infty}\frac{2\Gamma(n-D)\pi^{(D-n)/2}}{\Gamma(n+1)\Gamma(-D)(4+n-D)}s_{0}^{\frac{D-n-4}{2}}\right]\ .\nonumber\\
\end{eqnarray}
There is no pole in $2-D$. Since $I_{2}$ appears at second order in $2-D$ we
only need its value at $D=2$. Again, it is safe to truncate both series
appearing in (\ref{d.13}) at some finite values $k_{\mbox{\tiny max}}$ and
$n_{\mbox{\tiny max}}$, and $s_{0}$ has to be chosen from an appropriate
interval. Setting $k_{\mbox{\tiny max}}=n_{\mbox{\tiny max}}=20$ and
$s_{0}=10$ we obtain with {\em
  Mathematica$^{\bigcirc\hspace{-2.3mm}\mbox{\tiny R}}$\/}\ :
\begin{equation}
  \label{d.14}
  I_{2}=0.00109(1)+O((2-D))\ ,
\end{equation}
or-- due to the rescaling of the field by $S_{D}^{1/2}$--
\begin{equation}
  \label{d.14.1}
  \overline{\C_{c}^{2}}=0.043(1)+O((2-D))\ .
\end{equation}
The implementation of the summations with \mathematica is shown in the sequel.


\noindent
\begin{minipage}[t]{8cm}
   \raisebox{0mm}{\includegraphics[scale=0.5]{./eps/ovC.epsi}}
\end{minipage}
\hfill
\begin{minipage}[t]{8cm}
   \raisebox{0mm}{\includegraphics[scale=0.5]{./eps/ovCsq.epsi}}
\end{minipage}




\newpage \section{Exact results for polymers}
In this section we would like to derive some results for gaussian polymer
chains interacting with a $\delta$-like defect. The path integrals then turn
out to be factorizable and thus exactly solvable.\\
Let us start from the (bulk) polymer propagator being defined as the
conditional probability of finding the internal point
$s^{\prime}$ at the position $r_{f}$ when starting in $s$ at $r_{i}$.
The propagator  can be written as a functional integral for a restricted
partition function of the free chain according to
\begin{eqnarray}
  \label{a.3}
  \mathcal{Z}_{s,s^{\prime}}(r_{i}|r_{f}) & =&  \int \mathcal{D}[r]\
  \tilde \delta(r(s){-}r_{i})\tilde \delta(r(s^{\prime}){-}r_{f})\
  e^{-\mathcal{H}[r]}\nonumber\\
  & =& \ \int_{k}
  e^{-k^{2}|s^{\prime}{-}s|{+}ik(r_{f}{-}r_{i})}\ =\ |s^{\prime}{-}s|^{-d/2}\
  e^{-\frac{(r_{f}{-}r_{i})^{2}}{4|s^{\prime}{-}s|}}\ .
\end{eqnarray}
The propagator (\ref{a.3}) possesses the Markov property
\begin{equation}
  \label{a.4}
  \mathcal{Z}_{s,s^{\prime \prime}}(r_{i}|r_{f}) = \int \mbox{d}^{d}r_{m}\
  \mathcal{Z}_{s,s^{\prime}}(r_{i}|r_{m})\ \mathcal{Z}_{s^{\prime},s^{\prime
  \prime}}(r_{m}|r_{f})\ .
\end{equation}
\begin{comment}
\subsection{Markov property}
The propagator (\ref{a.3}) possesses the Markov property
\begin{equation}
  \label{a.4}
  \mathcal{Z}_{s,s^{\prime \prime}}(r_{i}|r_{f}) = \int \mbox{d}^{d}r_{m}\
  \mathcal{Z}_{s,s^{\prime}}(r_{i}|r_{m})\ \mathcal{Z}_{s^{\prime},s^{\prime
  \prime}}(r_{m}|r_{f})\ ,
\end{equation}
which will be proven as follows:
\begin{eqnarray}
  \label{a.4.1}
  \int \mbox{d}^{d}r_{m}\
  \mathcal{Z}_{s,s^{\prime}}(r_{i}|r_{m})\ \mathcal{Z}_{s^{\prime},s^{\prime
  \prime}}(r_{m}|r_{f})=\int \mbox{d}^{d}r_{m}\
  |s{-}s^{\prime}|^{-d/2}|s^{\prime}{-}s^{\prime\prime}|^{-d/2}\ \mbox{e}^{-\left(\frac{(r_{m}{-}r_{i})^{2}}{4|s{-}s^{\prime}|}{+}\frac{(r_{f}{-}r_{m})^{2}}{4|s^{\prime}{-}s^{\prime\prime}|}\right)}\nonumber\\
  =\int \mbox{d}^{d}r^{\prime}_{m}\
  |s{-}s^{\prime}|^{-d/2}|s^{\prime}{-}s^{\prime\prime}|^{-d/2}\
  \mbox{e}^{-\left(\frac{|s{-}s^{\prime}|{+}|s^{\prime}{-}s^{\prime\prime}|}{4|s{-}s^{\prime}||s^{\prime}{-}s^{\prime\prime}|}\right)\left(r_{m}^{\prime
  2}-\frac{(r_{i}|s^{\prime}{-}s^{\prime\prime}|{+}r_{f}|s{-}s^{\prime}|)^{2}}{4(|s{-}s^{\prime}|{+}|s^{\prime}{-}s^{\prime\prime}|)|s{-}s^{\prime}||s^{\prime}{-}s^{\prime\prime}|}\right)-\left(\frac{r_{i}^{2}|s^{\prime}{-}s^{\prime\prime}|{+}r_{f}^{2}|s{-}s^{\prime}|}{4|s{-}s^{\prime}||s^{\prime}{-}s^{\prime\prime}|}\right)}
  \ ,\nonumber\\
\end{eqnarray}
where
\begin{eqnarray}
  \label{a.4.2}
  r_{m}^{\prime}:=r_{m}{-}\frac{r_{i}|s^{\prime}{-}s^{\prime\prime}|{+}r_{f}|s{-}s^{\prime}|}{|s{-}s^{\prime}|{+}|s^{\prime}{-}s^{\prime\prime}|}\ .
\end{eqnarray}
Performing in (\ref{a.4.1}) the gaussian integral and taking into account that 
\begin{equation}
  \label{a.4.3}
  |s{-}s^{\prime}|{+}|s^{\prime}{-}s^{\prime\prime}|=|s{-}s^{\prime\prime}|
\end{equation}
leads to
\begin{eqnarray}
  \label{a.4.4}
  \int \mbox{d}^{d}r_{m}\
  \mathcal{Z}_{s,s^{\prime}}(r_{i}|r_{m})\ \mathcal{Z}_{s^{\prime},s^{\prime
  \prime}}(r_{m}|r_{f})&=&4^{d/2}|s{-}s^{\prime\prime}|^{-d/2}\
  \mbox{e}^{-\frac{r_{i}^{2}|s^{\prime}{-}s^{\prime\prime}|{+}r_{f}^{2}|s{-}s^{\prime}|}{4|s^{\prime}{-}s^{\prime\prime}||s{-}s^{\prime}|}{+}\frac{(r_{i}|s^{\prime}{-}s^{\prime\prime}|{+}r_{f}|s{-}s^{\prime}|)^{2}}{4|s{-}s^{\prime\prime}||s{-}s^{\prime}||s^{\prime}{-}s^{\prime\prime}|}}\nonumber\\
  &=&4^{d/2}|s{-}s^{\prime\prime}|^{-d/2}\
  \mbox{e}^{-\frac{(r_{f}{-}r_{i})^{2}}{4|s{-}s^{\prime\prime}|}}\ ,
\end{eqnarray}
which is easily seen from
\begin{displaymath}
  -\frac{r_{i}^{2}}{|s^{\prime}{-}s^{\prime}|}+\frac{r_{i}^{2}|s^{\prime}{-}s^{\prime\prime}|}{|s{-}s^{\prime\prime}||s{-}s^{\prime}|}=\frac{r_{i}^{2}(|s^{\prime}{-}s^{\prime\prime}|{-}|s{-}s^{\prime\prime}|)}{|s{-}s^{\prime}||s{-}s^{\prime\prime}|}=-\frac{r^{2}_{i}}{|s{-}s^{\prime\prime}|}\ ,
\end{displaymath}
a.s.f. thanks to eq. (\ref{a.4.3}). Absorbing a factor $2^{-d}$ into the definition
of $\mathcal{Z}_{s,s^{\prime}}(r_{i}|r_{f})$, which cancels out the prefactor of $4^{d/2}$,
one recovers (\ref{a.4}) from (\ref{a.4.4}).
\end{comment}
\subsection{Universal $1/r\ $- repulsion law for polymers}
We have shown  in section \ref{wall-force} by a scaling argument that as long as  $r\ll L^{\nu}$, the restricted partition function of
a manifold pinned at one of its internal points scales as
$
  \mathcal{Z}_{\infty}(r/L^{\nu})\sim
  (r/L^{\nu})^{\theta}$, with the contact-exponent given
by (\ref{2.19}). It is instructive to prove this  for the special case
of polymers in $d=1$. According to (\ref{2.0}) and (\ref{2.1}) this
corresponds to a polymer in $3d$-space interacting with a $\delta$-potential
on a  plane.\\
For this purpose we need the restricted partition function
(\ref{a.3}) in presence of a short-range interaction modeled through a
$\delta$-potential, which is situated at the origin of the embedding
space. The restricted partition function in the presence of the
interaction will be denoted by
$\mathcal{Z}^{g_{0}}_{s,s^{\prime}}(r_{i}|r_{f})$. We consider the case,
where the chain is fixed at its ends, such that $s{=}0$ and
$s^{\prime}{=}L$. We furthermore switch to a grand-canonical ensemble and
denote
\begin{equation}
  \label{a.5}
  \mathcal{Z}^{g_{0}}_{\tau}(r_{i}|r_{f}) :=
  \int_{0}^{\infty }\rmd L \, \mathcal{Z}^{g_{0}}_{0,L}(r_{i}|r_{f})\ e^{-\tau L}\ ,
\end{equation}
$\tau$ being some chemical potential. $\mathcal{Z}^{g_{0}}_{\tau}(r_{i}|r_{f})$ can be expanded in a
perturbation series in the coupling $g_{0}$ according to
\begin{equation}
  \label{a.6}
  \mathcal{Z}^{g_{0}}_{\tau}(r_{i}|r_{f})\ =\
  \sum_{n=0}^{\infty}(-g_{0})^{n}\mathcal{Z}^{g_{0},n}_{\tau}(r_{i}|r_{f})\ .
\end{equation}
The $\mathcal{Z}^{g_{0},n}_{\tau}(r_{i}|r_{f})$ are 
\begin{eqnarray}
  \label{a.7}
  \mathcal{Z}^{g_{0},n}_{\tau}(r_{i}|r_{f}) &:=& \int\limits_{0}^{\infty}\mbox{d}L\
  \int\limits_{0{<}x_{1}{<}\dots {<}x_{n}{<}L}
  \mathcal{Z}_{0,x_{1}}(r_{i}|0)\ \mathcal{Z}_{x_{1},x_{2}}(0|0)\times \dots
  \nonumber\\
  &&\dots \times \mathcal{Z}_{x_{n{-}1},x_{n}}(0|0)\
  \mathcal{Z}_{x_{n},L}(0|r_{f})\ \rme^{-\tau L}\nonumber \\
&=&\! \left[ \int_{0}^{\infty}\rmd x\ \rme^{-x \tau} 
 \mathcal{Z}_{0,x} (r_{i},0) \right]\!\!
\left[ \int_{0}^{\infty}\rmd x\ \rme^{-x \tau} 
 \mathcal{Z}_{0,x} (0,0) \right]^{n{-}1}\!
\left[ \int_{0}^{\infty}\rmd x\ \rme^{-x \tau} 
 \mathcal{Z}_{0,x} (0,r_{f}) \right]\!\! \nn\\
&=&  \mathcal{Z}_\tau ^{0} (r_{i}|0)\,  \mathcal{Z}_\tau ^{0} (0|0)^{n-1}\,  \mathcal{Z}_\tau ^{0} (0|r_{f}) \ ,
\end{eqnarray}
where we have used that after integration over $L$,
the integrals factorize; we further have defined
\begin{eqnarray}
\mathcal{Z}_{\tau } (r_{i}|r_{f}) &=& \int_{0}^{\infty } \rmd s
\mathcal{Z}_{0,s} (r_{i}|r_{f})   \rme^{-\tau s} \nn\\
\label{a:9}
&=&  \int_{0}^{\infty } \rmd s \int_{k}\rme^{- ( k^{2}+\tau )s+ ik
(r_{f}-r_{i})} =\int_{k}\frac{\rme^{ik (r_{f}-r_{i})}}{k^{2}+\tau }\ .
\end{eqnarray}
In $d{=}1$ the integration over the momenta $k$ in
(\ref{a:9}) leads to 
\begin{equation}
  \label{a.10}
  \mathcal{Z}_{\tau}(r_{i}|r_{f})_{|d{=}1}\ =\ \sqrt{\frac{\pi}{\tau}}\
  e^{-\sqrt{\tau}|r_{f}{-}r_{i}|}\ .
\end{equation} 
Let us now lock at what terms appear in the $n$-th order coefficient in
(\ref{a.7}): From the initial point of the chain a factor
$\mathcal{Z}_{\tau}(r_{i}|0)$ is contributed, while the final point is coming
up with $\mathcal{Z}_{\tau}(0|r_{f})$. Furthermore, the internal points give
$n{-}1$ powers of $\mathcal{Z}_{\tau}(0|0)$, where
\begin{equation}
  \label{a.11}
  \mathcal{Z}_{\tau}(0|0) = \sqrt{\frac{\pi}{\tau}}\ .
\end{equation}
The $(n{=}0)$-coefficient is special: Clearly, one has
\begin{equation}
  \label{a.12}
  \mathcal{Z}^{g_{0},0}_{\tau}(r_{i}|r_{f}) =
  \mathcal{Z}_{\tau}(r_{i}|r_{f})\ .
\end{equation}
Inserting (\ref{a.11})-(\ref{a.12}) into (\ref{a.7}) we obtain for (\ref{a.6}):
\begin{eqnarray}
  \label{a.13}
  \mathcal{Z}^{g_{0}}_{\tau}(r_{i}|r_{f}) &=& \sqrt{\frac{\pi}{\tau}}\
  \left[\rme^{-\sqrt{\tau}|r_{f}{-}r_{i}|}+ \rme^{-\sqrt{\tau}(|r_{f}|{+}|r_{i}|)}\
  \sum_{n{=}1}^{\infty}  \left(-g_{0}\sqrt{\frac{\pi}{\tau}}\right)^{n}\right]\nn\\
&=& \sqrt{\frac{\pi}{\tau}}\
  \left[\rme^{-\sqrt{\tau}|r_{f}{-}r_{i}|}- \rme^{-\sqrt{\tau}(|r_{f}|{+}|r_{i}|)}\
 \frac{1}{1+\frac{1}{g_0}\sqrt{\frac{\tau}{\pi}}}   \right]
\ .
\end{eqnarray} 
The limit of  $g_{0}\to \infty $ can be taken and leaves us with the
beautiful result
\begin{equation}
  \label{a.15}
  \mathcal{Z}^{\infty}_{\tau}(r_{i}|r_{f}) = \sqrt{\frac{\pi}{\tau}}\
  \left(\rme^{-\sqrt{\tau}|r_{f}{-}r_{i}|}-\rme^{-\sqrt{\tau}(|r_{f}|{+}|r_{i}|)}\right)\ .
\end{equation}
This is the propagator of a scalar field with Dirichlet boundary
conditions. It has applications when studying critical phenomena of
e.g.\ an Ising magnet in 
 half-space \cite{DiehlInDombGreen,EisenrieglerKremerBinder1982}. Performing the inverse Laplace-transformation of (\ref{a.15}) we arrive at
\begin{equation}
  \label{a.16}
  \mathcal{Z}^{\infty}_{L}(r_{i}|r_{f}) = 
  \frac{1}{L^{1/2}}\left(\rme^{-\frac{(r_{f}{-}r_{i})^{2}}{4|L|}}-\rme^{-\frac{(|r_{f}|{+}|r_{i}|)^{2}}{4|L|}}\right)\ .
\end{equation}
We  conclude that a $\delta$- potential acting in a
hyper-plane suppresses all configurations which penetrate or touch it,
when taking its amplitude to infinity.\\
Finally, in order to prove the universal repulsion law, 
 we start from (\ref{a.16}) and
integrate over all final positions $r_{f}$:
\begin{eqnarray}
  \label{a.17}
  \mathcal{Z}^{\infty}_{L}(r_{i}) = \int\limits_{-\infty}^{\infty}\mbox{d}r_{f}\
  \mathcal{Z}^{\infty}_{L}(r_{i}|r_{f}) = 2\sqrt{\pi}\left(1-\mbox{erf}
  \frac{|r_{i}|}{2\sqrt{L}}\right)\ ,
\end{eqnarray}
$\mbox{erf}$ denoting the error function. For small arguments we have
\begin{equation}
  \label{a.18}
  1-\mbox{erf}(x)\ \sim \ x\ .
\end{equation}
Furthermore, $\mathcal{Z}^{\infty}_{L}(r_{i})$ denotes the restricted
partition function of a chain  pinned at one of its ends in $r_{i}$. To
obtain $\mathcal{Z}_{\infty}(r/L^{\nu})$ in (\ref{2.17}) we have to
evaluate:
\begin{equation}
  \label{a.19}
  \mathcal{Z}_{\infty}(r/L^{\nu}) = \frac{1}{L}\int_{0}^{L}\mbox{d}s\
  \mathcal{Z}^{\infty}_{s}(r)\mathcal{Z}^{\infty}_{L{-}s}(r)\ \sim \ \left(\frac{r}{\sqrt{L}}\right)^{2}
\end{equation}
according to (\ref{a.18}) in the scaling regime $r{\ll}L^{1/2}$. We
thus find that $\theta =2$, which confirms (\ref{2.19}) for $D=1$ and $d=1$.
The above proof can be extended to arbitrary dimensions $0{<}d{<}2$, where
$\varepsilon{>}0$ for polymers. Then,
$\theta = 4-2d$. 

It would be nice to make the same arguments for membranes. 
However, the  proof is based on a drastic simplification, which only occurs in
$D{=}1$ and shows up in the factorizability of loop diagrams as in
(\ref{a.7}). This has no  extension to manifolds of internal dimension $D{>}1$.\\
\\
\subsection{Strong coupling expansion for the renormalized coupling $g$}
In the following we want to derive an expression for the renormalized coupling $g$ as defined in (\ref{2.3.1}) in terms of a strong coupling expansion in powers of the bare coupling. This is needed in the fixed membrane size ensemble as it is also considered in the $2-D$-expansion. In order to make matching our strong coupling series about $D=2$ with the known exact result for polymers, we shortly derive the latter both for an open as well as a closed chain.\\
Starting from the definition of the polymer-propagator let us point out that the full partition function then is defined as
\begin{eqnarray}
        \label{a.20}
        \mathcal{Z}^{O} &=& \int_{r_f}\int_{r_i}\mathcal{Z}_{s,s^{\prime}}(r_{                  i}|r_{f})\quad , \textrm{open end polymer}\\
        \mathcal{Z}^{C} &=& \int_{r}\mathcal{Z}_{s,s^{\prime}}(r|r)                     \quad , \textrm{closed polymer}         
\end{eqnarray}
In both cases the integrand is expanded in powers of the bare coupling according to (\ref{a.6}). Generally, the $n$th order coefficient in (\ref{a.7}) is coming up with $n-1$ powers of $\mathcal{Z}_{\tau}(0|0)$ to be here calculated for arbitrary $d<2$. Accordingly,
\begin{equation}
        \label{a.21}
        \mathcal{Z}_{\tau}(0|0)=\int_k\frac{1}{k^2+\tau}=\pi^{-d/2}\frac{S_{d}\frac{\pi}{2}\tau
        ^{-\E}}{\sin(\frac{d\pi}{2})}=\Gamma(\varepsilon)\tau^{-\varepsilon}\ ,
\end{equation}
where we have used the normalization of the integral according to
(\ref{3.8.3}) and
\begin{displaymath}
  \Gamma(\varepsilon)\Gamma(1{-}\varepsilon)=\frac{\pi}{\sin(\pi\varepsilon)}\qquad
  \textrm{and}\qquad S_{d}=2\frac{\pi^{d/2}}{\Gamma(\frac{d}{2})}\ .
\end{displaymath}
Note that here $\varepsilon=1-d/2$.
Furthermore, there is a $\mathcal{Z}_{\tau}(r_i|0)\mathcal{Z}_{\tau}(0|r_f)$ from the end-points contributing according to (\ref{a.20}) in the case of an open end chain
\begin{equation}
        \label{a.22}
        \int_{r_i}\int_{r_f}\mathcal{Z}_{\tau}(r_i|0)\mathcal{Z}_{\tau}(0|r_f)=\left(\int_{r}\int_{k}\frac{\mbox{e}^{-ikr}}{k^2+\tau}\right)^2=\tau^{-2}\ ,
\end{equation}
while for a closed chain one is obtaining:
\begin{equation}
        \label{a.23}
        \int_{r}\mathcal{Z}_{\tau}(r|0)\mathcal{Z}_{\tau}(0|r)=\int_r\int_{k_i}\int_{k_f}\frac{\mbox{e}^{ir(k_f-k_i)}}{(k_i^2+\tau)(k_f^2+\tau)}=\int_k\frac{1}{(k^2+\tau)^2}=\E\Gamma{(\E)}\tau^{-\E-1}
\end{equation}
The case $n{=}0$ is special. For an open polymer we have to evaluate
\begin{equation}
        \label{a.24}
        \int_{r_i}\int_{r_f}\mathcal{Z}_{\tau}^{g_0,0}(r_i|r_f)=\int_{r_i}\int_{r_f}\int_k\frac{\mbox{e}^{ik(r_f-r_i)}}{k^2+\tau}=\mathcal{V}_{\R^d}\tau^{-1}\ ,
\end{equation}
while a closed chain is yielding
\begin{equation}
        \label{a.25}
        \int_r\mathcal{Z}_{\tau}^{g_0,0}(r|r)=\mathcal{V}_{\R^d}\int_k\frac{1}{k^2+\tau}=\mathcal{V}_{\R^d}\Gamma(\E)\tau^{-\E}
\end{equation}
Diagrammatically, the $n$th order contribution to the partition function
(\ref{a.20}) can be written according to (\ref{a.21}-\ref{a.25}) as shown in
fig. (\ref{fa.0}).\\

\begin{figure}[t]
\begin{center}
    \leavevmode
    { \setlength{\unitlength}{1cm}
    \begin{picture}(10,5)
    \put(-1,0){\raisebox{0mm}{\includegraphics[scale=0.4]{./eps/daisyo.eps}}} 
    \put(6,0){\raisebox{0mm}{\includegraphics[scale=0.4]{./eps/daisyc.eps}}}
      \end{picture} }
\end{center}
    \caption{"Daisy"-diagrams contributing to $\mathcal{Z}^{O}$ (left) and
    $\mathcal{Z}^{C}$ (right), (\ref{a.20}), each leaf representing the integration
    over an internal distance.}
    \label{fa.0}
\end{figure}

Let us now calculate $\mathcal{Z}^0-\mathcal{Z}^{g_0}$ for a closed chain:
\begin{eqnarray}
        \label{a.25.1}
        \mathcal{Z}^0-\mathcal{Z}^{g_0}=-\E\tau^{-1}\sum_{n=1}^{\infty}(-\Gamma(\E)(g_0\tau^{-\E})^n=\E\tau^{-1}\frac{\Gamma(\E)g_0\tau^{-\E}}{1+\Gamma(\E)g_0\tau^{-\E}}\nonumber\\
        =\E\tau^{-1}\frac{1}{1+\tau^{\E}(\Gamma(\E)g_0)^{-1}}=\E
        \tau^{-1}\sum_{n=0}^{\infty}\left(-\frac{\tau^{\E}}{\Gamma(\E)g_0}\right)^n=\E
        \sum_{n=0}^{\infty}\left(-\frac{1}{\Gamma(\E)g_0}\right)^n\tau^{n\E-1}\ .
\end{eqnarray}
Performing an inverse Laplace-transformation, we arrive at
\begin{eqnarray}
        \label{a.26}
        g_{C}:=(\mathcal{Z}^0-\mathcal{Z}^{g_0})(L)=\frac{1}{2\pi i}\int\limits_{\gamma-i\infty}^{\gamma+i\infty}\mbox{d}\tau\ \mbox{e}^{\tau L}(\mathcal{Z}^0-\mathcal{Z}^{g_0})\nonumber\\
        =\E\left[1+\sum_{n=1}^{\infty}\left(-\frac{1}{\Gamma(\E)g_0L^{\E}}\right)^n\frac{\Gamma(n\E)\sin(\pi
        n\E)}{\pi}\right]=\E\left[1+\sum_{n=1}^{\infty}\left(-\frac{1}{\Gamma(\E)g_0L^{\E}}\right)^n\frac{1}{\Gamma(1{-}n\E)}\right]\
        ,\nonumber\\
\end{eqnarray}
where we have used that for any $\mu>0$
\begin{eqnarray}
  \label{a.27}
  \frac{1}{2\pi i}\int\limits_{\gamma-i\infty}^{\gamma+i\infty}\mbox{d}\tau\
  \tau^{\mu-1}\mbox{e}^{\tau L}=\frac{\Gamma(\mu)\sin(\pi\mu)}{\pi}L^{-\mu}\ .
\end{eqnarray}
$\mathcal{Z}^0-\mathcal{Z}^{g_0}$ as given in (\ref{a.26}) is the
dimensionless renormalized coupling $g$ in the case of a closed chain. Note
that in the calculation of the partition function of an open end chain there
has to be performed one more integration over the field $r$, such that only
$(\mathcal{Z}^0-\mathcal{Z}^{g_0})L^{(\E-1)}=:g$, $1{-}\varepsilon$ being the
dimension of the field, becomes dimensionless.\\
Before we proceed let us shortly consider how to treat the limit of vanishing
bulk space dimension $d$ in this framework: Setting $d=0$ and equivalently
$\varepsilon=1$ for polymers, then, the renormalized coupling is obtained from
(\ref{a.25.1}) and (\ref{a.26}) as
\begin{equation}
  \label{a.27.1}
  g_{C}(d{=}0)=\frac{1}{2\pi
  i}\int\limits_{\gamma-i\infty}^{\gamma+i\infty}\frac{\mbox{d}\tau}{\tau}\
  \mbox{e}^{\tau L}\frac{g_{0}}{g_{0}+\tau}\ .
\end{equation}
This time, the inverse Laplace-transformation can be immediately evaluated in
terms of residues. First order poles of the integrand are located at $\tau=0$
and $\tau=-g_{0}$, therefore providing
\begin{equation}
  \label{a.27.2}
  g_{C}(d{=}0)=1-\mbox{e}^{-g_{0}L}\ ,
\end{equation}
which is consistent with what we have found within the expansion in $2-D$ (\ref{d0.0}).
Let us calculate the inverse Laplace-transformation of
$\mathcal{Z}^0-\mathcal{Z}^{g_0}$ for an open end chain:
\begin{eqnarray}
        \label{a.28}
        \mathcal{Z}^0-\mathcal{Z}^{g_0}=\tau^{-2}g_{0}\sum_{n=1}^{\infty}(-(g_{0}\Gamma(\varepsilon)\tau^{-\varepsilon})^{n-1}=\frac{g_{0}\tau^{-2}}{1+(g_{0}\Gamma(\varepsilon)\tau^{-\varepsilon})}\nonumber\\
        =\frac{\tau^{\E-2}}{\Gamma(\E)}\frac{g_{0}\Gamma(\E)\tau^{-\E}}{1+g_{0}\Gamma(\E)\tau^{-\E}}=\frac{\tau^{\E-2}}{\Gamma(\E)}\frac{1}{1+(g_{0}\Gamma(\E))^{-1}\tau^{\E}}=\frac{\tau^{\E-2}}{\Gamma(\E)}\sum_{n=0}^{\infty}\left(-\frac{1}{g_{0}\Gamma(\E)}\right)^{n}\tau^{n\E}\ ,
\end{eqnarray}
leading to
\begin{eqnarray}
        \label{a.29}
        (\mathcal{Z}^0-\mathcal{Z}^{g_0})(L)=\frac{1}{2\pi i}\int\limits_{\gamma-i\infty}^{\gamma+i\infty}\mbox{d}\tau\ \mbox{e}^{\tau L}(\mathcal{Z}^0-\mathcal{Z}^{g_0})\nonumber\\
        =-\frac{1}{\Gamma(\E)}\sum_{n=0}^{\infty}\left(-\frac{1}{g_{0}\Gamma(\E)}\right)^{n}\int\limits_{\gamma-i\infty}^{\gamma+i\infty}\mbox{d}\tau\
        \frac{L\ \tau^{(n+1)\E-1}}{(n+1)\E-1}\ \mbox{e}^{\tau L}\nonumber\\
        =-\frac{1}{\Gamma(\E)}\sum_{n=0}^{\infty}\left(-\frac{1}{g_{0}\Gamma(\E)}\right)^{n}L^{1-(n+1)\E}\frac{\Gamma((n+1)\E)}{(n+1)\E-1}\frac{\sin(\pi(n+1)\E)}{\pi}\nonumber\\
        =L^{1-\E}\left[\frac{\sin(\pi\E)}{(1-\E)\pi}+\sum_{n=1}^{\infty}\left(-\frac{1}{L^{\E}g_{0}\Gamma(\E)}\right)^{n}\frac{1}{\Gamma(2{-}n\E)}\right]\ ,
\end{eqnarray}
where we have used that for $0<\mu<1$
\begin{equation}
  \label{a.29.1}
  \int\limits_{\gamma-i\infty}^{\gamma+i\infty}\mbox{d}\tau\
  \tau^{\mu-2}\ \mbox{e}^{\tau
  L}=\left.\frac{\tau^{\mu-1}}{\mu-1}\ \mbox{e}^{\tau L}\right|_{\gamma-i\infty}^{\gamma+i\infty}-L\int\limits_{\gamma-i\infty}^{\gamma+i\infty}\mbox{d}\tau\
  \frac{\tau^{\mu-1}}{\mu-1}\ \mbox{e}^{\tau L}\ ,
\end{equation}
and the boundary term vanishes.
The dimensionless renormalized coupling then is defined as in (\ref{2.3.1}),
such that
\begin{eqnarray}
  \label{a.30}
  g_O:=(\mathcal{Z}^0-\mathcal{Z}^{g_0})(L)L^{\E-1}=\frac{\sin(\pi\E)}{(1-\E)\pi}+\sum_{n=1}^{\infty}\left(-\frac{1}{L^{\E}g_{0}\Gamma(\E)}\right)^{n}\frac{1}{\Gamma(2{-}n\E)}\ .
\end{eqnarray}
From the strong coupling expansions (\ref{a.26}) and (\ref{a.30}) one easily
obtains the fixed point values of $g$. For the closed chain we find
\begin{equation}
  \label{a.31}
  g^{*}_{C}=\E\ ,
\end{equation}
while for the open end chain we have
\begin{equation}
  \label{a.32}
  g^{*}_{O}=\frac{\sin(\pi\E)}{(1-\E)\pi}\ .
\end{equation}

\subsection{Weak coupling expansion}
In order to be complete let us turn the expansions (\ref{a.26}) and
(\ref{a.30}) into weak coupling expansions. This can be easily done starting
for instance from the expansion in the Laplace-transformed picture for the
closed chain as given in the top line in (\ref{a.25.1}). Then,
\begin{eqnarray}
  \label{a.33}
  g_{C}:=(\mathcal{Z}^0-\mathcal{Z}^{g_0})(L)&=&\frac{1}{2\pi
  i}\int\limits_{\gamma-i\infty}^{\gamma+i\infty}\mbox{d}\tau\ \mbox{e}^{\tau
  L}(\mathcal{Z}^0-\mathcal{Z}^{g_0})=\frac{-\E}{2\pi
  i}\sum_{n=1}^{\infty}(-g_{0}\Gamma(\E))^{n}\!\!\!\!\int\limits_{\gamma-i\infty}^{\gamma+i\infty}\!\!\!\!\mbox{d}\tau\
\tau^{-n\E-1}  \ \mbox{e}^{\tau
  L}\nonumber\\
   &=&-\E\sum_{n=0}^{\infty}\frac{(-g_{0}\Gamma(\E)L^{\E})^{n}}{\Gamma(1{+}n\E)}\ ,
\end{eqnarray}
where we have simply used the analytical continuation of (\ref{a.27}) to
arbitrary values of $\mu$. (\ref{a.27}) can be obtained from integrating along
a branch cut, which makes it necessary to have $\mu>0$. But $\mu$ can be always rendered positive through a sufficient
number $m$ of partial integrations according to
(\ref{a.29.1}). Having chosen $m$ appropriately, it drops out after the
inverse Laplace transformation has been performed.\\
Let us apply the same procedure to the weak coupling expansion of the
renormalized coupling of the open chain in the top line in (\ref{a.28}):
\begin{eqnarray}
  \label{a.34}
  (\mathcal{Z}^0-\mathcal{Z}^{g_0})(L)=\frac{1}{2\pi
  i}\int\limits_{\gamma-i\infty}^{\gamma+i\infty}\mbox{d}\tau\ \mbox{e}^{\tau
  L}(\mathcal{Z}^0-\mathcal{Z}^{g_0})&=&\frac{g_{0}}{2\pi
  i}\sum_{n=1}^{\infty}(-(g_{0}\Gamma(\varepsilon))^{n-1}\!\!\!\!\int\limits_{\gamma-i\infty}^{\gamma+i\infty}\!\!\!\!\mbox{d}\tau\
  \tau^{-(n-1)\E-2}  \ \mbox{e}^{\tau
  L}\nonumber\\
  &=&Lg_{0}\sum_{n=1}^{\infty}\frac{(-g_{0}\Gamma(\E)L^{\E})^{n-1}}{\Gamma(2{+}(n{-}1)\E)}\ ,\nonumber\\
\end{eqnarray}
from what follows that
\begin{eqnarray}
  \label{a.35}
  g_{O}:=L^{\E-1}(\mathcal{Z}^0-\mathcal{Z}^{g_0})=g_{0}L^{\E}\sum_{n=0}^{\infty}\frac{(-g_{0}\Gamma(\E)L^{\E})^{n}}{\Gamma(2{+}n\E)}\ .
\end{eqnarray}

\subsection{Semi-flexible polymers}
Theories consisting in some one-dimensional manifold interacting with a single
defect turn out to be always one-loop exact. This is due to the
factorizability of the path-integrals like (\ref{a.7}) to appear in the
perturbation theory. Let us now discuss another model taking once again
advantage of this property. Consider so-called semi-flexible polymers, whose configurations
are controlled by bending elasticity. For low temperatures or equivalently on
short scales their configurational space can be parametrized by some
$d$-dimensional field $\vec r$, being defined through the displacement of the
individual mass points with respect to some flat background, therefore allowing for no
overlaps or even self-intersections of the polymer. Then, the
curvature along the manifold can be approximated through the usual Laplacian
$\partial_{tt} \vec r(t)$ and the free Hamiltonian reads:
\begin{equation}
  \label{a.36}
  \mathcal{H}_{0}[\vec r]=\int\limits_{0}^{L}\mbox{d}t\ \frac{1}{2}(\Delta \vec
  r(t))^{2}\ .
\end{equation}
Let us now consider the interaction with some short-range, $\delta$-like defect, acting in the
displacement space and being located at the origin: A short distance analysis
shows that any two nearby intersections of the polymer with the
defect appear like a single,  parallel intersection \cite{BundschuhLaessig1999}. This is due to the
bending stiffness of the manifold suppressing short-wavelength
fluctuations. Taking the most relevant, orientational dependent
interaction into account, which is
\begin{equation}
  \label{a.37}
  \mathcal{H}_{int}[\vec r]=v_{0}\int\limits_{0}^{L}\mbox{d}t\
  \delta^{d}(\vec r(t))\delta^{d}(\nabla \vec r(t))\ ,
\end{equation}
let us shortly sketch, how to completely sum up the perturbation theory in
this case following the same lines as in the preceding sections. Once again, we need to
find the propagator of the corresponding Schr\"odinger-type equation, which is
defined through the path-integral:
\begin{equation}
  \label{a.38}
  \mathcal{G}_{t,t^{\prime}}(r_{i}, v_{i}|r_{f}, v_{f})=\int\mathcal{D}[\vec
  r]\ \delta^{d}(r(t){-}r_{i})\delta^{d}(\nabla
  r(t){-}v_{i})\delta^{d}(r(t^{\prime}){-}r_{f})\delta^{d}(\nabla
  r(t^{\prime}){-}v_{f})\ \mbox{e}^{-\mathcal{H}_{0}[\vec r]}\ ,
\end{equation}
where $r_{i}, v_{i}$ and $r_{f}, v_{f}$ denote position and slope of the
initial and final point, respectively.
Be $\mathcal{G}_{t,t^{\prime}}^{v_{0}}$ the corresponding propagator in presence of the interaction
(\ref{a.37}). Switching to a
grand-canonical ensemble,
\begin{equation}
  \label{a.39}
   \mathcal{G}_{\tau}^{v_{0}}(r_{i}, v_{i}|r_{f},
   v_{f})=\int\limits_{0}^{\infty}\mbox{d}L\  \mathcal{G}_{0,L}(r_{i},
   v_{i}|r_{f}, v_{f})\ \mbox{e}^{-\tau L}\ ,
\end{equation}
the perturbation series of $\mathcal{G}_{\tau}^{v_{0}}(r_{i}, v_{i}|r_{f},
v_{f})$ can then be written as
\begin{equation}
  \label{a.40}
  \mathcal{G}_{\tau}^{v_{0}}(r_{i}, v_{i}|r_{f}, v_{f})=\sum_{n=0}^{\infty}(-v_{0})^{n}\mathcal{G}_{\tau}^{v_{0},n}(r_{i}, v_{i}|r_{f},
v_{f})\ .
\end{equation}
However, solving the path-integral of the free problem (\ref{a.38}) poses
considerable problems, when treating the energyless tilt mode. The solution
has been obtained from the equivalent problem of the corresponding
Schr\"odinger-type equation
\cite{MaggsHuseLeibler1989,GompperBurkhardt1989,Burkhardt1993}. It reads:
\begin{eqnarray}
  \label{a.41}
  \mathcal{G}_{t,t^{\prime}}(r_{i}, v_{i}|r_{f},
  v_{f})=\left(\frac{\sqrt{3}}{\pi(t^{\prime}{-}t)^{2}}\right)^{d}\
  \mbox{e}^{-\frac{6}{(t^{\prime}{-}t)^{3}}S_{t,t^{\prime}}(r_{i}, v_{i}|r_{f},
  v_{f})}\ ,
\end{eqnarray}
where
\begin{equation}
  \label{a.42}
  S_{t,t^{\prime}}(r_{i}, v_{i}|r_{f},
  v_{f})=(r_{f}{-}r_{i}-v_{i}(t^{\prime}{-}t))^{2}{+}\frac{(t^{\prime}{-}t)^{2}}{3}(v_{f}{-}v_{i})^{2}{-}(t^{\prime}{-}t)(r_{f}{-}r_{i}{-}v_{i}(t^{\prime}{-}t))(v_{f}{-}v_{i})
\end{equation}
is the action of the minimal energy path.\\
In analogy to (\ref{a.7}) the path-integral to be calculated to obtain $\mathcal{G}_{\tau}^{v_{0},n}(r_{i}, v_{i}|r_{f},
v_{f})$ factorizes after "time-ordering" the distances:
\begin{eqnarray}
  \label{a.43}
  \mathcal{G}_{\tau}^{v_{0},n}(r_{i}, v_{i}|r_{f},
v_{f})&=&\left[\int_{0}^{L}\mbox{d}L\ \mbox{e}^{-\tau L}\mathcal{G}_{0,L}(r_{i}, v_{i}|0,
0)\right]\left[\int_{0}^{L}\mbox{d}L\ \mbox{e}^{-\tau L}\mathcal{G}_{0,L}(0, 0|0,
0)\right]^{n{-}1}\times\nn\\
  &\times&\left[\int_{0}^{L}\mbox{d}L\ \mbox{e}^{-\tau L}\mathcal{G}_{0,L}(0, 0|r_{f},
v_{f})\right]\ ,\qquad n\geq 1\ .
\end{eqnarray}
The $n=0$ contribution is obtained from
\begin{eqnarray}
  \label{a.44}
  \mathcal{G}_{\tau}^{v_{0},0}(r_{i}, v_{i}|r_{f},
v_{f})=\int_{0}^{L}\mbox{d}L\ \mbox{e}^{-\tau L}\mathcal{G}_{0,L}(r_{i}, v_{i}|r_{f},
v_{f})\ .
\end{eqnarray}
It is hard to calculate the Laplace-transform of the propagator (\ref{a.41}),
but the renormalized coupling $v$ can be easily obtained from
$\mathcal{Z}^{0}-\mathcal{Z}^{v_{0}}$, $\mathcal{Z}^{v_{0}}$ being the full
partition function given by
\begin{equation}
  \label{a.45}
  \mathcal{Z}^{v_{0}}(\tau)=\int\mbox{d}^{d}r_{i}\ \mbox{d}^{d}v_{i}\ \mbox{d}^{d}r_{f}\ \mbox{d}^{d}v_{f}\ \mathcal{G}_{\tau}^{v_{0}}(r_{i}, v_{i}|r_{f}, v_{f})
\end{equation}
in the Laplace-transformed picture. Since
\begin{eqnarray}
  \label{a.46}
  \int\limits_{0}^{\infty}\mbox{d}L\ \mbox{e}^{-\tau L}\int\mbox{d}^{d}r_{i}\ \mbox{d}^{d}v_{i}\ \mathcal{G}_{0, L}(r_{i}, v_{i}|0, 0)=3^{-d/2}\frac{\Gamma(1{-}2d)}{\tau^{1{-}2d}}
\end{eqnarray}
and
\begin{eqnarray}
  \label{a.47}
  \int\limits_{0}^{\infty}\mbox{d}L\ \mbox{e}^{-\tau L}\mathcal{G}_{0, L}(0,
  0|0,
  0)=\left(\frac{\sqrt{3}}{\pi}\right)^{d}\frac{\Gamma(1{-}2d)}{\tau^{1{-}2d}}\
  ,
\end{eqnarray}
we find:
\begin{eqnarray}
  \label{a.48}
  (\mathcal{Z}^{0}-\mathcal{Z}^{v_{0}})(\tau)=-\Gamma(1{-}2d)\
  \tau^{-(1{-}2d)}\left(\frac{3\sqrt{3}}{\pi}\right)^{-d}\sum_{n=1}^{\infty}(-v_{0})^{n}\left(\frac{3^{d/2}\Gamma(1{-}2d)}{\pi^{d}}\tau^{-(1{-}2d)}\right)^{n}\nn\\
  =\frac{3^{-d}\Gamma^{2}(1{-}2d)\
  v_{0}\tau^{-2(1{-}2d)}}{1+v_{0}\tau^{-(1{-}2d)}\frac{3^{d/2}\Gamma(1{-}2d)}{\pi^{d}}}=\left(\frac{3\sqrt{3}}{\pi}\right)^{-d}\Gamma(1{-}2d)\tau^{-(1{-}2d)}\sum_{n=0}^{\infty}\left(-v_{0}\tau^{-(1{-}2d)}\frac{3^{d/2}\Gamma(1{-}2d)}{\pi^{d}}\right)^{-n}\ , \nn\\
\end{eqnarray}
where we have turned the series into a strong coupling expansion. Performing
an inverse Laplace transformation,
\begin{eqnarray}
  \label{a.49}
  &&\frac{1}{2\pi i}\int\limits_{\gamma-i\infty}^{\gamma+i\infty}\mbox{d}\tau\
  \mbox{e}^{\tau L}(\mathcal{Z}^{0}-\mathcal{Z}^{v_{0}})(\tau)\nn\\
  &=&\left(\frac{3\sqrt{3}}{\pi}\right)^{-d}\Gamma(1{-}2d)L^{1{-}2d}\sum_{n=0}^{\infty}\left(\frac{3^{d/2}\Gamma(1{-}2d)}{\pi^{d}}\right)^{-n}\frac{(-v_{0}L^{1{-}2d})^{-n}}{\Gamma(1+(n{-}1)(1{-}2d))}\ ,
\end{eqnarray}
the dimensionless renormalized coupling $v$ reads:
\begin{eqnarray}
  \label{a.50}
  v:=(\mathcal{Z}^{0}-\mathcal{Z}^{v_{0}})(L)L^{-(1{-}2d)}=\left(\frac{3\sqrt{3}}{\pi}\right)^{-d}\Gamma(1{-}2d)\sum_{n=0}^{\infty}\left(\frac{3^{d/2}\Gamma(1{-}2d)}{\pi^{d}}\right)^{-n}\frac{(-v_{0}L^{1{-}2d})^{-n}}{\Gamma(1+(n{-}1)(1{-}2d))}\ .\nn\\
\end{eqnarray}
The fixed point is obtained sending the dimensionless bare coupling
$z:=v_{0}L^{1{-}2d}$ to $\infty$:
\begin{eqnarray}
  \label{a.51}
  v^{*}:=\lim_{z\to
  \infty}v(z)=\left(\frac{3\sqrt{3}}{\pi}\right)^{-d}\frac{\Gamma(1{-}2d)}{\Gamma(2d)}\ .
\end{eqnarray}
From the strong coupling expansion (\ref{a.50}) one may immediately read off the
value of the correction to scaling exponent $\omega$ at the fixed point:
\begin{equation}
  \label{a.52}
  \omega(v^{*})=1-2d\ .
\end{equation}




\begin{comment}
\newpage \section{Steric repulsion in the limit $D\to 2$}
In this appendix we will present some formulas that might be useful  in
studying the scaling behavior in the limit $D=2$ of the general restricted partition function as
given in (\ref{2.8}). This might be interesting, since the scaling arguments
as discussed in subsection \ref{wall-force} break down in this limit. The
coefficient $\theta$ (\ref{2.19}) in the force-law (\ref{2.20}) turns to
infinity in $D=2$. Unfortunately, a detailed analysis of the limit of infinite
coupling for a pinned membrane at fixed position $\vec r\not=0$ in (\ref{2.8})
is still lacking, but the collection of the formulas presented in the sequel
might be a starting point.\\ 
Let us consider the perturbation expansion of
\begin{equation}
  \label{12.17}
  \mathcal{Z}_{p}\ =\ \mathcal{Z}\left< e^{ipr(s)}\right>\ =\
  \sum_{n=0}^{\infty}\frac{(-g_{0})^{n}\mathcal{Z}_{p}^{(n)}}{n!}\ .
\end{equation}
After integrating out the global translation from the path integral the $n$th
order coefficient reads
\begin{equation}
  \label{12.18}
  \mathcal{Z}_{p}^{(n)}\ =\ \int \prod_{i,j=1}^{n}\mbox{d}^{d}x_{i}\
  \mbox{d}^{d}k_{i}\ \delta^{d}((\sum_{i=1}^{n}k_{i})+p)\
  \exp\left[\frac{1}{2}\sum_{i,j=1}^{n{+}1}k_{i}k_{j}C(x_{i}{-}x_{j})\right]\ ,
\end{equation}
where $k_{n{+}1}{=}p$ and $x_{n{+}1}{=}s$. Since we ultimately want to
calculate the Fourier-transform of $\mathcal{Z}_{p}$ with respect to $p$ we
may shift the momentum variable according to
\begin{equation}
  \label{12.19}
  p\ \to \ p-\sum_{i=1}^{n}k_{i}\ ,
\end{equation}
such that
\begin{equation}
  \label{12.20}
  \delta^{d}((\sum_{i=1}^{n}k_{i})+p)\ \to \ \delta^{d}(p)
\end{equation}
and
\begin{equation}
  \label{12.21}
  \frac{1}{2}\sum_{i,j=1}^{n{+}1}k_{i}k_{j}C(x_{i}{-}x_{j})\ \to \
  \sum_{i=1}^{n}pk_{i}C(x_{i}{-}s)-\sum_{i,j=1}^{n}k_{i}k_{j}\frac{C(x_{i}{-}s){+}C(x_{j}{-}s){-}C(x_{i}{-}x_{j})}{2}\ .
\end{equation}
Abbreviating the quadratic form in (\ref{12.21}) as $D_{ij}$ and performing
the Fourier-transformation with respect to $p$ we arrive at
\begin{eqnarray}
  \label{12.22}
  \mathcal{Z}(\vec r,g_{0})\ &=&\ \int \mathcal{D}[\vec r^{\prime}]\
  \delta^{d}(\vec r(s){-}\vec r)\mbox{e}^{-\mathcal{H}[\vec r^{\prime}]}\nonumber\\
  &=&\ \sum_{n=0}^{\infty}\frac{(-g_{0})^{n}}{n!} \int_{p}\mbox{d}^{d}p\ \int \prod_{i,j=1}^{n}\mbox{d}^{d}x_{i}\
  \mbox{d}^{d}k_{i}\ \delta^{d}(p)\
  \exp\left[\sum_{i=1}^{n}pk_{i}C(x_{i}{-}s)-k_{i}D_{ij}k_{j}-i(p{-}\sum_{i=1}^{n}k_{i})\vec r\right]\nonumber\\
  &=&\ \sum_{n=0}^{\infty}\frac{(-g_{0})^{n}}{n!} \int \prod_{i,j=1}^{n}\mbox{d}^{d}x_{i}\
  \mbox{d}^{d}k_{i}\ \exp\left[-k_{i}D_{ij}k_{j}+i(\sum_{i=1}^{n}k_{i})\vec r\right]
\end{eqnarray}
Let us denote by $\tilde{\vec r_{i}}$ the $i$th-component of the $n$-component vector, whose identical entries are
the $d$-component vector $\vec r$. Then, we may write
\begin{eqnarray}
  \label{12.23}
  \left[-k_{i}D_{ij}k_{j}+i(\sum_{i=1}^{n}k_{i})\vec r\right]\ =\
  \left[-k_{i}D_{ij}k_{j}+ik_{i}\tilde{\vec r_{i}}\right]\nonumber\\
  =\
  -\left[(k_{i}{-}i\frac{D_{il}^{-1}}{2}r_{l})D_{ij}(k_{j}{-}i\frac{D_{jl}^{-1}}{2}r_{l})+\frac{\tilde{\vec r_{i}}D_{il}^{-1}\tilde{\vec r_{l}}}{4}\right]\ ,
\end{eqnarray}
such that after performing the integrations over the momenta we arrive at
\begin{eqnarray}
  \label{12.24}
  \mathcal{Z}(\vec r,g_{0})\ =\
  \sum_{N=0}^{\infty}\frac{(-g_{0})^{N}}{N!}\int
  \prod_{i=1}^{N}\mbox{d}^{d}x_{i}\ (\mbox{det}D_{ij})^{-d/2}\
  \exp\left[-\frac{\tilde{\vec r_{i}}D_{il}^{-1}\tilde{\vec r_{l}}}{4}\right]\nonumber\\
  &&\ .
\end{eqnarray}
In the limit $D\to 2$ we have
\begin{equation}
  \label{12.25}
  \mbox{det}\ D\ =\ \det D^{(0)}\ =\ \frac{1{+}N}{2^{N}}
\end{equation}
and
\begin{equation}
  \label{12.26}
  D^{-\mathbf{1}}\ =\ D^{(0)-\mathbf{1}}\ =\ 2\left(\mathbf{1}-\frac{N}{N{+}1}\mathbf{P}\right)\ ,
\end{equation}
where
\begin{equation}
  \label{12.27}
  \mathbf{P}=\frac{1}{N}\left(\begin{array}{ccc}
    1 & \ldots & 1 \\
    \vdots & \ddots & \vdots \\
    1 & \ldots & 1
  \end{array}\right)\ .
\end{equation}
Therefore,
\begin{equation}
  \label{12.28}
  D^{(0)-\mathbf{1}}\tilde{\vec r}\ =\ 2\left(\tilde{\vec
  r}-\frac{N}{N{+}1}\tilde{\vec r}\right)=\frac{2}{N{+}1}\tilde{\vec r}\  
\end{equation}
and
\begin{equation}
  \label{12.39}
  \tilde{\vec r}\ D^{(0)-\mathbf{1}}\tilde{\vec r}=2\frac{N}{N{+}1}\vec r^{2}\
  .
\end{equation}
Absorbing like in (\ref{6.26}) the factor $2^{d/2}$ from the determinant into
the coupling and introducing the dimensionless bare coupling $z$ (\ref{12.24})
can be rewritten in the limit $D\to 2$ as
\begin{eqnarray}
  \label{12.40}
  \mathcal{Z}(\vec
  r,z)=1+\sum_{k=0}^{\infty}\sum_{N=1}^{\infty}\frac{(-z)^{N}\ (-\vec r^{2})^{k}\
  N^{k}}{N!\ (1{+}N)^{d/2+k}k!}\ .
\end{eqnarray}
The inner sum can be rewritten in terms of the $f_{k}^{d}$s. Since
\begin{eqnarray}
  \label{12.41}
  \sum_{N=1}^{\infty}\frac{(-z)^{N}\
  N^{k}}{N!\
  (1{+}N)^{d/2+k}}=(-z)\sum_{N=0}^{\infty}\frac{(-z)^{N}(N{+}1)^{k-1}}{N!\ (2{+}N)^{d/2+k}}
\end{eqnarray}
and
\begin{eqnarray}
  \label{12.42}
  (N{+}1)^{k-1}=\sum_{l=0}^{k-1}(N{+}2)^{k-1-l}(-1)^{l}\left(
  \begin{array}[h]{c}
    k{-}1\nn\\
    l
  \end{array}\right)
\end{eqnarray}
we have
\begin{eqnarray}
  \label{12.43}
  \sum_{N=1}^{\infty}\frac{(-z)^{N}\
  N^{k}}{N!\
  (1{+}N)^{d/2+k}}&=&-\frac{1}{z}\sum_{l=0}^{k-1}(-1)^{l}\left(
  \begin{array}[h]{c}
    k{-}1\nn\\
    l
  \end{array}\right)z^{2}\sum_{N=0}^{\infty}\frac{(-z)^{N}}{N!\
    (2{+}N)^{d/2+l+1}}\nn\\
  &=&-\frac{1}{z}\sum_{l=0}^{k-1}(-1)^{l}\left(
  \begin{array}[h]{c}
    k{-}1\nn\\
    l
  \end{array}\right)\ f_{2}^{d+2(l{+}1)}(z)\ ,
\end{eqnarray}
such that (\ref{12.40}) becomes
\begin{eqnarray}
  \label{12.44}
  \mathcal{Z}(\vec
  r,z)=\frac{1}{z}\left(f_{1}^{d}(z)-\sum_{k=1}^{\infty}\sum_{l=0}^{k-1}\frac{(-1)^{l}(-\vec r^{2})^{k}}{k!}\left(\begin{array}[h]{c}
    k{-}1\nn\\
    l
  \end{array}\right)\ f_{2}^{d+2(l{+}1)}(z)\right)
\end{eqnarray}
The last expression can be further simplified making use of the integral
representation of the $f_{k}^{d}$s with the help of (\ref{6.28}). Then,
\begin{eqnarray}
  \label{12.45}
  f_{2}^{d+2(l{+}1)}(z)=\frac{z^{2}}{\Gamma(\frac{d}{2}{+}l{+}1)}\int_{0}^{\infty}\mbox{d}r\
  r^{d/2+l}\exp[-z\ \mbox{e}^{-r}-2r]\ ,
\end{eqnarray}
and the inner sum of (\ref{12.44}) can be rewritten in terms of the Kummer
confluent hypergeometric function $F_{11}(1{-}k,1{+}\frac{d}{2};z)$, which is
defined as
\begin{equation}
  \label{12.45.1}
  F_{11}(a,b;r):=\sum_{n=0}^{\infty}\frac{(a)_{n}}{(b)_{n}}r^{n}/n!=1+\frac{a}{b}r+\frac{a(a{+}1)}{b(b{+}1)}r^{2}/2!+\cdots\ ,
\end{equation}
such that
\begin{eqnarray}
  \label{12.46}
  \sum_{l=0}^{k-1}(-1)^{l}\left(\begin{array}[h]{c}
    k{-}1\nn\\
    l
  \end{array}\right)\frac{r^{d/2+l}}{\Gamma(\frac{d}{2}{+}l{+}1)}=\sum_{l=0}^{k-1}(-1)^{l}\frac{r^{d/2+l}\Gamma(k)}{\Gamma(k{-}l)\Gamma(l{+}1)\Gamma(\frac{d}{2}{+}l{+}1)}=\frac{r^{d/2}}{\Gamma(\frac{d}{2}{+}1)}F_{11}(1{-}k,1{+}\frac{d}{2};r)\ .
\end{eqnarray}
Inserting this into (\ref{12.44}) we obtain
\begin{eqnarray}
  \label{12.47}
  \mathcal{Z}(\vec
  r,z)=\frac{1}{z}\left(f_{1}^{d}(z)-\frac{z^{2}}{\Gamma(\frac{d}{2}{+}1)}\int_{0}^{\infty}\mbox{d}r\ r^{d/2}
  \sum_{k=1}^{\infty}\frac{(-\vec r^{2})^{k}F_{11}(k{-}1,1{+}\frac{d}{2};r)}{\Gamma(k{+}1)}\exp[-z\ \mbox{e}^{-r}-2r]\right)\nn\\
\end{eqnarray}
\end{comment}




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\newpage \section{Conformal mapping of the sectors}
\label{conformal} 
In order to calculate the two-loop diagram efficiently, one wants to
write it as an integral over a finite domain only.  To do so we need the technique of conformal
mapping of the sectors, which also serves for analytically continuing the
measure of integration to internal dimensions $D<1$. This  technique has
been extensively used and well documented in the context of  self-avoiding tethered membranes
\cite{WieseDavid1995,WieseDavid1997,WieseHabil}, but we repeat the
presentation here for completeness.\\  
Generally, in evaluating the three-point divergences we need to
integrate over some domain in the upper half-plane (see
\ref{4.18}). The measure of integration reads
\begin{equation}
  \label{b.0}
  \int_{y}\ =\
  \frac{S_{D{-}1}}{S_{D}}\int\limits_{-\infty}^{\infty}\mbox{d}y_{1}\
  \int\limits_{0}^{\infty}\mbox{d}y_{2}\ (y_{2})^{D{-}2}
\end{equation}
and the integrand is a function $f$ of the three distances $a,b$ and $c$
between the points $O=(0,0)$, $H=(-L,0)$ and $Y=(y_{1},y_{2})$, given
explicitly by (see fig.\ \ref{fb.0})
\begin{equation}
  \label{b.1}
  a=L=\textrm{fixed},\quad b=\sqrt{(y_{1})^{2}+(y_{2})^{2}}\ ,\quad
  c=\sqrt{(L{-}y_{1})^{2}+(y_{2})^{2}}\ .
\end{equation}%
\begin{figure}[t]
\begin{center}
    \leavevmode
    { \setlength{\unitlength}{1cm}
    \begin{picture}(8,5)
    \put(0,0){\raisebox{0mm}{\includegraphics[scale=0.5]{./eps/sectors.eps}}} 
    \put(3.8,1.5){$\mathcal{A}$}
    \put(1.8,3.5){$\mathcal{B}$}
    \put(5.8,3.5){$\mathcal{C}$}
      \end{picture} }
\end{center}
    \caption{The sectors $\mathcal{A},\mathcal{B}$ and $\mathcal{C}$}
    \label{fb.0}
\end{figure}%
In our problem $f$ is homogeneous, of degree $\lambda$, but not necessarily
symmetric:
\begin{equation}
  \label{b.2}
  f(\kappa a,\kappa b,\kappa c)\ =\ \kappa^{-\lambda}f(a,b,c)\ .
\end{equation}
Let us
now explicitly show, how sectors can be mapped onto each other. Specializing
to the mapping $\mathcal{B}\to \mathcal{A}$ we choose a coordinate system as
given in figure (\ref{fb.0}). The mapping $\mathcal{B}\to \mathcal{A}$ is
mediated by a special conformal transformation, the inversion with respect to
the circle around $O$ and radius $L$. In complex coordinates this is
($\bar Y$ is the complex conjugate of $Y$)
\begin{equation}
  \label{b.3}
  Y \longrightarrow \tilde Y = \frac{L^{2}}{\bar Y} = Y\,\frac{L^{2}}{b^{2}}\ \Leftrightarrow \ Y =
  \tilde Y \frac{L^{2}}{\tilde b^{2}}
\end{equation}
such that
\begin{eqnarray}
  \label{b.4}
  \left(
  \begin{array}{cc}
      y_{1} \\
      y_{2}
    \end{array}\right)\ \longrightarrow \  \left(
  \begin{array}{cc}
      \tilde y_{1} \\
      \tilde y_{2}
    \end{array}\right)\ =\ \left(
  \begin{array}{cc}
      \frac{y_{1}L^{2}}{b^{2}} \\
      \frac{y_{2}L^{2}}{b^{2}}
    \end{array}\right)\ \Leftrightarrow \ \left(\begin{array}{cc}
      y_{1} \\
      y_{2}
    \end{array}\right)\ =\ \left(
  \begin{array}{cc}
      \frac{\tilde y_{1}L^{2}}{\tilde b^{2}} \\
      \frac{\tilde y_{2}L^{2}}{\tilde b^{2}}
    \end{array}\right)\ .
\end{eqnarray}
This change of coordinates gives a Jacobian for the measure (\ref{b.0})
\begin{equation}
  \label{b.5}
  \left(\frac{L^2}{\tilde b^{2}}\right)^{D}\ .
\end{equation}
It can  easily be seen that the
mapping $\mathcal{B}\to \mathcal{A}$ is one to one: First, be $Y{=}(y_{1},y_{2})\in
\mathcal{B}$, that is $b>L$ and $b>c$. Then,
\begin{equation}
  \label{b.6}
  \tilde c^{2}\ =\ ((\tilde y_{1}{-}L)^{2}+(\tilde y_{2})^{2})\ =\
  \frac{L^{4}-2y_{1}L^{3}+b^{2}L^{2}}{b^{2}}\ =\ L^{2}\frac{c^{2}}{b^{2}}<L^{2}
\end{equation}
and
\begin{equation}
  \label{b.7}
  \tilde b^{2}\ =\ L^{2}\frac{L^{2}}{b^{2}}<L^{2}\ .
\end{equation}
Since $\tilde b^{2},\tilde c^{2}<L^{2}$, $\tilde Y \in \mathcal{A}$. Second, for the
inverse mapping, $Y(\tilde Y\in
\mathcal{A})\in \mathcal{B}$ is checked as follows:
\begin{equation}
  \label{b.8}
  c^{2}\ =\ b^{2}+L^{2}-2y_{1}L\ =\ b^{2}+L^{2}-2\tilde
  y_{1}\frac{L^{3}}{\tilde b^{2}}< b^{2}
\ ,
\end{equation}
since
\begin{displaymath}
 \tilde c^{2}=\tilde b^{2}-2\tilde
  y_{1}L+L^{2}<L^{2}\Leftrightarrow L^{2}-2\tilde
y_{1}\frac{L^{3}}{\tilde b^{2}} <0 
\end{displaymath}
and
\begin{equation}
  \label{b.9}
  b^{2}=L^{2}\frac{L^{2}}{\tilde b^{2}}>L^{2}\ .
\end{equation}
Since $b>L$ and $b>c$, $Y\in \mathcal{B}$. Finally, let us look at how
the whole integral transforms. First inserting the transformed
variables including the Jacobian (\ref{b.5}) and second using the homogeneity (\ref{b.2}), we arrive at
\begin{equation}
  \label{b.10}
  \int\limits_{Y\in\mathcal{B}}f(a=L,b,c) = \int\limits_{\tilde Y\in
  \mathcal{A}}\left(\frac{L}{\tilde b}\right)^{2D}f\left(L,\frac{L^{2}}{\tilde
  b},\frac{\tilde c L} {\tilde b}\right)=
 \int\limits_{\tilde Y\in
  \mathcal{A}}\left(\frac{L}{\tilde b}\right)^{2D-\lambda}f(\tilde b,\tilde
  a=L,\tilde c)\ .
\end{equation}
Let us restate the above calculation in a completely geometric interpretation as
sketched in the figure. The mapping consists of two steps:
\begin{itemize}
\item The rescaling with respect to $O$ by a factor $L/b$ which maps $Y$ onto
  $Y^{\prime}$ and $H$ onto $H^{\prime}$. (\ref{b.2}) implies that $f$ is
  changed by a factor $\left(\frac{L}{b}\right)^{-\lambda}$.
\item A mirror operation, which maps $Y^{\prime}$ onto $H$ and $H^{\prime}$
  onto $\tilde Y$, leaving invariant the origin $O$. This operation is a
  permutation of the first two arguments of $f$.
\end{itemize}
The mapping $Y \to \tilde Y$ corresponds to the special conformal
transformation (\ref{b.3}).\\
Analogously, we find that
\begin{equation}
  \label{b.11}
  \int\limits_{Y\in\mathcal{C}}f(a=L,b,c)\ =\ \int\limits_{\tilde Y\in
  \mathcal{A}}\left(\frac{L}{\tilde c}\right)^{2D-\lambda}f(\tilde c,\tilde
  b,\tilde a=L)\ .
\end{equation}

\end{appendix}





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\newpage \section*{Acknowledgements}
Allen, die zum Gelingen dieser Arbeit beigetragen haben, m\"ochte ich herzlich
danken: Hans Werner Diehl, der mir die Mitarbeit in seiner Arbeitsgruppe
erm\"oglicht hat und mit zahlreichen Anregungen und Kritik beigetragen
hat. Kay J. Wiese, der auch nach seinem Fortgang an die Str\"ande
Kaliforniens (ITP, Santa Barbara) die Arbeit stets engagiert betreut hat und bei dem meine
zahlreichen Fragen fast immer Geh\"or fanden. Mein zweiw\"ochiger Aufenthalt
in Santa Barbara bleibt in guter Erinnerung. Insbesondere auch eine lange
Diskussion mit Andreas Ludwig, der sich eingehend f\"ur den Limes $D\to 2$ und
die Entwicklung in $2-D$ interessiert hat.\\
F\"ur anregende Diskussionen oder Tips danke ich weiterhin besonders Lothar Sch\"afer
und Ralf Blossey.\\
Dar\"uber hinaus danke ich allen weiteren Mitarbeitern der Arbeitsgruppe und des
Fachbereichs Physik, die den Aufenthalt in Essen interessant oder angenehm gemacht
haben, sei es wissenschaftlich gesehen oder sei es aufgrund anderer
Ablenkungen. Insbesondere Andrea Ostendorff, Anja Gerwinski, Alexandra Apel,
Boris Shalaev, Frau Th\"alker, Florian Jasch, Hannes Hager, Hans Werner Diehl, Harald Karl,
Ira Terwyen, Jens Eggers,
Jouni Kallounki, Kay Wiese, Knut Elsner, Lothar Sch\"afer, Martin Smock, Mykola Sphot, Philipp
Kuhn, Ralf Blossey, Reinhard
Leidl, Rona Rangsch, R\"udiger Oberhage, Sutapa Mukherij und Ulla Lensen.\\
Diese Arbeit ist von der DFG im Rahmen des Leibniz Programms Di 378/2-1
unterst\"utzt worden.  



\begin{comment}
It is a pleasure to thank  R.\ Blossey, F.\ David, H.W.\ Diehl, 
M.\ Kardar,  and L.\ Sch\"afer  for useful discussions. The clarity of
this presentation has  profited from a critical reading by H.W.\ %
Diehl. We are grateful to Andreas Ludwig for  persisting questions,
and his never tiring efforts to understand the limit of $D\to 2$. 
  This work has  been supported  
by the DFG through the Leibniz program Di 378/2-1, under Heisenberg
grant Wi 1932/1-1, and  NSF grant  PHY99-07949. 
\end{comment}

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