%Paper: hep-th/9207021
%From: I_PESANDO@torino.infn.it
%Date: Wed, 8 JUL 92 17:16 GMT

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%operatore di Dirac
\def\Dir{\nabla\kern-7.8pt\Big{/}}

%insiemi di numeri
%\def\reali{{\hbox{\s@ l\kern-.5mm R}}}
\def\reali{{\hbox{l\kern-.5mm R}}}
%\def\naturali{{\hbox{\s@ l\kern-.5mm N}}}
\def\naturali{{\hbox{l\kern-.5mm N}}}
\def\interi{{\hbox{Z\kern-1.5mm Z}}}
%\def\interi{{\mathchoice
% {\hbox{\s@ Z\kern-1.5mm Z}}
% {\hbox{\s@ Z\kern-1.5mm Z}}
% {\hbox{{\s@b Z\kern-1.2mm Z}}}
% {\hbox{{\s@b Z\kern-1.2mm Z}}}  }}
\def\complessi{{\bf C}}
\def\unity{{\hbox{\s@ 1\kern-.8mm l}}}
\def\uno{{\hbox{ 1\kern-.8mm l}}}

\def\derp#1{{\partial~\over\partial #1}}
\def\ww{\wedge}
\def\part{\partial}
\def\cpm{C_{+-}}
\def\bk{\backslash}

\def\aa{\alpha}
\def\bb{\beta}
\def\cc{\chi}
\def\cb{\bar\chi}
\def\dd{\delta}
\def\DD{\Delta}
\def\ee{\epsilon}
\def\eb{\bar\epsilon}
\def\etab{\bar\eta}
\def\ff{\phi}
\def\FF{\Phi}
\def\vf{\varphi}
\def\gg{\gamma}
\def\GG{\Gamma}
\def\kk{\kappa}
\def\ll{\lambda}
\def\lb{\bar\lambda}
\def\oo{\omega}
\def\OO{\Omega}
\def\Ob{\bar\Omega}
\def\OB{\bar\Omega}
\def\pp{\psi}
\def\PP{\Psi}
\def\pb{\bar\psi}
\def\rr{\rho}
\def\rb{\bar\rho}
\def\ss{\sigma}
\def\SS{\Sigma}
\def\tt{\theta}
\def\tb{\bar\theta}
\def\taub{\bar\tau}
\def\xib{\bar\xi}
\def\zz{\zeta}
\def\zb{\bar\zeta}

\def\ib{{\bar\i}}
%\def\jb{{\bar\j}}
\def\jb{{\bar j}}
\def\kb{{\bar k}}
\def\mb{{\bar m}}

%
\begin{document}
\begin{titlepage}
\begin{flushright}
DFTT 29/92\\
July 1992\\
Preliminary Version\\
hep-th/9207021
\end{flushright}
\vspace*{0.5cm}
\begin{center}
{\bf
\begin{Large}
{\bf
POLYMERS AND TOPOLOGICAL FIELD THEORY :
A 2 LOOP COMPUTATION\\}
\end{Large}
}
\vspace*{1.5cm}
         {\large I. Pesando}\footnote{E-mail I\_PESANDO@TO.INFN.IT ,
                                             39163::I\_PESANDO}
         \\[.5cm]
         Dipartimento di Fisica Teorica dell'Universit\`{a} di Torino
            \footnote{Address after 1 October 1992: Niels Bohr Institute,
                      Blegdamsvej 17, 2100 Copenhagen, Denmark}\\
         Istituto Nazionale di Fisica Nucleare, Sezione di Torino\\
         via P.Giuria 1, I-10125 Torino, Italy
\end{center}
\vspace*{0.7cm}
\begin{abstract}
{\large
Within the Quantum Action Principle framework we show the perturbative
renormalizability of previously proposed topological lagrangian \`a la
Witten-Fujikawa describing polymers, then we perform a 2 loop
computation.
The theory turns out to have the same predictive power of De Gennes
theory, even though its running coupling constants exhibit a very
peculiar behaviour.
Moreover we argue that the theory presents two phases , a
topological and a non topological one.
}
\end{abstract}
\vfill
\end{titlepage}

\setcounter{footnote}{0}
\section*{Introduction.}
In a previous work \cite{PC} we proposed a topological theory \`a la
Fujikawa-Witten \cite{Fu,Wi,Bi} describing the self--avoiding walks
(hereafter SAW), i.e. the polymers in the De Gennes model \cite{DG,DCl}.

The aim of this approach was the exact computation of the critical
exponents of SAW; this would have been achieved by an exact computation
of the theory $\bb$-function(s) relying on the "topologicity" of the
theory.

In this article we want to prove that the theory we proposed is actually
perturbatively renormalizable and to perform the calculation of the
interesting quantities up to the second loop.
The explicit computation reveals that , in spite of the topologicity of
the theory, the hope for an exact computation of $\bb$-function is not
fulfilled, nonetheless the theory has some interesting features such as
the doubling of the coupling constants, which exhibit a very peculiar
behaviour under the RG flow, and the possibility for a spontaneous
breaking of the topological phase.

The article is divided as follows:
in section one we review the relation among our model and those of De
Gennes (\cite{DG,DCl}) and of Parisi-Sourlas-Mc Kane (\cite{MK,PS});
in section two we prove of the
perturbative renormalizability in the formalism of Quantum Action
Principle {\cite{QAP}-\cite{QAP1});in section three we explain the two loop
computation in the framework of background field method (\cite{bck});
 in section four we discuss the renormalization group flow,
finally we draw our conclusions.


\def\dag{\dagger}
\def\FP{\Phi\Pi}

\sect{The topological theory of polymers.}
%In a previous work \cite{PC} we proposed a topological theory
%describing the SAW , i.e. the polymers;
We start discussing briefly the relation between our model and the MPS
one.
The renormalization requires a
slight generalization of the previous lagrangian \cite{PC} that can
easily understood as the necessity of including all the BRST invariant
terms with the same dimension.
The lagrangian is now given by:
$${\cal L}=
\rr b^\dag b
+i b^\dag   (-\DD+m^2)\ff
+i \ff^\dag (-\DD+m^2)b
-i \xi^\dag (-\DD+m^2)\eta
+i \eta^\dag(-\DD+m^2)\xi
$$
\eq
+\ll\left(~ib^{\dagger}\ff-i~\ff^{\dagger}b
      -i~\xi^{\dagger}\eta-i~\eta^{\dagger}\xi\right)^2
+\nu\left(~b^{\dagger}\ff+\ff^{\dagger}b
      -\xi^{\dagger}\eta+\eta^{\dagger}\xi\right)^2
\lbl{laga}
\en
where $b=(b_1,\dots, b_N)^T,\ff=(\ff_1,\dots,\ff_N)^T$ are two vectors
of N complex Lorentz scalars (N being an arbitrary natural number),
$\xi=(\xi_1,\dots,\xi_N)^T$, $\eta=(\eta_1,\dots,\eta_N)^T$ are two
vectors of N complex Lorentz scalar ghosts, and $\rr$,
$\ll$ and $\nu$ are arbitrary positive numbers satisfying the condition
$\ll > \nu$.
The dimension and the BRST charge of the fields is given in the
following table:

\centerline{
\begin{tabular}{|c|c|c|c|c|} \hline
           & $\ff$  &  $b$   & $\xi$  & $\eta$ \\ \hline
dim        &   0    &  2     &    1   &    1   \\ \hline
$\FP$      &    0   &  0     &   -1   &    1   \\ \hline
\end{tabular}
}
\vskip.5cm
\noindent
and discussed in the next section.
The two adimensional coupling constants $\ll$ and $\nu$ are related to
the $O(n\rightarrow 0)$ coupling constant $g$ by
\eq
g=\ll-\nu
\lbl{rel}
\en
this implies that $\ll=\nu$ is a complicate way of describing a free
theory.
The proof of this relation and of the formal equivalence between De
Gennes theory and the present one is based upon the equality of the two
points Green function of the two theories; this can easily be achieved
by rewriting (\ref{laga}) using two auxiliary fields $\aa,\bb$ as
\eq
{\cal L}=
\rr b^\dag b
+i b^\dag O \ff    +i \ff^\dag O^\dag b
-i \xi^\dag O \eta +i \eta^\dag O^\dag \xi
+\aa^2+\bb^2
\lbl{prova}
\en
where
$$
O=   (-\DD+m^2+2i\sqrt{\ll}\aa+2\sqrt{\nu}\bb)
$$
and by using the McKane-Parisi-Sourlas trick \cite{MK,PS} on De Gennes
theory \cite{DG,DCl} along with (\ref{rel}), in such a way that De
Gennes theory can be rewritten as:
$${\cal L}_{O(n\rightarrow0)}=
\ff^\dag   O \ff
+\psi^\dag O \psi
+\aa^2+\bb^2
$$
We want to stress that this equivalence is true only if $\ll,\nu >0$
because otherwise, after integrating over $b$ in (\ref{prova}), the
remaining effective action would not be bounded from below.

\noindent
For computational purpose it is better to rewrite (\ref{laga}) as
$${\cal L}=
\rr~ \vf^\dag U\vf
+i \vf^\dag  Y(-\DD+m^2)\vf
- \psi^\dag X(-\DD+m^2)\psi
$$
\eq
+\ll(\vf^\dag X\vf -i \psi^\dag Y\psi)^2
+\nu(\vf^\dag Y\vf +i \psi^\dag X\psi)^2
\lbl{lag}
\en
where we introduced the following matrices
$$
X=\left(\begin{array}{cc}
         0 & i\uno_N \\
        -i\uno_N & 0
        \end{array}
   \right)=\ss_1\otimes\uno_N
{}~~~~
Y=\left(\begin{array}{cc}
         0 & \uno_N\\
         \uno_N & 0
        \end{array}
   \right)=\ss_2\otimes\uno_N
$$$$
U=\left(\begin{array}{cc}
         \uno_N & 0\\
         0      & 0
        \end{array}
   \right)
$$
and we defined
$$
\vf=\left( \begin{array}{c} b \\ \ff \end{array} \right )
{}~~~~
\psi=\left( \begin{array}{c} \xi \\ \eta \end{array} \right )
$$
Notice the symmetry in the four-field terms
$$ (\ll,X,Y)\leftrightarrow (\nu,Y,-X)$$
that turns out to be useful in performing the actual computation.

\def\L{{\cal L}}
\def\W{{\cal W}}
\def\S{{\cal S}}
\def\pam{\partial^\mu}
\def\pan{\partial^\nu}

\def\D{{\cal D}}
\def\F{{\cal F}}
\sect{Symmetries and renormalization}
In the previous section we gave the topological lagrangian along with
the dimension of the fields and their BRST charge, without motivating
these choices and the terminology.

Now we proceed in explaining the field dimensions, they are deduced
looking at the explicit form of the free theory propagators:
\eqa
<\ff^*_j (x)\ff_k (0)>=\dd_{jk}\int~d^4k~e^{ikx}{\rr\over(k^2+m^2)^2}
\nonumber\\
<\ff^*_j (x) b_k (0)>=\dd_{jk}\int~d^4k~e^{ikx}{-i\over(k^2+m^2)}
\nonumber\\
<\xi^*_j (x) \eta_k (0)>=\dd_{jk}\int~d^4k~e^{ikx}{-i\over(k^2+m^2)}
\ena
One could wonder about the dimensionality of the parameter $\rr$ and,
consequently, of the other fields, but what justifies setting to zero
the dimensionality of $\rr$ is just the power counting in which the
propagator $\ff\ff$ has dimension -4 (in unit of mass).
Just because of this noncanonical dimension of the $\ff\ff$ propagator
% it is worth redoing
we performed the usual power counting both with $\rr\ne0$ and $\rr=0$.
It is easy to show that the critical dimension is four (as it should be
to reproduce De Gennes theory) and that if we
indicate with $E_\ff,E_b,E_\xi,E_\eta$ the number of external legs
 of the fields $\ff,b,\xi,\eta$ in a truncated diagram,
at the critical dimensionality $D_{\mbox{cr}}=4$,
there are only the following superficially divergent diagrams:
%\centerline{
\eq
\begin{tabular}{|c|c|c|c|}\hline
$E_b$ &$E_\ff$ &$E_\xi=E_\eta$& D \\ \hline
 0     &  0     &    0         & 4 \\ \hline
 1     &  1     &    0         & 2 \\ \hline
 0     &  0     &    1         & 2 \\ \hline
 1     &  1     &    1         & 0 \\ \hline
 2     &  0,2   &    0         & 0 \\ \hline
 0     &  0     &    2         & 0 \\ \hline
\end{tabular}
\lbl{tabella}
\en
%}

We are ready to discuss both the symmetries and
the broken symmetries of the action:
\begin{enumerate}
\item the discrete symmetry:
\eq
      \ff\rightarrow\ff^*,b\rightarrow b^*,\xi\rightarrow\xi^*,
      \eta\rightarrow\eta^*
\en
      that is responsible for the non appearance
      in the lagrangian of a term
      $$
      \left(~ib^{\dagger}\ff-i~\ff^{\dagger}b
      -i~\xi^{\dagger}\eta-i~\eta^{\dagger}\xi\right)
      \left(~b^{\dagger}\ff+\ff^{\dagger}b
      -\xi^{\dagger}\eta+\eta^{\dagger}\xi\right)
      $$
       which is odd under such a transformation;
      this symmetry allows to construct the action
      only from the real part of functionals
      (possibly with complex coefficients)
      and from  even power of the imaginary part of functionals
      (possibly with complex coefficients);
\item $GL(N,\complessi)_{\mbox{bos}}$ broken to $U(N,\reali)$
      by the term proportional to $b^\dag b$:
\eqa
      \dd^{b}(\tt^\aa)\ff &=&i~G_\aa\ff~\tt^\aa
      ~~~~
      \dd^{b}(\tt^\aa)\ff^\dag =-i~\ff^\dag G^\dag_\aa\ff~\tt^{\aa *}
 \nonumber\\
      \dd^{b}(\tt^\aa) b^\dag &=& -i~b^\dag G_\aa~\tt^\aa
      ~~~~
      \dd^{b}(\tt^\aa) b = i~G^\dag_\aa b~\tt^{\aa *}
\ena
      where $G_\aa$ are the generators of $gl(N,\complessi)$
      and $\tt_{\aa}$ is the complex parameter of the transformation.

      \noindent
      We use the following explicit representation for the generators
      $$
      (G_\aa)_{pq}\equiv (G_{(ab)})_{pq}=-i\dd_{ap}\dd_{bq}
      ~~~~
      ( G_{(ab)}^\dag=-G_{(ba)} )
      $$
      \noindent
      Considering $gl(N)$ over the complex field allows to vary
      independently $(\ff,b^*)$ from $(\ff^*,b)$, in fact we can build
      the following generators of the decomplexified algebra:
\eqa
      \dd_{\aa}^{(+)}={1\over2}(\dd_{\aa}(\tt_{\aa}=1)
                        -i(\dd_{\aa}(\tt_{\aa}=i) )
\nonumber\\
      \dd_{\aa}^{(-)}={1\over2}(\dd_{\aa}(\tt_{\aa}=1)
                        +i(\dd_{\aa}(\tt_{\aa}=i) )
\ena
      whose action on the bosonic fields is given by
\eqa
      \dd_{(ab)}^{(+)}\ff_p=\dd_{ap}\ff_b,
      ~~~~
      \dd_{(ab)}^{(+)}b^*_p=-\dd_{bp} b^*_a,
      ~~~~
      \dd_{(ab)}^{(+)}\ff^*_p=\dd_{(ab)}^{(+)}b_p=0
 \nonumber\\
      \dd_{(ab)}^{(-)}\ff^*_p=\dd_{ap}\ff^*_b,
      ~~~~
      \dd_{(ab)}^{(-)}b_p=-\dd_{bp} b_a,
      ~~~~
      \dd_{(ab)}^{(-)}\ff_p=\dd_{(ab)}^{(-)}b^*_p=0
\ena

      This symmetry is however broken ,
      in fact we find immediately the breaking under the transformation
      by a complex parameter $\tt^\aa$:
      $$ \dd {\cal L}=
        i~\rr~b^\dag(G^\dag_\aa-G_\aa)b ~Re \tt^\aa
        +\rr~b^\dag(G^\dag_\aa+G_\aa)b ~Im \tt^\aa
      $$
      the breaking vanishes when restricting the
      symmetry to the $u(N,\reali)$ generated by $T_A$ whose explicit
      representation is given by:
      $$T_{(aa)}=iG_{(aa)}$$
      $$T_{1(ab)}=G_{(ab)}+G_{(ab)}^\dag
      ~~~~
      T_{2(ab)}=i(G_{(ab)}-G_{(ab)}^\dag)
      ~~~~
      \mbox{ with } a<b
      $$.
\item $GL(N,\complessi)_{\mbox{ferm}}$ with complex parameter $\tt_\aa$:
\eqa
      \dd^{f}(\tt^\aa)\eta &=& i~G_\aa\eta ~\tt^\aa
      ~~~~
      \dd^{f}(\tt^\aa)\eta^\dag =-i\eta^\dag G_\aa^\dag~\tt^{\aa *}
\nonumber\\
      \dd^{f}(\tt^\aa) \xi^\dag &=& -i~ \xi^\dag G_\aa~\tt^\aa
      ~~~~
      \dd^{f}(\tt^\aa) \xi = i~ G_\aa^\dag \xi~\tt^{\aa *}
\ena
      As in the previous case of the broken bosonic $GL(N,\complessi)$,
      it is possible to build $\dd^{f(\pm)}$ and vary independently
      the couple $(\eta,\xi^*)$ from $(\eta^*,\xi)$.
\item BRST-like transformations with complex parameter $\tt^\aa$:
\eqa
      \hat\dd(\tt^\aa) \ff=iG_\aa\eta~\tt^\aa ,
      ~~~~
      \hat\dd(\tt^\aa)\eta=0
\nonumber\\
      \hat\dd(\tt^\aa)  \xi^\dag=i b^\dag G_\aa~\tt^\aa ,
      ~~~~
      \hat\dd(\tt^\aa)  b=0
\ena
      In particular the generator of the "canonical" BRST is
      $s=\sum_{a=1}^N \hat\dd_{(aa)}$, in such a way we can rewrite the
      lagrangian (\ref{laga}) as
      $${\cal L}=s[\rr~b^\dag\xi
        +i \xi^\dag   (-\DD+m^2)\ff
        +i \ff^\dag (-\DD+m^2)\xi
      $$$$
     +\ll\left(~ib^{\dagger}\ff-i~\ff^{\dagger}b
      -i~\xi^{\dagger}\eta-i~\eta^{\dagger}\xi\right)
     \left(~i\xi^\dag\ff-i\ff^\dag\xi\right)
      $$$$
     +\nu\left(~b^{\dagger}\ff+\ff^{\dagger}b
      -\xi^{\dagger}\eta+\eta^{\dagger}\xi\right)
     \left(~\xi^\dag\ff+\ff^\dag\xi\right)]
      $$
    Notice that the explicit form and existence of $s$ justifies the
     dimensions and the charges of the fields  we gave.
     Exactly as before, we can build $\hat\dd^{(\pm)}$ and vary
    independently $(\ff,\xi^*)$ from $(\ff^*,\xi)$.

\item antiBRST-like transformations broken by $b^\dag b$:
\eqa
      \widehat{\bar\dd_\aa}\eta=iG^\dag_\aa\ff,
      ~~~~\widehat{\bar\dd_\aa}\ff=0 \nonumber\\
      \widehat{\bar\dd_\aa} b^\dag=-i \xi^\dag G_\aa^\dag,
      ~~~~\widehat{\bar\dd_\aa} \xi=0
\ena
      The breaking is :
      $$ \widehat{\bar\dd}(\tt^\aa){\cal L}=
       i~\rr~(b^\dag G^\dag_\aa\xi -\xi^\dag G_\aa b)~Re(\tt^\aa)
       +~\rr~(b^\dag G^\dag_\aa\xi +\xi^\dag G_\aa b)~Im(\tt^\aa)
       $$
\end{enumerate}

Since the broken symmetries are broken by a term of dimension four, it
would be very difficult to keep them under control, so we prefer to give
up these symmetries and to consider only the unbroken ones.

In order to implement the Ward-Takahashi identities (WTI) we
introduce the following functional operators:
\begin{enumerate}
\item
\eq
      u(N,\reali)\Longrightarrow
      \W^b_A=\int~i(T_A\ff)_p{\dd\over\dd\ff_p}
      -i(b^\dag T_A)_p{\dd\over\dd b^*_p}+(c.c.)
\en
\item
\eq
      gl(N,\complessi)\Longrightarrow
      \left\{
      \begin{array}{l}
      \W^{f(+)}_\aa=\int~i(G_\aa\eta)_p{\vec{\dd}\over\dd\eta_p}
      -i(\xi^\dag G_\aa)_p{\vec{\dd}\over\dd \xi^*_p}
      \\
      \W^{f(-)}_\aa=\int~i(G^\dag_\aa \xi)_p{\vec{\dd}\over\dd \xi_p}
      -i(\eta^\dag G^\dag_\aa)_p{\vec{\dd}\over\dd \eta^*_p}
      \end{array}
      \right.
\en
\item
\eq
      \mbox{BRST-like} \Longrightarrow
      \left\{
      \begin{array}{l}
      \S^{(+)}_\aa=\int~i(G_\aa\eta)_p{\dd\over\dd\ff_p}
      +i(b^\dag G_\aa)_p{\vec{\dd}\over\dd \xi^*_p}
      \\
      \S^{(-)}_\aa=\int~i(G^\dag_\aa b)_p{\vec{\dd}\over\dd\xi_p}
      +i(\eta^\dag G^\dag_\aa)_p{\dd\over\dd \ff^*_p}
      \end{array}
      \right.
\en
\end{enumerate}

\noindent
Notice that sine the fields are doublets under the BRST it is not
necessary to introduce external sources for different kind of
BRST multiplets.

\noindent
All the symmetries are contained in the following WTI:
\eqa
{[}\W^b_A,\W^b_B ]\GG
   &=&f_{AB}^{~~~C} \W^b_C \GG
\label{wtglbos}
\\
{[}\W^b_A,\W^{f(\pm)}_\bb ] \GG
   &=& 0
\\
{[}\W^b_A, \S^{(\pm)}_\bb] \GG
   &=& g_{A\bb}^{~~~\gg(\pm)} \S^{(\pm)}_\gg\GG
\\
 {[} \W ^{f(\pm)}_\aa,\W ^{f(\pm)}_\bb {]}\GG
   &=&  f_{\aa\bb}^{~~~\gg}\W^{f(\pm)}_\gg\GG
\\
 {[} \W ^{f(\pm)}_\aa,\W ^{f(\mp)}_\bb {]}\GG
   &=& 0
\\
 {[} \W ^{f(\pm)}_\aa,\S^{(\pm)}_\bb {]}\GG
   &=& g_{\aa |\bb}^{~~~\gg(\pm)}\W^{f(\pm)}_\gg\GG
\\
 {[} \W ^{f(\pm)}_\aa,\S^{(\mp)}_\bb {]}\GG
   &=&
\\
 {[} \S^{(\pm)}_\aa,\S^{(\pm)}_\bb ]_+\GG
&=&
 {[} \S^{(\pm)}_\aa,\S^{(\mp)}_\bb ]_+\GG
 =0
\lbl{wtss}
\ena
where
\eqa
{[}T_A,T_B]&=&i~f_{AB}^{~~~C}T_C
\nonumber\\
{[}G_\aa,G_\bb]&=&i~f_{\aa\bb}^{~~~\gg}G_\gg
\nonumber\\
T_AG_\aa &=&
i~g_{A\aa}^{~~~\bb(+)}G_\bb=
i~g_{A\aa}^{~~~\bb(-)*}G_\bb
\nonumber\\
G_\aa G_\bb &=&
-i~g_{\aa |\bb}^{~~~\gg(+)}G_\gg=
i~g_{\aa |\bb}^{~~~\gg(-)*}G_\gg
\nonumber
\ena
It can be shown (with a big amount of algebra) that these symmetries
are not anomalous.

After the discussion of the WTI we can discuss the stability
of the classical action $\GG_{\mbox{cl}}$.
This amount to impose the conditions (\ref{wtglbos}-\ref{wtss}) to the
perturbed action $\GG '=\GG_{\mbox{cl}}+\DD$.

The $U(1)^N_f$ symmetry ( $\W^f_{(aa)}\DD=0$) implies that every term of
$\DD$ has to be built using an equal number of conjugate ghost fields
 and  ghost fields\footnote{
This assertion is intuitively obvious, however a rigorous proof is
based on the observation that
$\W^f_{(aa)}=N(\eta_a)-N(\eta^*_a)+N(\xi^*_a)-N(\xi_a)$
where $N(\eta_a)=\int \eta_a{\vec{\dd}\over\dd\eta_a}$ can be interpreted
as an occupation number operator.},
in the mean time the ghost charge implies that
every term should contain an equal number of $\xi$ and $\eta$.
Taking also in account the discrete symmetry, the dimension of the fields
 and the fact that we are looking
at an integrated functional, the explicit most general form of $\DD$ is
\eqa
\DD&=&\int
\xi^*_a\eta_b ~f_{ab}[\ff,\ff^*,b,b^*]
+\xi_a\eta^*_b ~f_{ab}[\ff^*,\ff,b^*,b]
\nonumber\\
&&
+\pam\xi^*_a\eta_b ~g_{\mu ab}[\ff,\ff^*]
+\pam\xi_a\eta^*_b ~g_{\mu ab}[\ff^*,\ff]
\nonumber\\
&&
+\pam\xi^*_a\pan\eta_b ~h_{\mu\nu ab}(\ff,\ff^*)
+\pam\xi_a\pan\eta^*_b ~h_{\mu\nu ab}(\ff^*,\ff)
\nonumber\\
&&
+\xi^*_a\xi^*_b\eta_c\eta_d ~l_{1~ab|cd}(\ff,\ff^*)
+\xi_a\xi_b\eta^*_c\eta^*_d ~l_{1~ab|cd}(\ff^*,\ff)
\nonumber\\
&&
+\xi^*_a\xi_b\eta^*_c\eta_d ~l_{2~a|b|c|d}(\ff,\ff^*)
+n[\ff,\ff^*,b,b^*]
\ena
{}From Lorentz invariance and applying $\W^{f(\pm)}_{(ab)}~~(a\ne b)$
to this expression, it reduces to:
\eqa
\DD&=&\int
\xi^\dag   \eta ~f[\ff,\ff^*,b,b^*]
-\eta^\dag \xi ~f[\ff^*,\ff,b^*,b]
\nonumber\\
&&
+\pam\xi^\dag\eta ~
(\part_\mu\ff g^{(1)}(\ff,\ff^*)+\part_\mu\ff^* g^{(2)}(\ff,\ff^*))
\nonumber\\
&&
-\eta^\dag\pam\xi ~
(\part_\mu\ff^* g^{(1)}(\ff^*,\ff)+\part_\mu\ff g^{(2)}(\ff^*,\ff))
\nonumber\\
&&
+\pam\xi^\dag\part_\mu\eta ~h(\ff,\ff^*)
-\pam\eta^\dag\part_\mu\xi ~h(\ff^*,\ff)
+\xi^\dag\eta\eta^\dag\xi ~l_{2}(\ff,\ff^*)
\nonumber\\
&&
+(\xi^\dag\eta)^2 ~l_{1}(\ff,\ff^*)
+(\eta^\dag\xi)^2 ~l_{1}(\ff^*,\ff)
+n[\ff,\ff^*,b,b^*]
\ena
{}From $U(1)^N_b\otimes U(1)^N_f$ and
$\S^{(\pm)}\DD|_{\part^0}=0$ in the sector without derivatives, we get
$l_1,l_2$ constants and:
$$
\left. \DD\right|_{\part^0}=
\int -{1\over4}(2l_1+l_2)
\left(b^{\dagger}\ff-\ff^{\dagger}b
      -\xi^{\dagger}\eta-\eta^{\dagger}\xi\right)^2
$$$$
+ {1\over4}(2l_1-l_2)
\left(~b^{\dagger}\ff+\ff^{\dagger}b
      -\xi^{\dagger}\eta+\eta^{\dagger}\xi\right)^2
$$
\eq
+n_0 b^\dag b
-f_0 (b^\dag\ff+\ff^\dag b -\xi^\dag\eta+\eta^\dag\xi)
\en
where $f_0$ and $n_0$ are constants.
Examining the sector with two derivatives, it is easy to realize
that terms proportional to $m^2$, i.e. with the structure
$m^2\part^2\ff$, are absent;
then from $\S^{(\pm)}\DD|_{b\part^2}=\S^{(\pm)}\DD|_{b^*\part^2}=0$, it
is not difficult to prove that
$$
\left. \DD\right|_{\part^2}=
n~\int
b^\dag\DD\ff
+ \ff^\dag \DD b
- \xi^\dag \DD\eta
+ \eta^\dag \DD\xi
$$
Finally we can set immediately to zero the four derivatives part
of $\DD$ because it is impossible to have diagrams with $p^4$ behaviour
(\ref{tabella}).

\def\fc{\phi^\dagger}
\def\FC{\Phi^\dagger}
\def\PC{\Psi^\dagger}
\def\pc{\psi^\dagger}

\sect{The quantum corrections.}

In the following we will use the background field method (\cite{bck}).
To this aim we split the fields as follows:
\eq
\vf\rightarrow \FF+\vf_{\mbox{quant}}
{}~~~~
\pp\rightarrow \PP+\pp_{\mbox{quant}}
\lbl{redef}
\en
where $\FF,\PP$ are the classical background fields.
Performing this splitting  (and dropping the specification quant)
the lagrangian (\ref{lag}) becomes:
$${\cal L}_{\mbox{quant}}=
+i \vf^\dag  Y(-\DD)\vf
- \psi^\dag X(-\DD)\psi
$$$$
+\vf^T M_1 \vf
+\vf^\dag M_2 \vf
+\vf^\dag M_3 \vf^*
+\pp^T N_1 \pp
+\pp^\dag N_2 \pp
+\pp^\dag N_3 \pp^*
$$$$
+\vf^T \OB_1 \pp
+\vf^\dag \OB_2 \pp
+\pp^\dag \OO_1 \vf
+\pp^\dag \OO_2 \vf^*
$$$$
+A_{2ij\kb}\vf_i\vf_j\vf^*_\kb
+A_{3i\jb\kb}\vf_i\vf^*_\jb\vf^*_\kb
+{\cal A}_{2ij\kb}\pp_i\pp_j\pp^*_\kb
+{\cal A}_{3i\jb\kb}\pp_i\pp^*_\jb\pp^*_\kb
$$$$
+{\cal B}_{3i\jb k}\vf_i\vf^*_\jb\pp_k
+{\cal B}_{4i\jb\kb}\vf_i\vf^*_\jb\pp^*_\kb
+B_{3i\jb k}\pp_i\pp^*_\jb\vf_k
+B_{4i\jb\kb}\pp_i\pp^*_\jb\vf^*_\kb
$$
\eq
+C_{ij\kb\mb}\vf_i\vf_j\vf^*_\kb\vf^*_\mb
+D_{ij\kb\mb}\pp_i\pp_j\pp^*_\kb\pp^*_\mb
+{\cal E}_{i\jb k \mb}\vf_i\vf^*_\jb\pp_k\pp^*_\mb
\lbl{lagbk}
\en
where all the coefficients can be obtained easily from (\ref{lag}) using
(\ref{redef});
explicitly  we get
%for instance:
\eqa
\vf^T M_1 \vf&=&
     \ll \FC X \vf ~\FC X \vf
    +\nu \FC Y \vf ~\FC Y \vf
%+( (\ll,X,Y)\leftrightarrow (\nu,Y,-X))
\nonumber\\
M_2&=&a~U
   +2\ll(\FC X\FF-i\PC Y\PP) X
\nonumber\\
&&
   +[i~m^2+2\nu(\FC Y\FF+i\PC X\PP)] Y
\nonumber\\
  && +2\ll X\FF \FC X
   +2\nu Y\FF \FC Y
\nonumber\\
\vf^\dag M_3 \vf^*&=&
    \ll \fc X \FF ~\fc X \FF
   +\nu \fc Y \FF ~\fc Y \FF
\nonumber\\
\pp^T N_1 \pp&=&
    -\ll \PC Y \pp ~\PC Y \pp
    -\nu \PC X \pp ~\PC X \pp
\nonumber\\
N_2&=&
   -2i\ll(\FC X\FF-i\PC Y\PP) Y
\nonumber\\
  && +[-m^2+2i\nu(\FC Y\FF+i\PC X\PP)] X
\nonumber\\
  && -2\ll Y\PP \PC Y
   -2\nu X\PP \PC X
\nonumber\\
\pp^\dag N_3 \pp^*&=&
   -\ll \pc Y \PP ~\pc Y \PP
   -\nu \pc X \PP ~\pc X \PP
\nonumber\\
\vf^T \OB_1 \pp&=&
    2i\ll \FC X \vf ~\PC Y \pp
   -2i\nu \FC Y \vf ~\PC X \pp
\nonumber\\
\vf^\dag \OB_2 \pp&=&
    2i\ll \fc X \FF ~\PC Y \pp
   -2i\nu \fc Y \FF ~\PC X \pp
\nonumber\\
\pp^\dag \OO_1 \vf&=&
    2i\ll \FC X \vf ~\pc Y \PP
   -2i\nu \FC Y \vf ~\pc X \PP
\nonumber\\
\pp^\dag \OO_2 \vf^*&=&
    2i\ll \fc X \FF ~\pc Y \PP
   -2i\nu \fc Y \FF ~\pc X \PP
\nonumber\\
A_{2ij\kb}\vf_i\vf_j\vf^*_\kb&=&
    2\ll \fc X \vf ~\FC X \vf
   +2\nu \fc Y \vf ~\FC Y \vf
\nonumber\\
A_{3i\jb\kb}\vf_i\vf^*_\jb\vf^*_\kb&=&
    2\ll \fc X \vf ~\fc X \FF
   +2\nu \fc Y \vf ~\fc Y \FF
\nonumber\\
{\cal A}_{2ij\kb}\pp_i\pp_j\pp^*_\kb&=&
   -2\ll \pc Y \pp ~\PC Y \pp
   -2\nu \pc X \pp ~\PC X \pp
\nonumber\\
{\cal A}_{3i\jb\kb}\pp_i\pp^*_\jb\pp^*_\kb&=&
   -2\ll \pc Y \pp ~\pc Y \PP
   -2\nu \pc X \pp ~\pc X \PP
\nonumber\\
{\cal B}_{3i\jb k}\vf_i\vf^*_\jb\pp_k&=&
   -2i\ll \fc X \vf ~\PC Y \pp
   +2i\nu \fc Y \vf ~\PC X \pp
\nonumber\\
{\cal B}_{4i\jb\kb}\vf_i\vf^*_\jb\pp^*_\kb&=&
   -2i\ll \fc X \vf ~\pc Y \PP
   +2i\nu \fc Y \vf ~\pc X \PP
\nonumber\\
B_{3i\jb k}\pp_i\pp^*_\jb\vf_k&=&
   -2i\ll \FC X \vf ~\pc Y \pp
   +2i\nu \FC Y \vf ~\pc X \pp
\nonumber\\
B_{4i\jb\kb}\pp_i\pp^*_\jb\vf^*_\kb&=&
   -2i\ll \fc X \FF ~\pc Y \pp
   +2i\nu \fc Y \FF ~\pc X \pp
\nonumber\\
C_{ij\kb\mb}\vf_i\vf_j\vf^*_\kb\vf^*_\mb&=&
   \ll \fc X \vf ~\fc X \vf
   +\nu \fc Y \vf ~\fc Y \vf
\nonumber\\
D_{ij\kb\mb}\pp_i\pp_j\pp^*_\kb\pp^*_\mb&=&
   -\ll \pc Y \pp ~\pc Y \pp
   -\nu \pc X \pp ~\pc X \pp
\nonumber\\
{\cal E}_{i\jb k \mb}\vf_i\vf^*_\jb\pp_k\pp^*_\mb&=&
   -2i\ll \fc X \vf ~\pc Y \pp
   +2i\nu \fc Y \vf ~\pc X \pp
\ena
\noindent
We performed the computation of Feynman graphs
in $D=4-2\ee$ and within the $\overline{\mbox{MS}}$ scheme.
In fig.s 1,2,3,4 \footnote{ Dashed lines are $\pb \pp$ propagators,
continuous lines are $\vf^* \vf$ propagators.}
two representatives for each of
four different kinds of graphs involved in the computation are given.
Here we want only to point out that the diagrams like those of
of Fig.~2 give $Z_\vf$, those similar to those of Fig.~3
(i.e. those containing at least either one factor $M_2$
 or one $N_2$) generate and cancel the overlapping divergences
proportional to currents similar to
$\FC (x) X \FF (x)~\FC (y) X \FF (y)$
while the graphs (like those ) of Fig. 4 cancel the overlapping
divergences containing currents of the kind
$\FC (x) X \FF (y)~\FC (y) X \FF (x)$.

\noindent
The one loop computation (graphs of Fig.1 ) yields :
\eqa
\dd_{(1)}\ll&=&{1\over (4\pi)^2\ee}4\ll(\ll-3\nu)
\nonumber\\
\dd_{(1)}\nu&=&-{1\over (4\pi)^2\ee}4(\ll^2-\ll\nu+2\nu^2)
\nonumber\\
\dd_{(1)}m^2&=&{1\over (4\pi)^2\ee}2m^2(\ll-\nu)
\nonumber\\
\dd_{(1)}\rr &=&-{1\over (4\pi)^2\ee}2\rr (\ll+\nu)
\lbl{1loop}
\ena

\noindent
The two loops computation yields:
\eqa
\dd_{(2)}\ll&=&{1\over (4\pi)^4}
               \left[{4\ll\over\ee^2}(10\ll^2-24\ll\nu+30\nu^2)
                    +{4\ll\over\ee}(-5\ll^2+18\ll\nu-21\nu^2)\right]
\nonumber\\
\dd_{(2)}\nu&=&-{1\over (4\pi)^4}
               \left[{4\over\ee^2}(6\ll^3-24\ll^2\nu+18\ll\nu^2-16\nu^3)
                    +{4\over\ee}(-6\ll^2+15\ll^2\nu-12\ll\nu^2+11\nu^3)
               \right]
\nonumber\\
\dd_{(2)}m^2&=&{1\over (4\pi)^4}m^2
               \left[{10\over\ee^2}-{6\over\ee}\right](\ll-\nu)^2
\nonumber\\
\dd_{(2)}\rr &=&-{1\over (4\pi)^4}2\rr
               \left[-{1\over\ee^2}(\ll^2+6\ll\nu+5\nu^2)
                    +{1\over\ee}(-\ll^2+2\ll\nu+3\nu^2)
               \right]
\nonumber\\
Z_\vf^{(2)}&=&
Z_b^{(2)}=Z_\xi^{(2)}=Z_\eta^{(2)}=
-{1\over (4\pi)^4\ee}(\ll-\nu)^2
\lbl{2loop}
\ena
where $Z_\vf$ is the wave function renormalization (
$\vf_{bare}=Z^{1/2}_\vf \vf_{ren}$).

\noindent
One could wonder why setting $Z_\vf=Z_b=Z_\xi=Z_\eta$ when $b$ and
$\vf$ have a different dimension; the answer lies in the fact that
renormalization fixes $Z_\rr Z_b$, $Z_b Z_\vf=Z_\xi Z_\eta$,
$Z_\ll (Z_b Z_\vf)^{1/2}$ and $Z_\nu (Z_b Z_\vf)^{1/2}$, while leaving
two free parameters ( $Z_b$ and $Z_\eta$, for instance).
This arbitrariness is however easily understood as the possibility of
redefining $b$ and $\eta$ inside the path integral; because of this
interpretation, this arbitrariness does not affect the physics.
Notice that there is also another natural choice for the free parameters:
$Z_\rr=1, Z_b=Z_\xi$, so that $\rr$ becomes a free constant
and not a coupling constant;
we want to stress that even in the delta gauge
($\rr=0$, \cite{Bi}) the theory does not become finite and the quantum
corrections to $\ll, \nu$ do not change.


\section{The RG flow.}
As it is easy to see $Z_\ff$, $\dd m^2$ and $\dd\ll-\dd\nu$ are
expressible as a function of $g=\ll-\nu$, in fact
from (\ref{1loop},\ref{2loop}) we get:
\eqa
Z_\ff&=&-{1\over (4\pi)^4\ee}g^2
\nonumber\\
\dd m^2&=&{1\over (4\pi)^2\ee}2m^2g
         +{1\over (4\pi)^4}m^2g^2
          \left({10\over\ee^2}-{6\over\ee}\right)
\nonumber\\
\dd g=\dd \ll -\dd \nu&=&
      {1\over (4\pi)^2\ee}8g^2
      +{1\over (4\pi)^4}g^3({64\over\ee^2}-{44\over\ee})
\ena
These are exactly the quantum corrections obtainable in the theory
$O(n\rightarrow 0)$ and this strongly suggests, even if it does
not prove, that the
perturbative expansion of $Z_\ff,\dd m^2$ and $\dd\ll-\dd\nu$ in our theory is
equal to that of the corresponding quantities in $O(n\rightarrow 0)$
theory \footnote{
Would we have chosen $Z_\rr '=1, Z_b '=Z_\xi '$, this would not have
been completely true, nevertheless what really matters, the physical
quantities, would have behaved exactly as $O(n\rightarrow 0)$ theory:
for instance, the two points function $<\ff^* b>$ depends only on
$Z_\ff 'Z_b '=Z_\ff Z_b=Z^2_\ff(g)$
}.
Nevertheless the two coupling constants exhibit a very peculiar
behaviour under the RG flow, moreover there are not acceptable fixed
point beside the trivial one ($\ll=\nu=0$).
In order to show this explicitly, let us compute the $\bb$ and $\gg$
functions, we get in $D=4-2\ee$:
\eqa
\bb_\ll&=&
      -2\ee\ll
      +{1\over (4\pi)^2}8(\ll^2-3\ll\nu)
      +{1\over (4\pi)^4}16(-5\ll^3+18\ll^2\nu-21\ll\nu^2)
\nonumber\\
\bb_\nu&=&
      -2\ee\nu
      -{1\over (4\pi)^2}8(\ll^2-\ll\nu+2\nu^2)
      +{1\over (4\pi)^4}16(6\ll^3-15\ll^2\nu+12\ll\nu^2-11\nu^3)
\nonumber\\
\bb_\rr&=&
      -{1\over (4\pi)^2}4\rr(\ll+\nu)
      +{1\over (4\pi)^4}8\rr(\ll^2-2\ll\nu-3\nu^2)
\nonumber\\
\gg_m&=&
      {1\over (4\pi)^2}4m^2(\ll-\nu)
      -{1\over (4\pi)^4}24m^2(\ll-\nu)^2
\ena
Integrating the $\bb$ differential equations at one loop we get easily
(integrating firstly $g(\mu)$, then $\ll(\mu)$ and finally getting
$\nu(\mu)$ as the difference of the previous two functions):
%\eqa
%\ll(\mu)&=&{ g_0\over2}
%         {1\over 1-{g_0\over \pi^2}\log({\mu\over\mu_0})}
%         {1\over 1-\left( 1-{g_0\over \pi^2}\log({\mu\over\mu_0})
%                        \right)^{1/2}
%                   \left( 1-{g_0\over 2\ll_0} \right) }
%\nonumber\\
%\nu(\mu)&=&{ g_0\over2}
%         {1\over 1-{g_0\over \pi^2}\log({\mu\over\mu_0})}
%\left(  {1\over2}
%         {1\over 1-\left( 1-{g_0\over \pi^2}\log({\mu\over\mu_0})
%                        \right)^{1/2}
%                   \left( 1-{(g_0\over 2\ll_0} \right) }
%     -1
%\right)
%\nonumber\\
%\ena
\eqa
\ll(\mu)&=&{ g_0\over2}
         {1\over 1-{g_0\over \pi^2}x}
         {1\over 1-\left( 1-{g_0\over \pi^2}x
                        \right)^{1/2}
                   \left( 1-{g_0\over 2\ll_0} \right) }
\nonumber\\
\nu(\mu)&=&{ g_0\over2}
         {1\over 1-{g_0\over \pi^2}x}
\left(  {1\over2}
         {1\over 1-\left( 1-{g_0\over \pi^2}x
                        \right)^{1/2}
                   \left( 1-{g_0\over 2\ll_0} \right) }
     -1
\right)
\nonumber\\
\ena
where $\ll_0=\ll(\mu_0),\nu_0=\nu(\mu_0)$ and $g_0=g(\mu_0)=\ll_0-
\nu_0$ and $x=log({\mu\over\mu_0})$.



It is easy to see that $\ll(\mu)$ has two singularities (fig. 5):
one is the usual Landau pole of $\ff^4$ at
$x_L=\log({\mu_L\over\mu_0})={\pi^2\over g_0}$
and the other is at
$$x_P=\log({\mu_P\over\mu_0})=
  {\pi^2\over g_0}
\left(1-{1\over
         {(1-{g_0\over 2\ll_0})^2} } \right)
$$
This latter singularity is shared also by $\nu(\mu)$ because it takes
place at a finite value of $g(\mu)$. What is the meaning of this singularity?
There are two possibilities; it could be
 either a breakdown of the perturbative expansion ( and in this case
it is probably related to the specific formulation of the theory)
or a problem intrinsic to the theory (\cite{BCCZ}).
What makes more reliable the first possibility is that this pole is
present even when the theory is free, i.e. setting $\ll=\nu$, in this
case $\bb_\ll=\bb_\nu=-{1\over\pi^2}\ll^2$ would lead to a singularity
in $x_P=\log({\mu_P\over\mu_0})=-{\pi^2\over \ll_0}$

\noindent
There is also an other singular point of the perturbative expansion of
the theory:
it happens when $\nu(\mu)$ crosses the zero and then it becomes
negative, in that case
the theory is not bounded from below anymore as can easily seen from
(\ref{prova}); this happens for

$$\log({\mu\over\mu_0})>x_Z={\pi^2\over g_0} \left(1-{1\over
{(2-{g_0\over\ll_0})^2} }\right) $$

The singular points of $\rr$ are at $x_L$ where it diverges and
at $x_P$ where it vanishes,
but differently from the previous singular behaviours, these
can be eliminated setting $\rr=0$, that, as shown by
(\ref{1loop},\ref{2loop}), does not change the physics.

\sect{Conclusion.}
In this paper we have demonstrated that the topological theory
we proposed is renormalizable and we have explicitly computed its
two loop perturbative expansion,
however the main aim of our approach, the exact computation
of the critical indexes of SAW, has revealed unreachable, nevertheless
this topological theory reveals interesting features:
\begin{enumerate}
\item even in the delta gauge it is not finite;
\item it has two phases, one of which has an explicit breaking of
      the topological character.
\end{enumerate}

There is an heuristic way to see immediately the existence of two
phases. It consists of a mean field approximation in which all the
fields are constants, whence the action can be written as
$$S=( \rr |b|^2+4\ll w^2-2i~m^2z+4\nu z^2)~ V$$
where $V$ is the volume, $w=Im (b^*\ff)+i~ Re(\xi^*\eta)$ and
$z=Re (b^*\ff)-i~ Im(\xi^*\eta)$. If we try to minimize
this action and
we consider that it is limited from below, we get immediately that
$b=w=0$ and $z=-{i m^2\over 4\nu}$, that implies
$$ <b^*\ff+\ff^* b>=0 \ne <\xi^*\eta-\eta^*\xi>={i m^2\over 2\nu}$$
while they should be equal in order not to break
the BRST symmetry.

\vskip 1cm
{\large \bf Acknowledgements.}

\noindent
I want to thank M. Caselle, D. Cassi, F. Gliozzi, R. Iengo and
N. Maggiore for useful discussions.

\begin{thebibliography}{99}
%\bibitem{ID}
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% InterEditions//Editions du CNRS (Paris 1989)
\bibitem{PC}
I. Pesando and D. Cassi, \MPL{B6} (1992) 485
\bibitem{Bi}
D. Birmingham, M. Blau, M. Rakowski and G. Thompson, \PRE{209} (1991) 129
and references therein.
\bibitem{Wi}
E. Witten, \CMP{117} (1988) 353\\
E. Witten, \CMP{118} (1988) 411
\bibitem{Fu}
K. Fujikawa, \NP{B223} (1983) 218
\bibitem{DG}
P.G. De Gennes, \PL{38A} (1972) 339
\bibitem{DCl}
P.G De Gennes, Scaling concepts in polymer physics, Cornell University Press
(Ithaca 1985)\\
J. des Cloizeaux, G. Jannink, Les polym\`eres en solution,
Les Editions de Physique (Paris 1987)
\bibitem{MK}
A.J. McKane, \PL{76A} (1980) 22
\bibitem{PS}
G. Parisi, N. Sourlas, \JPL{41} (1980) L403
\bibitem{QAP}
Y.M.P. Lam, \PR{D6} (1972)2145 and 2161\\
T.E. Clarck and J.H. Lowenstein, \NP{B113} (1976) 109 \\
P. Breintenlohner and D. Maison, \CMP{52} (1977) 11
\bibitem{QAP1}
O. Piguet and A. Rouet, \PRE{76} (1981) 1\\
C. Becchi, "Lectures on the renormalization of gauge theories" in Les
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\bibitem{bck}
J. Honerkamp, \NP{B36} (1972) 130\\
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\end{thebibliography}

\end{document}




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/cols charinfo 3 get 256 mul charinfo 4 get add def
/rows charinfo 5 get 256 mul charinfo 6 get add def
/hoff charinfo 7 get 256 mul charinfo 8 get add dup 32767 ge{65536 sub} if def
/voff charinfo 9 get 256 mul charinfo 10 get add dup 32767 ge{65536 sub}if def
/prelen 11 def}
{/dm charinfo 1 makelong 65536 div def
/dn charinfo 5 makelong 65536 div def
/cols charinfo 9 makelong def
/rows charinfo 13 makelong def
/hoff charinfo 17 makelong def
/voff charinfo 21 makelong def
/prelen 25 def
} ifelse
} ifelse
/llx hoff neg .5 sub def
/lly voff 1 add rows sub .5 add def
dm dn llx lly llx cols add lly rows add setcachedevice
cols
rows
true
fontdict /ImageMaskMatrix get dup 4 llx neg put dup 5 rows lly add put
{ charinfo prelen charinfo length prelen sub getinterval }
imagemask
} bind def


/EmptyEncoding 256 array def 0 1 255 {EmptyEncoding exch /.notdef put} for

/DefinePKFont {
4 array astore /bbox exch def
/data exch def
/encoding exch def
/vppp exch def
/hppp exch def
/ds exch def

9 dict dup begin
/FontType 3 def
/Encoding encoding def
/BuildChar /BuildPK load def
/ImageMaskMatrix [1 0 0 -1 0 0] def
/FontMatrix
[65536 ds 1048576 div hppp mul div 0 0 65536 ds 1048576 div vppp mul div 0 0]
def
/FontBBox bbox def
/CharData data def
CharData /.notdef <000000000000> put
end
definefont pop
} def


/Locate {
8 dict begin
[/newury /newurx /newlly /newllx /ury /urx /lly /llx]
{exch def} forall
newllx newlly translate
newurx newllx sub urx llx sub div
newury newlly sub ury lly sub div
scale
llx neg lly neg translate
end
} bind def

systemdict /setpacking known {savepacking setpacking} if

end

dvitops begin
%%BeginFont: cmr10-360
/cmr10-360 10485760 326455 326455
EmptyEncoding 256 array copy
dup 8 /c8 put
dup 10 /c10 put
dup 40 /c40 put
dup 41 /c41 put
dup 46 /c46 put
dup 49 /c49 put
dup 50 /c50 put
dup 51 /c51 put
dup 52 /c52 put
dup 70 /c70 put
dup 84 /c84 put
dup 97 /c97 put
dup 98 /c98 put
dup 99 /c99 put
dup 100 /c100 put
dup 101 /c101 put
dup 102 /c102 put
dup 103 /c103 put
dup 104 /c104 put
dup 105 /c105 put
dup 107 /c107 put
dup 108 /c108 put
dup 109 /c109 put
dup 110 /c110 put
dup 111 /c111 put
dup 112 /c112 put
dup 114 /c114 put
dup 115 /c115 put
dup 116 /c116 put
dup 117 /c117 put
dup 118 /c118 put
dup 119 /c119 put

33 dict dup begin
/c8<00241D22FD2103FFFE0003FFFE00000F8000000F8000000F8000000F8000000F8000003F
E00001FFFC0007EFBF001F8F8FC03E0F83E03E0F83E07C0F81F0FC0F81F8FC0F81F8FC0F
81F8FC0F81F8FC0F81F8FC0F81F87C0F81F03E0F83E03E0F83E01F8F8FC007EFBF0001FF
FC00003FE000000F8000000F8000000F8000000F8000000F800003FFFE0003FFFE00>def
/c10<0024
1F23FE22001FF000007FFC0001F83F0003E00F80078003C00F8003E01F0001F03F0001F8
3E0000F87E0000FC7E0000FC7E0000FC7E0000FC7E0000FC7E0000FC7E0000FC3E0000F8
3F0001F83F0001F81F0001F01F0001F00F0001E00F8003E0078003C00380038003C00780
C1C00706C1C0070660C0060C60C0060C60E00E0C7FE00FFC7FE00FFC3FE00FF83FE00FF8
>def
/c40<00130C32FD240030006000C001800380070006000E000E001C001C003800380038007800
700070007000F000F000F000F000F000F000F000F000F000F000F000F000F000F0007000
7000700078003800380038001C001C000E000E00060007000380018000C000600030>def
/c41<0013
0C32FD24C0006000300018001C000E000600070007000380038001C001C001C001E000E0
00E000E000F000F000F000F000F000F000F000F000F000F000F000F000F000F000E000E0
00E001E001C001C001C0038003800700070006000E001C00180030006000C000>def
/c46<000E0606
FC0578FCFCFCFC78>def
/c49<00191121FD2000600000E00003E000FFE000FDE00001E00001E00001
E00001E00001E00001E00001E00001E00001E00001E00001E00001E00001E00001E00001
E00001E00001E00001E00001E00001E00001E00001E00001E00001E00001E00001E0007F
FF807FFF80>def
/c50<00191421FE2003F8000FFE001C1F00300F806007C06003E0F803E0FC03F0FC
01F0FC01F07801F03003F00003F00003E00003E00007C0000780000F00001E00001C0000
380000700000E00001C0000380300700300E00300C00701800603FFFE07FFFE0FFFFE0FF
FFE0>def
/c51<00191422FE2003F8000FFE001E0F803807C03807C07C07E07C03E07C03E03C07E018
07E00007C00007C0000F80000F00003E0003FC0003F800001E000007800007C00003E000
03E00003F03003F07803F0FC03F0FC03F0FC03E0F803E06007C03007801C0F000FFE0003
F800>def
/c52<00191621FF20000380000380000780000F80000F80001F80003F8000378000678000
E78000C7800187800387800307800607800E07800C0780180780380780300780600780E0
0780FFFFFCFFFFFC00078000078000078000078000078000078000078000FFFC00FFFC>def
/c70<00
211C22FE21FFFFFFE0FFFFFFE007C007E007C001E007C000E007C0006007C0007007C000
7007C0003007C0003007C0303007C0303007C0300007C0300007C0700007C0F00007FFF0
0007FFF00007C0F00007C0700007C0300007C0300007C0300007C0300007C0000007C000
0007C0000007C0000007C0000007C0000007C0000007C00000FFFF0000FFFF0000>def
/c84<00241F
22FE217FFFFFFC7FFFFFFC7C07C07C7007C01C6007C00C6007C00CE007C00EE007C00EC0
07C006C007C006C007C006C007C0060007C0000007C0000007C0000007C0000007C00000
07C0000007C0000007C0000007C0000007C0000007C0000007C0000007C0000007C00000
07C0000007C0000007C0000007C0000007C0000007C00003FFFF8003FFFF80>def
/c97<00191615FE
141FF0003FFC007C1E007C0F007C078038078000078000078003FF800FFF801F07803E07
807C0780F80780F8078CF8078CF8078C7C0F8C7C3FDC3FF3F80FC1E0>def
/c98<001C1823FF220F00
00FF0000FF00001F00000F00000F00000F00000F00000F00000F00000F00000F00000F00
000F00000F0FC00F7FE00FE0F80F803C0F003C0F001E0F001E0F001F0F001F0F001F0F00
1F0F001F0F001F0F001F0F001E0F003E0F003C0F80780EE0F00E7FE00C1F80>def
/c99<00161215FE
1401FE0007FF000F0F801E0F803C0F807C0700780000F80000F80000F80000F80000F800
00F80000F800007C00007C00C03C00C01E01800F830007FE0001F800>def
/c100<001C1823FE220000
F0000FF0000FF00001F00000F00000F00000F00000F00000F00000F00000F00000F00000
F00000F001F8F007FEF00F07F01E01F03C00F07C00F07800F0F800F0F800F0F800F0F800
F0F800F0F800F0F800F07800F07800F03C00F03C01F01F07F807FCFF03F0FF>def
/c101<00161315FF
1401FC0007FF000F0F801E03C03C03C07C03E07801E0FFFFE0FFFFE0F80000F80000F800
00F80000F800007800007C00603C00601E00C00F838007FF0000FC00>def
/c102<000F11230022001E
00007F0001E78001CF8003CF800787000780000780000780000780000780000780000780
00078000FFF800FFF8000780000780000780000780000780000780000780000780000780
000780000780000780000780000780000780000780000780007FFC007FFC00>def
/c103<00191621FF
1500007803F8FC0FFF9C1E0F1C3C07983C07807C07C07C07C07C07C07C07C03C07803C07
801E0F001FFE0033F8003000003800003800003FFF001FFFC01FFFF03FFFF87800F87000
3CE0001CE0001CE0001CE0001C7000383800701E01E00FFFC001FE00>def
/c104<001C1923FF220F00
0000FF000000FF0000001F0000000F0000000F0000000F0000000F0000000F0000000F00
00000F0000000F0000000F0000000F0000000F0FC0000F3FE0000FE0F0000FC078000F80
78000F0078000F0078000F0078000F0078000F0078000F0078000F0078000F0078000F00
78000F0078000F0078000F0078000F0078000F007800FFF3FF80FFF3FF80>def
/c105<000D0B220021
0E001F003F003F001F000E0000000000000000000000000000000F00FF00FF001F000F00
0F000F000F000F000F000F000F000F000F000F000F000F000F000F00FFE0FFE0>def
/c107<001A1723
FF220F0000FF0000FF00001F00000F00000F00000F00000F00000F00000F00000F00000F
00000F00000F00000F0FFC0F0FFC0F07E00F03800F07000F0E000F1C000F38000F78000F
FC000FBE000F1E000F1F000F0F800F07800F07C00F03C00F03E00F01F0FFE7FEFFE7FE>def
/c108<00
0D0C2300220F00FF00FF001F000F000F000F000F000F000F000F000F000F000F000F000F
000F000F000F000F000F000F000F000F000F000F000F000F000F000F000F000F000F00FF
F0FFF0>def
/c109<002B2815FF140F0FE03F80FF3FF0FFC0FF607981E01FC03F00F00F803E00F00F00
3C00F00F003C00F00F003C00F00F003C00F00F003C00F00F003C00F00F003C00F00F003C
00F00F003C00F00F003C00F00F003C00F00F003C00F00F003C00F00F003C00F0FFF3FFCF
FFFFF3FFCFFF>def
/c110<001C1915FF140F0FC000FF3FE000FFE0F0001FC078000F8078000F007800
0F0078000F0078000F0078000F0078000F0078000F0078000F0078000F0078000F007800
0F0078000F0078000F0078000F007800FFF3FF80FFF3FF80>def
/c111<00191615FF1400FC0007FF80
0F03C01E01E03C00F0780078780078F8007CF8007CF8007CF8007CF8007CF8007CF8007C
7800787C00F83C00F01E01E00F03C007FF8000FC00>def
/c112<001C181FFF140F0FC0FF7FE0FFE0F8
0F807C0F003C0F003E0F003E0F001F0F001F0F001F0F001F0F001F0F001F0F001F0F003E
0F003E0F007C0F80780FE1F00F7FE00F1F800F00000F00000F00000F00000F00000F0000
0F00000F0000FFF000FFF000>def
/c114<00141115FF140F3E00FF7F00FFCF801F8F800F8F800F8700
0F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F0000
0F0000FFF800FFF800>def
/c115<00140F15FE140FCC3FFC703C601CE00CE00CF000FC007FE03FF01F
F807FC007EC01EC00EE00EE00EF00CF83CFFF887E0>def
/c116<00130E1FFF1E030003000300030007
00070007000F001F003F00FFF8FFF80F000F000F000F000F000F000F000F000F000F000F
0C0F0C0F0C0F0C0F0C0F0C079803F001E0>def
/c117<001C1915FF140F007800FF07F800FF07F8001F
00F8000F0078000F0078000F0078000F0078000F0078000F0078000F0078000F0078000F
0078000F0078000F0078000F0078000F00F8000F01F8000783FC0003FE7F8001FC7F80>def
/c118<00
1A1715FF14FFE3FEFFE3FE1F00F80F00600F00600780C00780C007C1C003C18003C18001
E30001E30001E30000F60000F60000FE00007C00007C00007C00003800003800>def
/c119<00242115
FF14FFCFFCFF80FFCFFCFF801F01E03E000F01E01C000F01F018000F01F0180007837030
000783783000078378300003C638600003C63C600003C63C600001EC1CC00001EC1EC000
01EC1EC00000F80F800000F80F800000F80F8000007007000000700700000070070000>def
end 0 -13 41 37 DefinePKFont
%%EndFont
/_cmr10-360 /cmr10-360 FF
/F43 _cmr10-360 786432 SF
%%BeginFont: cmmi10-360
/cmmi10-360 10485760 326455 326455
EmptyEncoding 256 array copy
dup 65 /c65 put
dup 66 /c66 put
dup 68 /c68 put
dup 69 /c69 put
dup 77 /c77 put
dup 78 /c78 put
dup 88 /c88 put
dup 120 /c120 put
dup 121 /c121 put

10 dict dup begin
/c65<00
252123FE22000001C000000001C000000003C000000003C000000007C000000007C00000
000FC00000001FC00000001FC000000037E000000033E000000063E0000000E3E0000000
C3E000000183E000000183E000000303E000000303E000000603E000000E03E000000C03
E000001803E000001803F000003FFFF000003FFFF000006001F00000C001F00000C001F0
00018001F000018001F000030001F000070001F0000F0001F000FFE01FFF80FFE03FFF80
>def
/c66<00262322FE2100FFFFFC0000FFFFFF000007C00F800007C007C0000F8007C0000F8003C0
000F8003E0000F8003E0001F0003C0001F0007C0001F0007C0001F000F80003E001F0000
3E003E00003E007C00003FFFF000007FFFF000007C00F800007C007C00007C003E0000F8
003E0000F8001E0000F8001F0000F8001F0001F0001E0001F0003E0001F0003E0001F000
7C0003E0007C0003E000F80003E003F00007E007C000FFFFFF8000FFFFFC0000>def
/c68<00292522
FE2100FFFFF80000FFFFFF000007C01F800007C007C0000F8003E0000F8001E0000F8000
F0000F8000F0001F0000F0001F0000F8001F0000F8001F0000F8003E0000F8003E0000F8
003E0000F8003E0000F8007C0000F0007C0001F0007C0001F0007C0001F000F80003E000
F80003E000F80003C000F80007C001F000078001F0000F0001F0001F0001F0001E0003E0
003C0003E000F80003E001E00007E00FC000FFFFFF0000FFFFF80000>def
/c69<00252322FE2100FF
FFFFE000FFFFFFE00007C003E00007C001E0000F8000E0000F8000E0000F8000C0000F80
00C0001F0000C0001F00C0C0001F00C0C0001F00C0C0003E018000003E018000003E0780
00003FFF8000007FFF0000007C070000007C070000007C07000000F806000000F8060180
00F806018000F800030001F000030001F000060001F000060001F0000E0003E0000C0003
E0001C0003E000780007E001F800FFFFFFF800FFFFFFF000>def
/c77<00303122FE2100FFE00003FF
8000FFE00003FF800007E00007F0000007E0000DF000000DE0000FE000000DE0001BE000
000DE0001BE000000DE00033E0000019E00067C0000019E00067C0000018F000C7C00000
18F000C7C0000030F0018F80000030F0030F80000030F0030F80000030F0060F80000060
F0061F00000060F00C1F00000060F0181F0000006078181F000000C078303E000000C078
303E000000C078603E000000C078C03E0000018078C07C0000018079807C000001807980
7C000001807B007C000003007E00F8000003003E00F8000003003C00F800000F803C01F8
0000FFF0383FFF8000FFF0303FFF8000>def
/c78<00282822FE2100FFE00FFF00FFE00FFF0007E000
F00007F000E0000DF000C0000DF000C0000CF800C0000CF800C00018F8018000187C0180
00187C018000187C018000303E030000303E030000303F030000301F030000601F060000
601F860000600F860000600F860000C00FCC0000C007CC0000C007CC0000C003EC000180
03F800018003F800018001F800018001F800030001F000030000F000030000F0000F8000
F000FFF0006000FFF0006000>def
/c88<00292822FF21007FFE0FFF007FFE1FFF0003F003F00003F0
03C00003F003000001F006000001F80E000001F81C000000F838000000FC30000000FC60
0000007CC00000007F800000003F000000003F000000003F000000001F000000001F8000
00003F800000006F80000000CFC00000018FC000000387C000000707E000000E03E00000
0C03E000001803F000003001F000006001F00000C001F80001C000F80007E001FC00FFF8
0FFFC0FFF80FFFC0>def
/c120<001C1615FE1403E0F00FF3F81C3F1C303E3C303C7C603C7C603C7800
783000780000780000780000F00000F00038F00878F00CF9E018F9E018F1E0306370E07E
3FC03C1F00>def
/c121<0018181FFF140F800C1FC01E31E01E61E03CC1E03CC1E03CC3C03C03C07807
80780780780780780F00F00F00F00F00F00F00F00F01E00F01E00F03E00787E003FFC001
F3C00003C00003C00007803C07007C0F007C1E00783C007078003FE0001F8000>def
end 0 -10 51 35 DefinePKFont
%%EndFont
/_cmmi10-360 /cmmi10-360 FF
/F44 _cmmi10-360 786432 SF
%%BeginFont: cmr8-300
/cmr8-300 8388608 272046 272046
EmptyEncoding 256 array copy
dup 49 /c49 put
dup 50 /c50 put
dup 51 /c51 put
dup 52 /c52 put

5 dict dup begin
/c49<00120C15
FE1403000F00FF00F7000700070007000700070007000700070007000700070007000700
070007007FF07FF0>def
/c50<00120D15FE141F803FE071F0F8F0F878F87870780078007800F000E0
01C0038007000E000C18181830387FF0FFF0FFF0>def
/c51<00120F15FF140FC01FF03078783C783C
783C103C007800F007E007E00078003C001E701EF81EF81EF83C70383FF00FC0>def
/c52<00120F15
FF14007000F000F001F003F0077006700E701C70187030707070E070FFFEFFFE00700070
0070007003FE03FE>def
end 0 0 16 21 DefinePKFont
%%EndFont
/_cmr8-300 /cmr8-300 FF
/F13 _cmr8-300 524288 SF
%%BeginFont: line10-300
/line10-300 10485760 272046 272046
EmptyEncoding 256 array copy
dup 0 /c0 put
dup 64 /c64 put

3 dict dup begin
/c0<00002C2C012A0000000000300000000000700000000000E000000000
01C0000000000380000000000700000000000E00000000001C0000000000380000000000
700000000000E00000000001C0000000000380000000000700000000000E00000000001C
0000000000380000000000700000000000E00000000001C0000000000380000000000700
000000000E00000000001C0000000000380000000000700000000000E00000000001C000
0000000380000000000700000000000E00000000001C0000000000380000000000700000
000000E00000000001C0000000000380000000000700000000000E00000000001C000000
0000380000000000700000000000E00000000000C00000000000>def
/c64<00002C2C012AC0000000
0000E000000000007000000000003800000000001C00000000000E000000000007000000
000003800000000001C00000000000E000000000007000000000003800000000001C0000
0000000E000000000007000000000003800000000001C00000000000E000000000007000
000000003800000000001C00000000000E000000000007000000000003800000000001C0
0000000000E000000000007000000000003800000000001C00000000000E000000000007
000000000003800000000001C00000000000E00000000000700000000000380000000000
1C00000000000E000000000007000000000003800000000001C00000000000E000000000
0070000000000030>def
end -1 -1 43 43 DefinePKFont
%%EndFont
/_line10-300 /line10-300 FF
/F68 _line10-300 655360 SF
%%BeginFont: cmsy8-300
/cmsy8-300 8388608 272046 272046
EmptyEncoding 256 array copy
dup 121 /c121 put

2 dict dup begin
/c121<00100B1DFE1604000E000E000E000E000E000400040075C0FFE075C0
04000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E00
>def
end 0 -6 13 23 DefinePKFont
%%EndFont
/_cmsy8-300 /cmsy8-300 FF
/F15 _cmsy8-300 524288 SF
%%BeginFont: circle10-300
/circle10-300 10485760 272046 272046
EmptyEncoding 256 array copy
dup 110 /c110 put

2 dict dup begin
/c110<00004040201F0000003FFC000000000003FFFFC0000000001FE007F8000000007E00007E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>def
end -32 -32 32 32 DefinePKFont
%%EndFont
/_circle10-300 /circle10-300 FF
/F70 _circle10-300 655360 SF
end
%%EndProlog
%%BeginSetup
%%PaperSize: a4
%%EndSetup

%%Page: 1 1
dvitops begin
/#copies 1 def BP
25400000 473628672 1000 72 72 842 SC
26212 1637500 27061505 28612922 R
26212 1637500 27061505 27918622 R
1637500 26213 22249869 27905516 R
1637500 26213 22944169 27905516 R
26212 1637500 16521245 28612922 R
26212 1637500 16521245 27918622 R
26212 7320280 5897145 28265772 R
1637500 26213 9195719 34388706 R
1637500 26213 9890019 34388706 R
26212 1637500 14055825 28612922 R
26212 1637500 14055825 27918622 R
26212 1637500 3421245 28612922 R
26212 1637500 3421245 27918622 R
26212 927480 24487355 16887138 R
26212 927480 22657285 16887138 R
26212 927480 20827215 16887138 R
26212 927480 18997145 16887138 R
26212 1637500 27155825 17234288 R
26212 1637500 27155825 16539988 R
26212 1637500 16521245 17234288 R
26212 1637500 16521245 16539988 R
26212 927480 11387355 16887138 R
26212 927480 9557285 16887138 R
26212 927480 7727215 16887138 R
26212 927480 5897145 16887138 R
26212 1637500 14055825 17234288 R
26212 1637500 14055825 16539988 R
26212 1637500 3421245 17234288 R
26212 1637500 3421245 16539988 R
26212 1637500 27155825 6685777 R
26212 1637500 27155825 5991477 R
26212 1637500 16521245 6685777 R
26212 1637500 16521245 5991477 R
26212 1637500 14054515 6685777 R
26212 1637500 14054515 5991477 R
26212 1637500 3421245 6685777 R
26212 1637500 3421245 5991477 R
BO
F13 F
4991678 5460985(2)Z
15178148(1)X
18281130(2)X
28474052(3)X
5023254 16009496(3)Z
15211034(4)X
18044315(3)X
28232095(4)X
5007466 27388130(3)Z
15200928(2)X
11221352 32522020(2)Z
18154829 27388130(2)Z
28379732(2)X
EO
BO
F15 F
12974342 36005375(y)Z
17488324(y)X
EO
BO
F43 F
4423323 5343021(\012)Z
14609793(\012)X
3768323 11515900(Fig.1)Z
5767176(Tw)X
6882040(o)X
7536651(diagrams)X
10907394(con)X
12069621(tributing)X
15387882(to)X
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