%Paper: 
%From: fanchiot@nxth04.cern.ch
%Date: Wed, 9 Mar 1994 12:00:04 +0000 (WET)

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\newcommand{\smallz}{{\scriptscriptstyle Z}} %  a smaller Z
\newcommand{\smallw}{{\scriptscriptstyle W}} %
\newcommand{\smallh}{{\scriptscriptstyle H}} %
\newcommand{\LT}{{\scriptscriptstyle L.T.}} %
\newcommand{\mz}{m_\smallz}
\newcommand{\mw}{m_\smallw}
\newcommand{\mh}{m_\smallh}
\newcommand{\mt}{m_t}
\newcommand{\dr}{\mbox{$ \delta \rho$}}
\newcommand{\dro}[1]{\mbox{$ \delta \rho^{{\scriptscriptstyle
(#1)}}$}}
\newcommand{\azz}{\mbox{$ A_{\smallz \smallz} $}}
\newcommand{\Azz}[1]{\mbox{$ A_{\smallz \smallz}^{{\scriptscriptstyle
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% (x_1) \,
%                     #4 (x_2) \right] \, | \,0 >}
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% (0)
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\begin{document}
\begin{titlepage}
\begin{flushright}
	\small
 	CERN-TH.7180/94\\
        DFPD 94/TH/12\phantom{94}\\
        NYU-Th-94/02/01
\end{flushright}

\begin{center}
{\Large Two-loop next-to-leading $\mt$ corrections to the $\rho$
parameter}
 \\
\vspace{1.5cm}
Giuseppe Degrassi\\
{\em Dipartimento di Fisica, Universit\`a  di Padova,
Sezione INFN di Padova \\
Via Marzolo 8, 35131 Padova, Italy} \\
Sergio Fanchiotti \\
{\em Theory Division, CERN, CH-1211 Geneva 23, Switzerland}\\
Paolo Gambino\\
{\em Department of Physics, New York University, New York, NY 10003,
USA}
\end{center}
\vspace{2cm}
\begin{center}
{\bf Abstract}
\end{center}
\vspace{0.2cm}
The $O(\gmuq)$ correction to the $\rho$ parameter is computed within
the
Standard Model using the current algebra formulation of radiative
corrections.
This approach is proved to be equivalent to the effective Lagrangian
method
proposed by Barbieri {\em et al.} Using the same framework, the
$O(\gmud)$
correction to the ratio of neutral-to-charged current amplitudes is
analysed in an $SU(2)$ model. The $O(\gmud)$ contribution is shown to
be
numerically comparable to the leading $O(\gmuq)$ term for realistic
values
of the top mass. The resummation of higher-order effects is
discussed.\par
\vspace{0.2cm}\nopagebreak
{\small \noindent CERN-TH.7180/94\\ February~1994}
\end{titlepage}
\setcounter{footnote}{0}
\renewcommand{\thefootnote}{\arabic{footnote}}

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\section{Introduction}
The key role played by a heavy top quark in the analysis of \ew\ data
has
by now been emphasized for more than fifteen years \cite{b1}.
One-loop
virtual top effects are present in the \selfs\ of the $W$ and $Z$
vector
bosons \cite{b1,b2} and in the $Zb \bar{b}$ vertex \cite{b3}.
Self-energy
contributions affect important \ew\ quantities like the $\rho$
parameter,
which measures the relative strength, at low momentum transfer,
between the
neutral- and charged-current interactions
\be
\rho = 1 + \dr \label{e1i}
\ee
and $\Delta r$ \cite{b4}, the correction relevant to the $\mw$--$\mz$
interdependence.

As is  well known, at the one-loop level the top contribution to \dr\
is given by  \cite{b1}
\ben \label{e2i} \beq
\dr =& N_c x_t \\ \label{e2ai}
x_t =& \frac{G_\mu \mt^2}{8\, \pi^2\, \sqrt2} \label{e2bi}
\eeq \een
where $N_c$ is the colour factor,
while the leading top behaviour in $\Delta r$ can be described by
\cite{b2,b4}
\be
\Delta r \simeq {c^2 \over s^2} \dr \,, \label{e3i}
\ee
where $s$ is an abbreviation for $\sin \theta_W \equiv g^\prime /
\sqrt{g^{\prime 2} + g^2}$,  $c^2= 1-s^2$, and $g$ and $g^\prime$ are
the
coupling constants of $SU(2)$ and $U(1)$ respectively.

Since \dr\ increases rapidly with $\mt$, it is possible to get an
upper
bound on the top mass using \ew\ precision measurements. A recent
analysis
of the available precision data yields $\mt= 164^{+16+18}_{-17-21}$
GeV
with $\mh= 300^{+700}_{-240}$ GeV \cite{b5}. This bound applies
within
the Standard Model (SM) or assuming that virtual effects, due to
other
kinds of undiscovered physics, contribute very little to \ew\
quantities.

With the increasing evidence that the top is heavy, attention has
been paid to
two-loop corrections involving this quark. Following the work of van
der
Bij and Hoogeveen on the two-loop leading top effect to the $\rho$
parameter,
namely the $O(\gmuq)$ contribution, QCD corrections to the one-loop
$\mt^2$
term have been computed both for the \selfs\ of the vector bosons
\cite{b7}
and for the $Z \rightarrow b \bar{b}$ decay width \cite{b8}.
Concerning the
latter,
the \ew\ two-loop $\mt^4$ corrections are now also available
\cite{b9,b10,b11}.

The first calculation \cite{b6} of the one-particle irreducible (1PI)
$O(\gmuq)$ contribution to the $\rho$ parameter was performed a few
years ago
in the approximation of a small Higgs mass, $\mh \ll \mt$. Recently
it was
extended by Barbieri {\em et al.} to the case of arbitrary $\mh$.
Subsequently,
a simple explicit analytic expression for the Higgs-mass correction
was derived
\cite{b11}.
%%% An interesting feature pointed out in \efe{b9} is that the
%%% $O(\gmuq)$ term has very little to do with the gauge sector of
% SM.
An interesting feature of the leading $\mt$ corrections, first noted
in
\efe{b11bis} and then developed to two-loops in \efe{b9}, is that
these terms
have very little to do with the gauge sector of the SM.
Indeed the calculations presented in \efe{b9}\ and \efe{b11}\ were
performed
using
a Yukawa Lagrangian involving a Higgs doublet plus the top and bottom
fields.
The connection between physical quantities and the renormalization
constants
of the Yukawa Lagrangian was obtained by exploiting the Ward
identities
of the SM Lagrangian with the gauge interaction switched off. The use
of
this ``gaugeless'' limit greatly simplifies the two-loop calculation,
making it quite manageable.

The aim of this paper is twofold. First, we derive explicitly the
connection
between the SM and the effective Yukawa Lagrangian used in \efe{b9}.
We
consider the SM diagrams contributing to \dr\ to leading order in
$\mt$
and we evaluate them employing techniques used in the current-algebra
formulation of radiative corrections \cite{b12}. This framework
allows
us to easily enforce the relevant Ward identities, leading us to a
result
that differs from the one presented in \efs{b9}{b11} by subleading
contributions
$O(\gmud)$. These terms are clearly due to the gauge sector of the
theory,
as can be  seen by noticing that the $O(\gmuq)$ contribution
corresponds
to effects proportional to $g_t^4$, where $g_t$ is the top Yukawa
coupling,
while a $O(\gmud)$ term scales as $g_t^2 g^2$, with $g$ the gauge
coupling.
As a second result we perform a complete calculation (reducible plus
irreducible parts) through $O(\gmud)$ of the $\rho$ parameter in an
$SU(2)$ theory. Although this result has no direct implication
in the analysis of \ew\ precision experiments, it is meant to provide
an
estimate of the size of subleading two-loop contributions involving
the top.

The plan of this paper is as follows. Section 2 presents the
derivation
of the 1PI $O(\gmuq)$ term in the $\rho$ parameter within the SM,
using
the formalism of current correlation functions and their associated
current algebra. In section 3 we derive the 1PI $\mt$ \selfs\
contribution
to \dr\ in an  $SU(2)$ theory. Section 4 is devoted to the
calculation of the
$\rho$ parameter in $SU(2)$. Finally, in section 5 we discuss the
resummation
of higher-order effects and draw our conclusions.
\section{One-particle irreducible O(\gmuq) correction  to \dr\ in
\newline
        $SU(2) \times U(1)$}
In this section we derive the 1PI\footnote{As the mass counterterm
diagrams
are needed to cancel divergences in the irreducible two-loop
corrections,
we include them in the latter.}
two-loop $m_t^4$ contribution
to \dr\ in the SM, keeping the Higgs mass arbitrary. As far as only
leading terms are concerned, we can identify \dr\ with $\Delta$,
where
\be
\Delta \equiv \frac{\aww (0)}{\mw^2} - \frac{\azz (0)}{\mz^2} .
\label{e1.0}
\ee
In \equ{e1.0} $A(q^2)$ represents the transverse part of the \self,
i.e.\ the
cofactor
of $g^{\mu \nu}$ in the \self\ tensor $\Pi^{\mu \nu} (q^2)$.

Just by power-counting inspection it is easy to realize that the only
diagrams that can contribute to the leading $m_t^4$ term are those
containing,
besides top and bottom, the physical and unphysical scalars. A
suitable
choice of the tadpole counterterm allows one to neglect the tadpole
diagrams in the calculation \cite{b13}. Enforcing it, the only
topologies in the \selfs\
contributing to the leading term are those depicted in fig.~1. In the
figure, wavy lines represent vector bosons, dashed lines scalars
(physical or unphysical) while solid lines are fermions.
\par In order to fix our notation we write the part of the
SM Lagrangian density that describes the interaction of the $W,\,Z$
and  scalars with fermions as \cite{b12}
\be
{\cal L}_{int}  = - \frac{g}{\sqrt{2}}(W^{\dagger}_\mu J_W^\mu +
{\rm h.c.})
        -  \frac{g}{c}Z_\mu J^\mu _Z
        - \frac{g}{2\, \mw} \left[ \Phi_1 S_1 + \Phi_2 S_2
        + \sqrt2 \,( \Phi^\dagger S + {\rm h.c.}) \right] \label{e1}
\ee
where  $\mw$ stands for the mass of the $W$ boson,
%$c$ is an abbreviation for $\cos \theta_W =g/ \sqrt{g^2+g^{\prime
% 2}}$
%($g^\prime$
%being the $U(1)$ coupling constant),
$J_Z^\mu$ and $J_W^\mu$ are the
fermionic currents coupled to $Z$ and $W$ respectively,  $W^\dagger$
is the
field that creates a $W^+$ meson, $\Phi_1$ is the physical Higgs
boson,
$\Phi_2$ and $\Phi$ the unphysical
counterparts associated with the $Z$ and $W$ and
\ben \label{e2} \beq
S_1 =&  \bar{\psi}\, m^0\, \psi     \label{e2a0} \\
S_2 =& 2 \,\partial_\mu J^\mu_Z = -i \bar{\psi}\, m^0\, C_3 \gamma_5
\psi
                   \label{e2a} \\
S =& -i \, \partial_\mu J^\mu_W = \bar{\psi} \Gamma \psi. \label{e2b}
\eeq
\een
In \eqs{e2}, $\psi$ represents the column vector $\psi \equiv
(t,b)^T, \:
m^0, \: C_3$ and $\Gamma$ are the $2 \times 2$ matrices
\ben \label{e2cde} \beq
m^0 =& \left(  \begin{array}{cc} m_t^0 & 0 \\
                                  0 & 0 \end{array} \right)
\label{e2c} \\
C_3 =& \left(  \begin{array}{cr} 1 & 0 \\
                                 0 & -1 \end{array} \right)
\label{e2d} \\
\Gamma =& \left(  \begin{array}{cc} 0 & 0 \\
                   -m_t^0 \, a_+ & 0 \end{array} \right) ,
\label{e2e}
\eeq \een
$a_+ \equiv \frac{1+\gamma_5}{2}$, and the superscript
$0$ on $m_t$ refers to the bare mass.
As is evident from \Eqs{e2c}{e2e} we are considering only the third
generation and taking the bottom quark as massless.

We begin by studying \azz (0). Using current correlation functions we
can
combine the amplitudes of fig.~1a,b, where the continuous line
represents a top
and the dashed one a Higgs or $\Phi_2$, into the expression
\beq
\lefteqn{\Pi_{(a,b)}^{\mu \nu}(q^2) = \sum_{j=1}^2  \frac{g^4}{4
\,c^2\,\mw^2}
\frac{1}{2} \ik \frac1{k^2 - m_j^2}} \nonumber \\
& \times  \tint \quaesteso{J_Z^\mu}{J_Z^\nu}{S_j}{S_j} ,\label{e3}
\eeq
where $n$ is the space-time dimension, $\mu$ is the 't Hooft mass
scale, $T^*$ is the covariant time-ordered product and $m_1 \equiv
\mh$
and $m_2 \equiv \mz$. In the case of the unphysical scalar,
eq.(\ref{e3})
is valid in the 't Hooft--Feynman gauge.  As the leading $m_t^4$ term
does
not depend upon the gauge sector we have chosen this gauge  for
practical
convenience.

In order to trigger  Ward identities we contract $\Pi_{(a,b)}^{\mu
\nu}(q^2)$
with $q^\mu\, q^\nu$. Contraction of a  current operator with its
four-momentum gives rise to a term involving the divergence of a
current plus
an equal time commutator that reduces the number of operators inside
the
time-ordered product by one unit. Noticing that
\be
\Dq \left\{ q_\mu q_\nu \Pi^{\mu \nu} (q^2) \right\}_{q^2=0} = A(0),
\label{e4}
\ee
and introducing the short-hand notation $\langle ...\rangle $
for the vacuum expectation value (v.e.v.) of the covariant
time-ordered
product
$\langle 0\, |\, T^* ... |\, 0 \rangle$,
we can write
\beq
\lefteqn{A_{(a,b)} (0) = \frac{g^4}{8\, c^2 \mw^2} \Dq \ik \int d^n y
e^{-iq \cdot y} } \nonumber \\
&\times \left\{ \sum_{j=1}^2 \int d^n x_1 e^{i k \cdot x_1} \int d^n
x_2
e^{-ik \cdot x_2} \frac{\qua{S_2}{S_2}{S_j}{S_j}}{4(k^2 - m_j^2)}
\right. \nonumber\\
&\left. \mbox{}+ i \uint \tre{S_2}{S_1}{S_2} \left[ \frac1{k^2 -
m_1^2} -
\frac1{(k-q)^2 - m_2^2} \right] \right\} _{q^2=0} \nonumber\\
&\mbox{}+ \frac{g^4}{16\, c^2 \mw^2} \ik \left\{ \left(1 - {4 \over
n}
\frac{k^2}{k^2- m_1^2} \right) \uint \frac{\due{S_2}{S_2}}{(k^2 -
m_1^2)^2}
+ (1\leftrightarrow 2)
%\right. \nonumber \\
%&\mbox{}+ \left. \left(1 - {4 \over n}
%\frac{k^2}{k^2- m_2^2} \right) \uint \frac{\due{S_1}{S_1}}{(k^2 -
% m_2^2)^2}
\right\} . \label{e5}
\eeq
In \equ{e5}\ the notation $ (1 \leftrightarrow 2)$ represents a term
obtained
by the  previous one inside the curly bracket by the substitution
$1 \leftrightarrow 2$.

We now examine the contributions that are described by the topology
shown in fig.~1c. Let us consider the case when one dashed line
represents a $\Phi_1$ and the other one a $\Phi_2$. We write
\beq
\Pi_{(c)}^{\alpha \nu}(q^2) =& - \frac{g^4}{4\, c^2 \mw^2} \ik
    \frac{(2k-q)^\nu}{[k^2-m_1^2][(k-q)^2-m_2^2]} \nonumber \\
 & \times \dint  \tre{J_Z^\alpha}{S_1}{S_2} \label{e6}.
\eeq
Contraction with $q^\alpha$ gives
\beq
\lefteqn{q_\alpha \Pi_{(c)}^{\alpha \nu} (q^2)=  \frac{i g^4}{8\, c^2
\mw^2}
\ik  \frac{(2k-q)^\nu}{[k^2-m_1^2][(k-q)^2-m_2^2]}} \nonumber \\
&  \times \left\{ \dint  \tre{S_2}{S_1}{S_2} \right. \nonumber\\
&\mbox{}+i \left. \int d^n x \left[ e^{i(k-q) \cdot x} \due{S_2}{S_2}
- e^{i k \cdot x} \due{S_1}{S_1} \right] \right\}.
\label{e7}
\eeq
Recalling that
\be
\dq \left(q_\alpha \Pi^{\alpha \nu} (q^2) \right) = \Pi^{\mu
\nu}(q^2) +
q_\alpha \dq \Pi^{\alpha \nu} (q^2) \label{e8}
\ee
it follows that the contribution of fig.~1c to $\Pi^{\mu
\nu}_{ZZ}(0)$ is
obtained by differentiating \equ{e7} with respect to $q^\mu$ and then
setting $q^2=0$. Consider the first term in \equ{e7},
\be
T^\nu = \ik  \frac{(2k-q)^\nu}{[k^2-m_1^2][(k-q)^2-m_2^2]}
\big\{...\big\}
\label{e9}
\ee
where $\big\{...\big\}$
 represents the three-point correlation function times the
appropriate
constants. By Lorentz invariance, $T^\nu$ should have the form
$T^\nu = \alpha (q^2) q^\nu$, % \label{e9bis}
and therefore
\be
\dq T^\mu = 2\frac{\partial\alpha}{\partial q^2}( q^2)\
 q^\mu q^\nu + \alpha (q^2)\  g^{\mu \nu}. \label{e10}
\ee
The contribution to $A(q^2)$ is then given by $\alpha(q^2)$. It is
easy
to show that
\be
\alpha (0) = \left. \Dq ( q_\nu T^\nu) \right|_{q^2=0}, \label{11}
\ee
or  explicitly
\be
\alpha(0) = \Dq \left. \ik
\left[ \frac1{(k-q)^2- m_2^2} - \frac1{k^2- m_1^2}
           + \frac{m_1^2 - m_2^2}
{(k^2 - m_1^2)[(k-q)^2 - m_2^2]} \right]
\  \big\{...\big\}\right|_{q^2=0}. \label{e13}
\ee

Using \equ{e13}, %and deriving the rest of
we can write for the transverse part of $\Pi_{(c)}^{\mu \nu}\,$:
\beq
\lefteqn{A_{(c)} (0) = \frac{i g^4}{8\, c^2 \mw^2} \Dq \left\{
    \dint \tre{S_2}{S_1}{S_2}
\right. } \nonumber\\& \times \left.
\ik
\left[\frac1{(k-q)^2- m_2^2} - \frac1{k^2- m_1^2}+
 \frac{ m_1^2 - m_2^2}
{(k^2 - m_1^2)[(k-q)^2 - m_2^2]} \right]
              \right\}_{q^2=0} \nonumber \\
 &\mbox{}- \frac{g^4}{8\, c^2 \mw^2} \ik \left\{ \left(1 - {4 \over
n}
       \frac{k^2}{k^2- m_1^2} \right) \uint
       \frac{\due{S_2}{S_2}}{(k^2 - m_1^2)(k^2- m_2^2)}
+ (1 \leftrightarrow 2)
 \right\} .\nonumber \\
& \  \label{e14}
\eeq
The contribution of $\Phi_1$ and $\Phi_2$ to fig.~1d,e can be
similarly
computed:
\beq
A_{d,e} (0) =& \frac{g^4}{16\, c^2 \mw^2} \ik \left\{ \left(1 - {4
\over n}
       \frac{k^2}{k^2- m_2^2} \right) \uint
       \frac{\due{S_1}{S_1}}{(k^2 - m_1^2)^2}
+ (1\leftrightarrow 2 )
\right\} . \nonumber \\
& \ \label{e15}
\eeq
An analogous  analysis for the $\Phi$ contribution to  $\azz (0)$
gives
\beq
\lefteqn{A_{(a-e)} (0) = \frac{g^4}{8\, c^2 \mw^2} \Dq \left\{ \ik
\frac1{k^2 - \mw^2} \right. } \nonumber \\
& \times \left. \tint \qua{S_2}{S_2}{S^\dagger}{S} \right\}_{q^2=0}.
\label{e16}
\eeq

Summing eqs.~(\ref{e5}), and (\ref{e14})--(\ref{e16}) we get,
for the transverse part of the $Z$ \self\ at $q^2=0$,
\ben \label{e17} \beqs
\lefteqn{\frac{\azz (0)}{\mz^2} =
{g^4\over 8 \mw^4} \Dq \ik \left\{ \tint \right. } \\
&\times \left[ { \qua{S_2}{S_2}{S^\dagger}{S} \over k^2-\mw^2} +
\sum_{j=1}^2 { \qua{S_2}{S_2}{S_j}{S_j} \over 4 (k^2-m_j^2)} \right]
\\
&\left. +i(\mh^2-\mz^2)  \dint { \tre{S_2}{S_1}{S_2} \over
(k^2-\mh^2)((k-q)^2-\mz^2)} \right\}_{q^2=0} \\
&\mbox{}+ {g^4\over 16 \mw^4} \ik
{(\mh^2-\mz^2)^2 \over (k^2-\mh^2)^2(k^2-\mz^2)^2}
\\& \times
\left\{ \left( 1- {4 \over n}{k^2 \over k^2 - m_2^2} \right) \uint
\due{S_1}{S_1} + (1\leftrightarrow 2)
%  \left( 1- {4 \over n}{k^2 \over k^2 - \mh^2} \right) \uint
%\due{S_2}{S_2}
\right\} \yesnumber \label{e17a} .
\eeqs

The discussion of the $W$ \self\ can be performed on the same footing
as
the $Z$ case. Here we present only the result
\beqs
\lefteqn{\frac{\aww (0)}{\mw^2} =
{ g^4\over 8 \mw^4 } \Dq \ik \left\{ \tint \right. } \\
&\times\left[ { \langle S^\dagger (y)S(0)\left(S^\dagger (x_1)S(x_2)+
{\rm h.c.}
\right) \rangle \over(k^2-\mw^2)} +
\sum_{j=1}^2 { \qua{S^\dagger}{S}{S_j}{S_j} \over 2(k^2-m_j^2)}
\right] \\
&\mbox{}+ \dint \left[  i(\mh^2-\mw^2) {\langle \left(
S^\dagger(y)S(0)+ {\rm h.c.}\right)S_1(x)\rangle \over
(k^2-\mh^2)((k-q)^2-\mw^2)} \right.\\
&\hphantom{\dint [} \mbox{}+ \left. \left. (\mz^2-\mw^2) {\langle
\left( S^\dagger(y)S(0) -{\rm h.c.} \right) S_2(x)\rangle
\over (k^2-\mz^2)((k-q)^2-\mw^2)}\right] \right\}_{q^2=0}\\
&\mbox{}+ {g^4\over 16 \mw^4 } \ik \left\{
{(m_1^2-\mw^2)^2 \over (k^2-m_1^2)^2(k^2-\mw^2)^2}
\left[ \left( 1- {4 \over n}{k^2 \over k^2 - \mw^2} \right) \uint
\due{S_1}{S_1} \right.\right.\\
& \hphantom{\dint [}
+ \left.\left. 2 \left( 1- {4 \over n}{k^2 \over k^2 - m_1^2} \right)
\uint
\due{S^\dagger}{S} \right]+  (1\leftrightarrow 2) \right\}
\yesnumber \label{e17b} .
\eeqs
\een

Before discussing the diagrams involving the counterterms,
it is interesting to consider
the limit $g,\,g^\prime \rightarrow 0$ in \eqs{e17} and the
connection with
the result
presented in \efe{b9}. The authors of \efe{b9}\ observed that, for
what concerns
the leading one- and two-loop $\mt$ effects, there is a Ward identity
that
relates, at $q^2=0$,  the $W$ and $Z$ \selfs\ to the ones of the
unphysical
counterparts. Explicitly
\ben \label{e18} \beq
\frac{\aww (0)}{\mw^2} =& - \left. \Dq \Pi_{\Phi \Phi} (q^2)
\right|_{q^2=0}
                        \label{e18a} \\
\frac{\azz (0)}{\mz^2} =& - \left. \Dq \Pi_{\Phi_2 \Phi_2} (q^2)
                       \right|_{q^2=0} .               \label{e18b}
\eeq \een
As noted in \efe{b9}, \eqs{e18} have the advantage that the r.h.s.\
can be
evaluated by switching off the gauge interaction in the SM
Lagrangian, namely
using only the Yukawa and Higgs part of it. In fig.~2 we show the
diagrams
belonging to the scalar \selfs\ that contribute to the leading
$m_t^4$ term.

In order to understand the connection between \eqs{e17} and the
r.h.s.\
of \eqs{e18}
consider for example the four-point correlation functions in
\equ{e17a}.
In the limit $g,\,g^\prime \rightarrow 0$ or $\mw,\, \mz \rightarrow
0$,
their contribution to \dr\ is proportional to $g_t^4$. In fact every
$S$ operator contains an $\mt=g_t v$ factor, where $v$ is the
v.e.v., while the coefficient in front is proportional
to $1/v^4$. In this limit the terms involving four-point functions
represent exactly the contribution to $\left. \Dq \Pi_{\Phi_2 \Phi_2}
(q^2)
\right|_{q^2=0}$ of
the diagrams shown in
figs.~2a and 2b. A similar connection can be established
between the three- and two-point correlation functions in
\equ{e17a} and the
diagrams 2c and 2d respectively. We notice that figs.~2c and 2d
involve trilinear scalar couplings proportional to $\mh^2$. Although
no such   coupling is present in the diagrams of fig.~1, we recover
these terms using \equ{e13} and simple algebraic manipulations.
Equation (\ref{e17b}) shows an analogous connection between its
various
contributions and $\left. \Dq \Pi_{\Phi \Phi} (q^2) \right|_{q^2=0}$
computed
in the Yukawa theory. We observe that, in obtaining \eqs{e17}, we
took
advantage of
several commutation relations that were derived using an
anticommuting
$\gamma_5$.

To complete the calculation of the leading $\mt^4$ term, we have to
include
the counterterm diagrams. The counterterms that can contribute to the
order
we are interested in are the mass counterterm for the top, Higgs and
unphysical scalars\footnote{The mass counterterm for the unphysical
scalars is related to the tadpole counterterm and therefore
determined
from the tadpole diagrams \cite{b13}.}. However, it is easy to see
that the
diagrams containing the counterterm associated to the scalars do not
give $\mt^4$
terms in \dr\ because of the cancellation of the leading part between
the
two \selfs. We are left with the mass counterterm of the top quark,
which we define on-shell. As we are studying the  $\mt^4$ term, it is
sufficient to include in the top counterterm only diagrams involving
scalars.
In the 't Hooft--Feynman gauge we write the contribution
to $\Delta$ arising from the diagrams containing the top counterterm
as
\beq
\Delta_{ct}=& N_c x_t^2 \left\{ 3 \left[ \frac1{\epsilon} + 2\,{\cal
C}
        - 2 \ln \frac{\mt^2}{\mu^2} \right] + \frac92 -ht
\frac{6-ht}2
         \ln ht - ht \right. \nonumber  \\
& \left.   + \frac{ht-4}2 \sqrt{ht}\, g(ht) + zt \left( \frac{zt}2-1
\right)
  \ln zt - zt +{ zt^{3/2} \over 2} g(zt) - 2 N(\sqrt{wt}) \right\}\,,
  \label{e19}
\eeq
where $n= 4- 2 \epsilon$ and ${\cal C}= -\gamma_E +\ln 4 \pi$,
$ht= \mh^2 / \mt^2, \: zt=\mz^2 / \mt^2,\: wt=\mw^2 / \mt^2$,
the function $N$ is defined in Appendix A of \efe{b14}, and
\beq
g(x) =&  \left\{
          \begin{array}{lr}
           \sqrt{4-x}\,\left
	      (\pi - 2 \arcsin{\sqrt{x/4}}
	               \right) & 			0 < x \leq 4 \\
	   {}\\
	   2 \sqrt{x/4-1}\,\ln\left(
	   	\frac{1-\sqrt{1-4/x}}{1+\sqrt{1-4/x}}
		              \right) & 		x > 4 \,.
	  \end{array}
	\right.  \label{e19a}
\eeq

The 1PI $\mt^4$ contribution to \dr\ is obtained by working out
explicitly
the correlation functions in \eqs{e17} and then evaluating the
resulting
integrals and \equ{e19} in the limit $\mw=\mz=0$. In the actual
computation
we have used an anticommuting $\gamma_5$ and
some recent contributions on two-loop integrals \cite{b15}. We
find perfect agreement with the analytic results presented in
\efe{b11}.

\section{Self-energy irreducible contribution to \dr\ in SU(2)}
In the previous section we applied current algebra techniques to
simplify
diagrams contributing to the leading $\mt^4$ term: in this way we
obtained
expressions (\eqs{e17}) containing not only the leading term, but
also
subleading contributions. Obviously, there are other 1PI diagrams
that
can contribute to subleading order.

However, when one tries to go beyond the leading $\mt$ term in \dr,
the calculation of  $\Delta$ is no longer sufficient and one has
to investigate  the ratio of neutral-to-charged-current scattering at
zero
momentum transfer. As a consequence, vertex and box diagrams should
also be
taken
into account and a complete two-loop calculation becomes a very
difficult
task.

Here we present a first study of the next-to-leading-order two-loop
$\mt$ corrections to \dr\ in an $SU(2)$ model obtained by setting
$g^\prime =0$
in the SM.   This amounts
to computing the 1PI and reducible \selfs, vertex and box diagrams
present in
neutral and charged scattering processes to $O(\gmud)$. We recall
that in the
$SU(2)$ model the bare masses of the charged and neutral vector
bosons
are equal, there is no weak mixing angle, and electromagnetic
interactions
are not present. The breaking of the symmetry by a non-degenerate
$(t,b)$
doublet is responsible for the splitting in the values of the
physical
masses of the vector bosons.

In this section we study the 1PI two-loop \selfs \footnote{The
treatment
of the counterterm diagrams and tadpoles is analogous to
what was done in section 2.};
in particular we concentrate on\footnote{We indicate the charged and
neutral
vector bosons of $SU(2)$ as
$W$ and $Z$, respectively. Their \selfs\ are correspondingly labelled
$\aww$ and $\azz$.}
$\Delta_2 \equiv [\aww(0)- \azz(0)]/ \mz^2$.
Using the results of \efe{b15}, the
$\mt$ contribution to it can be actually computed without any
approximation.
However, we report only the first  correction, $O(zt)$, to the
leading term
in order to be consistent with the accuracy we compute the
remaining parts contributing to \dr. We begin by analysing the
contribution
of the scalars to the
\selfs. Beside the diagrams of fig.~1, to $O(\gmud)$ there are other
graphs
containing the Higgs boson that have to be considered. They are shown
in
fig.~3.
To evaluate these new contributions we use the same
techniques as employed in the previous section, namely we  express
groups
of Feynman diagrams in terms of current correlation functions and
then
contract the currents inside with their four-momenta to enforce Ward
identities. In order to use the results of the previous section we
also employ in the calculation of the subleadings the 't
Hooft--Feynman
gauge.
We combine \eqs{e17} and (\ref{e19}), evaluated in the limit
$\mw=\mz$,  $c=1$, with the diagrams of fig.~3 , and obtain
\ben \label{e20} \beqs
\Delta_2^s &= N_c x_t^2 \left\{ 25 -4ht -8zt +{4 zt \over ht} + \pi^2
\left( \frac12
- \frac1{ht} +\frac{2 zt}3 - {2 zt \over 3 ht^2} + {zt \over ht}
\right)
+ \frac{(ht-4) \sqrt{ht} \,g(ht)}2 \right. \\
&\mbox{}+ {(ht-1)^2 (6ht- 3ht^2 +4 zt +2ht zt) Li_2 (1-ht) \over
ht^2}
- \left( 6+6 ht - \frac{ht^2}2 - 2 zt + {4 zt \over ht} \right) \ln
(ht) \\
& \left. -7 zt \ln (zt) +\left( -15 +9 ht - {3 ht^2 \over 2} - 2 zt -
{8 zt \over ht} + ht zt \right) \phi \left( \frac{ht}4 \right)
\right\} ,
\yesnumber  \label{e20a}
\eeqs
for the two-loop contribution to $\Delta_2$ involving the top and the
scalars to $O(zt)$, in the region $ht \gg zt$  ,
while for $ht \ll zt \ll 1$:
\beq
\Delta_2^s = N_c x_t^2 \left\{ 19 - 2 \pi^2 -4 \pi \sqrt{ht} - 16 zt
+
          {2 \pi^2 \over 3} zt - zt \ln zt \right\} \,, \label{e20b}
\eeq \een
In \equ{e20a}
\ben \label{e21} \beq
Li_2 (x) =& - \int_0^x dt {\ln (1-t) \over t}     ,   \label{e21b}
\eeq
and
\be
\phi(z) =
       \begin{cases}
       4 \sqrt{{z \over 1-z}} Cl_2 ( 2 \arcsin \sqrt z ) & $ 0 < z
\leq 1$\\
       { 1 \over \lambda} \left[ - 4 Li_2 ({1-\lambda \over 2}) +
       2 \ln^2 ({1-\lambda \over 2}) - \ln^2 (4z) +\pi^2/3 \right]
       & $z >1 $\,,
       \end{cases}
       \label{e21c}
\ee
where $Cl_2(x)= {\rm Im} \,Li_2 (e^{ix})$ is the Clausen function
and
\be
\lambda = \sqrt{1 - {1 \over z}}. \label{e21d}
\ee
\een
We notice that when $zt \rightarrow 0$ in \equ{e20a}, $\Delta_2^s$
becomes
identical to eq.~(12) of \efe{b11}.

The remaining diagrams belonging to the 1PI \selfs\ are obtained by
replacing all dashed lines in fig.\ 1 by wavy ones, namely
substituting the
unphysical scalars by their associated vector bosons. We consider
this
contribution together with the part of the counterterm diagrams not
included
in \eqs{e20} as there is a partial cancellation in the ultraviolet
poles.
Concerning the counterterms, we showed in section 2 that in the SM
the
diagrams containing the mass counterterm of the scalars
do not contribute
to the leading $\mt^4$ term. In the restricted $SU(2)$ model we
are now considering, the bare masses, $\mw^0$ and $\mz^0$, are equal
and only
one mass counterterm is available in the gauge-boson sector.
Consequently,
vector boson and Higgs mass counterterm graphs cancel out exactly in
$\Delta_2$
and we are left only with diagrams involving the top-mass
counterterm.
Equations (\ref{e20}) include diagrams containing the part of the
top-mass
counterterm due to the scalars; to $O(\gmud)$ one has to consider in
the counterterm also the contribution of the vector bosons. The
diagrams
involving this part of the top counterterm together with the diagrams
of
fig.~1, assuming the internal dashed lines to represent vector
bosons, give
to $O(zt)$
\be
\Delta_2^{v.b.} = N_c x_t^2 \left\{ 16 zt \left( {1 \over \epsilon} -
2 \ln \left( {\mt^2 \over \mu^2} \right)
+ 2\, {\cal C} \right) + 56 zt + 6 zt \ln zt - 3 \pi^2 zt
                        \right\} .       \label{e23}
\ee
We notice that $\Delta_2^{v.b.}$ contains diagrams involving
vector bosons coupled to a fermionic triangle. However, the group
$SU(2)$
is automatically-anomaly free and therefore these graphs
are not affected by the anomaly problem.

The calculation of the two-loop $\mt$ part in $\Delta_2$ has given a
somewhat surprising result. The answer is not finite but a $1 /
\epsilon$ pole has appeared (cf.\ \equ{e23}). This pole is the
confirmation
that, beyond the leading term, $\Delta_2$ cannot be identified  with
\dr,
but the ratio of the complete neutral-to-charged amplitudes has to be
considered. As
we will see in the following section, the vertex contributions
contain
a similar pole (with opposite sign!) and, as expected, \dr\ comes out
finite.


\section{The $\rho$ parameter to $O(\gmud)$ in $SU(2)$}
In this section we compute the ratio of neutral-to-charged-current
amplitudes
to two-loop $O(\gmud)$ contributions in an $SU(2)$ model. We begin by
recalling that in $SU(2)$, concerning the gauge sector, there are
only
two bare parameters, $g_0$, the coupling constant, and $v_0$, the
v.e.v.
Therefore two renormalization conditions are needed. Our strategy is
to
trade the renormalized parameters $g$ and $v$ for the physical
quantities
$G_\mu$ and $\mz$, where $\mz$ is the on-shell mass of the neutral
vector
boson. The simplest way to achieve this is to take
$\delta \mz^2={\rm Re}\azz(\mz^2)$
and adjust $\delta g$ in such a way that the relation
\be
\frac{G_\mu}{\sqrt{2}}=\frac{g^2}{8\mz^2} \label{e25}
\ee
holds up to second order.

Equation (\ref{e25}) implies that the counterterm $\delta g$ should
cancel the
one and two-loop contributions to muon decay. Defining
\be
\delta Y = {\rm Re} \left[\aww(0)-\azz(\mz^2)\right] \label{e26}
\ee
and referring to fig.~4 for the one-loop diagrams contributing to
muon decay
we have
\be
2 \dg = \Vwu{1} +\Bwu{1} - \dy \,, \label{e27}
\ee
where the superscript $(1)$ reminds us that we are considering
one-loop
contributions. In \equ{e27} \Vwu{1}\ and \Bwu{1}\ represent the
vertex and box
diagrams. Their expressions can be gleaned from section 4 of
\efe{b12}.
Explicitly
\ben \label{e28} \beq
\Vwu{1} &= \frac{g^2}{16\pi^2}\left(\frac{4}{\epsilon}+4{\cal C} -4
\log\frac{\mz^2}{\mu^2}\right) \label{e28a} \\
\Bwu{1} &=\frac{5}{2} \frac{g^2}{16\pi^2}. \label{e28b}
\eeq\een

Consider now the two-loop contributions to the process. Some of the
relevant diagrams are shown in fig.~5. The group of graphs
represented
in fig.~5a gives a contribution
\ben \label{e30}
\be
M_{5a} = \left( \dy \right)^2 M_0 \,, \label{e30a}
\ee
where $M_0$ is the zeroth-order amplitude. Analogously, we have for
fig.~5b
\be
M_{5b} = - \dy \Vwu{1} M_0 \,. \label{e30b}
\ee
We now discuss 1PI vertex and box diagrams. The only graphs that can
give
an
$\mt^2$ contribution are those where a fermionic \self\ is inserted
in a vector
boson propagator line (fig.~5c). It is easy to show, using arguments
similar
to the ones developed in section 4 of \efe{b16}, that the $O(\gmud)$
part
of these diagrams can be obtained by considering the \self\ insertion
with
momentum transfer set equal to zero. Therefore these diagrams, plus
the
corresponding ones with a mass counterterm for the vector bosons,
give
to $O(\gmud)$:
\beq
M_{5c} =& \left( \Vwu{2} + \Bwu{2} \right) \dy M_0 \label{e30c} \\
\Vwu{2} =& - 2 \frac{g^2}{16 \pi^2} \label{e30d}\\
\Bwu{2} =& - \frac54 \frac{g^2}{16 \pi^2}. \label{e30e}
\eeq
To complete the calculation of $2 \delta g / g$, we have to consider
the reducible diagrams. Figure~5d gives a contribution
\be
M_{5d} = \left(  \dg \right)^2 M_0 \,, \label{e30f}
\ee
while expanding the coupling constants in the one-loop diagrams of
fig.~4
one obtains
\be
M_{exp} = - 4 \dg \left( \Vwu{1} + \Bwu{1} - \dy \right) M_0
\,.\label{e30g}
\ee
\een
Putting together \eqs{e30}\ and taking into account the two-loop 1PI
diagrams
analogous to fig.~4a,b, we can write for $\delta g / g$ to the order
we
are interested in
\beq
\frac{2\delta g}{g}=&
\left.\frac{{\rm Re}\azz(\mz^2)-\aww(0)}{\mz^2}\right|_{1PI}
 + \Vwu{1}+\Bwu{1} \nonumber\\
& +\left(\dg \right)^2 + \left( 3\Vwu{1}+\Vwu{2}+ 4 \Bwu{1} +
\Bwu{2} - \dy \right) \dy \,, \label{e31}
\eeq
where the first term includes both one- and two-loop 1PI
contributions to
the \selfs.

Having defined our counterterms we are now ready to consider
neutral-current
processes; in particular we will focus on neutrino--electron
scattering. With
our choice of renormalization conditions the interaction strength of
the
charged current is set equal to $G_\mu / \sqrt{2}$, therefore the
neutral-current amplitude
divided by $G_\mu / \sqrt{2}$ will give us the $\rho$ parameter
directly.

%The diagrams contributing to neutrino-electron scattering have the
% same
%topologies as the ones depicted in fig.4 and fig.5. Referring to
%fig.(4) for the one-loop contributions we have
We refer again to fig.~4 and fig.~5 for one- and two-loop diagrams,
in this
case relevant to neutrino-electron scattering. The one-loop
contributions
of fig.~4 give
\bea
\dro{1} &=&  \frac{{\rm Re} \Azz{1} (\mz^2)-
            \Azz{1} (0)}{\mz^2} + \Vzu{1} +\Bzu{1}
       - 2 \dg \nonumber \\
        &=&  \frac{\Aww{1}(0)-
            \Azz{1} (0)}{\mz^2} + \Vzu{1} - \Vwu{1}
          +\Bzu{1} - \Bwu{1} \,, \label{e32}
\eea
where $\Vwu{1}=\Vzu{1}$ and
\be
\Bzu{1} = \frac{g^2}{16 \pi^2} \left( \frac52 -\frac94 I_3 \right).
\label{e33}
\ee
In \equ{e33}\ we have explicitly shown the part of the box diagrams
that is
process-dependent, i.e.\ the contribution proportional to $I_3$, the
isospin
of the target ($I_3 =-1$ for electrons).

We now proceed to examine the neutral process at the two-loop level.
We can neglect
diagrams 5a and 5b because the combination ${\rm
Re}\azz(\mz^2)-\azz(0)$ does
not contain $\mt^2$ terms. The sum of all the other diagrams gives a
contribution to $\rho$ equal to
%%\clearpage
\beq
\dr =& \left. \frac{{\rm Re}\azz(\mz^2)-\azz(0)}{\mz^2} \right|_{1PI}
+ V_\smallz
      + B_\smallz +\left( \dg \right)^2 \nonumber \\
&      - 4 \dg \left( \frac{{\rm Re}\Azz{1}(\mz^2)-
       \Azz{1} (0)}{\mz^2} + \Vzu{1} +\Bzu{1} \right)
       - 2 { \delta g \over g }, \label{e34}
\eeq
where $V_\smallz = \Vzu{1} +\Vzu{2} \dy$, $B_\smallz = \Bzu{1}
+\Bzu{2}
\dy$,
and
\ben \label{e35} \beq
\Vzu{2} =& - 4 \frac{g^2}{16 \pi^2} \label{e35a}\\
\Bzu{2} =& -  \frac{g^2}{16 \pi^2} \left[ \frac52 - \frac32 I_3
\right] .
           \label{e35b}
\eeq
\een
Recalling that
\be
\dy \simeq N_c x_t \left[ 1 + \epsilon \left( \frac12 + {\cal C} -
\ln
      \frac{\mt^2}{\mu^2} \right) \right] \label{e36}
\ee
we can write, after some simple algebra, the one- and two-loop
contribution
to the $\rho$ parameter up to $O(\gmud)$ terms as
\ben \label{e37}
\beq
\dro{1} =& x_t \left( N_c - 9 I_3 zt \right) \label{e37a} \\
\dro{2} =& \left.
\frac{\Aww{2}(0)- \Azz{2}(0)}{\mz^2}\right|_{1PI} \nonumber\\
& + N_c x_t^2  \left[ N_c -4 zt \left( {4\over\epsilon}+
                  8{\cal C} - 4 \ln{\mz^2\over\mu^2}
-4\ln{m_t^2\over\mu^2}
                +{41\over4} + 3 I_3 \right) \right] \,. \label{e37b}
\eeq
\een
Substituting the sum of eqs.~(24) and (27) in the first term of
eq.~(\ref{e37b}) we get our final result. The $1/ \epsilon $ pole
present in the
first term of \equ{e37b}\ (cf.\ \equ{e23}) is cancelled by an
analogous
contribution coming from the vertex part.

In table 1 we present $\dro{2}$ for $zt=0,\,0.2,\,0.25,\,0.3$ and
for several values
of $ht$. The entries in the table report the case $N_c=3$ and
$I_3=-1$.
The column $zt=0$ coincides with the results obtained in
\efs{b9}{b11}.
In all entries, the difference by a factor of 3
between the numbers reported in our
table and the ones presented in \efs{b9}{b11} is due to the fact that
\equ{e37b}
contains also the leading reducible contribution, i.e.\ the term
$(N_c x_t)^2$,
while the tables in \efs{b9}{b11} present only the two-loop
irreducible part.
In order to avoid numerical problems in the region $ht \simeq zt$ we
have
prepared the table employing the exact expressions for the diagrams
involving
the Higgs. The remaining contributions have been included to $O(zt)$.
Using
the asymptotic expressions of \eqs{e20}, we get a difference with
respect to the
numbers of table 1 by, at most, 3\%.

As the table shows, the $O(zt)$ terms have a dramatic effect
especially for
light Higgs-mass values.
This is easily understood by looking at \equ{e37b}. In fact the
finite
part\footnote{The term proportional to ${\cal C}$ is not included in
the finite
              part.}  of the term proportional to $zt$ in the second
row of
\equ{e37b} amounts, with $\mu=\mt$, to $-10.9,\, -12.8,\, -14.5$ for
$zt =0.2,\,0.25,\,0.3$, by far the largest contribution to $\dro{2}$
for small
$ht$. As $ht$ increases, the contribution of the \selfs\ starts to be
more
important. It is interesting to consider the asymptotic expansion of
the finite
part of the two-loop $\mt$ contribution to $\Delta_2$ for large $ht$
\beqs
\Delta_2 \simeq & N_c x_t^2 \left\{
          \frac{49}4 + \pi^2 - \frac{27}2 \ln ht + \frac32 \ln^2 ht
          + zt \left( 46 - \frac73 \pi^2 - \ln zt \right) \right\} \\
&+ \frac1{ht} \left[ \frac23 - 4 \pi^2 - 4 \ln ht - 9 \ln^2 ht
    + zt \left( \frac32 - 9 \ln ht - 3 \ln^2 ht \right) \right] \\
& + \frac1{ht^2} \left[ \frac{1613}{48} - 5 \pi^2 + \frac{125}4 \ln
ht
    - 15 \ln^2 ht \right. \\
& \hphantom{\frac1{ht^2} \frac{1613}{48}}
+ \left. \left. zt \left( \frac{124}{9} - 4 \pi^2 + \frac{16}3 \ln ht
- 10 \ln^2 ht
  \right)  \right] + \dots  \right\}. \yesnumber \label{e40}
\eeqs
We note that in the limit $ht \rightarrow \infty$ the inclusion of
subleading
contributions to the \selfs\ gives rise to an $\mh$-independent term.


\section{Conclusions}
In the previous sections we have shown that the use of current
correlation
functions and their associated current algebra provides, in the
fermionic
sector, an efficient way to enforce the relevant Ward identities
while
discussing at the same time several Feynman diagrams. In particular,
we
have seen in section 2 that, for the leading term, the formalism is
equivalent to the effective Lagrangian approach of \efe{b9}, sharing
with it the same advantages from a computational point of view.
Although
with respect to \efe{b9} we had to pay the price to explicitly derive
the Ward identities, this effort was rewarded by gaining a formalism
that could deal at the same time with subleading terms.

The calculation of \dr\ to $O(\gmud)$ shows some interesting
features.
Differently from the one-loop case where $\Delta_2$ is finite
and gauge-invariant, at the two-loop level,
beyond the leading $\mt^4$ term,
neither of these two properties is kept and therefore $\Delta_2$
cannot
be identified with a physical observable. As a comparison,
a similar situation happens in the SM already at the one-loop level.
In
fact,
there, one-loop gauge-dependent bosonic contributions do not cancel
in
the difference of the two \selfs.

Two, related, consequences of the above fact are: {\it i)} to define
a physical
observable, we have to resort to physical processes and this
introduces
process-dependent quantities. \linebreak
{\it ii)} Concerning the $\mt^4$ contribution,
the two-loop reducible and irreducible parts are separately finite.
Furthermore, the reducible term is equal to $(N_c x_t)^2$, namely
the square of the one-loop contribution. This fact suggested to the
authors
of \efe{b17} that a possible way to take into account higher-order
effects is to write $\rho$ as
\be
\rho \equiv \frac1{1 -\dr_{irr}} \,, \label{e50}
\ee
where
\be
\dr_{irr} = \dr^{(1)}_{irr} + \dr^{(2)}_{irr}\,,  \label{e51}
\ee
with $\dr^{(1)}_{irr}$ and $\dr^{(2)}_{irr}$ the 1PI one- and
two-loop
contribution to $\Delta$. There is no actual justification why
\equ{e50}
should be the correct way to resum higher-order top effects, its
validity
relying upon
the fact that up to two-loops it reproduces the correct result. We
have
seen that once subleading contributions are included, $\Delta_2$,
the equivalent of $\dr^{(2)}_{irr}$ in the $SU(2)$ model,
is not  finite. We expect that if we were to evaluate
$\dr^{(2)}_{irr}$ in the
SM to subleading order a similar situation would prevail. It then
seems that
\equ{e50} cannot work at the subleading level.
The simplest choice of replacing $\dr^{(2)}_{irr}$
by $\dro{2}$ in \equ{e50} cannot be theoretically justified. In fact
the scattering amplitude contains vertex and box
contributions (cf.\ \equ{e37b}), and it is not yet
clear how this type of diagrams can be resummed. To avoid this
problem,
one can
consider to resum only the leading top contributions. However, at the
two-loop
level there are diagrams (cf.\ fig.~1) that contribute both to the
leading
and the subleading part. The choice to split a diagram contribution
into two parts, one of which is resummed, seems quite arbitrary.
Probably the ``safest'' way to resum
higher-order effects is to assume
\be
\rho =\frac1{1 -\dr_{irr}^{(1)}} + \dr_{irr}^{(2)} \,, \label{e52}
\ee
where the resummation of $\dr_{irr}^{(1)}$ can be theoretically
justified
on the basis of $1/ N_c$ expansion arguments \cite{b50}.
Although there is no practical implication in choosing \equ{e52}\
instead of \equ{e50}, the two expressions have a different behaviour
with
respect to the limit $\mt \rightarrow \infty$. While in \equ{e50}, as
$\mt$ grows the two-loop term starts to cancel a large part
of the one-loop contribution and, eventually, for reasonable values
of
$\mh$, overwhelms it, leaving us with an upper bound on the $\rho$
parameter,
the same does not happen in \equ{e52}. Indeed \equ{e52} does not
allow
for screening of heavy physics by higher-order effects and the
$\rho$ parameter, in this case, is not bounded from above.

Finally, we want to comment on the size of subleadings effects.
As shown in table~1, the leading $O(\gmuq)$ contribution and the
subleading $O(\gmud)$
one are numerically comparable for realistic values of the top mass
and,
moreover, the two contributions have the same sign. The effect of
$O(\gmuq)$ correction, for $\mt= 200$ GeV, has been estimated
\cite{b51}
to decrease the value of the predicted $W$ mass in the SM by $\sim
24$ MeV.
Assuming that subleading effects were comparable to the leading
contribution, the shift in the prediction of the $W$ mass would be
almost
as large as the envisaged experimental accuracy $(\delta \mw)_{exp}
= \pm 50$ MeV. However, it is not correct to directly apply this
result
to LEP physics for several reasons. First, the calculation
of the subleading effects
we presented was performed in an $SU(2)$ model, where electromagnetic
interactions are not present. Secondly, for what concerns LEP
physics,
it is useful to compute $\sin^2 \theta_W$ as accurately as possible.
The
relation between the standard input parameters, $G_\mu,\, \mz$,
$\alpha$, and $\sin^2 \theta_W$ involves a correction factor
($\Delta r$ in the on-shell scheme, $\Delta \hat{r}$ in the
$\overline{MS}$
scheme
\cite{b52}) that contains \selfs\ evaluated at momentum transfers
equal
to the mass of the vector bosons and not at $q^2=0$. This fact
introduces
additional subleading $O(\gmud)$ terms that are not clearly taken
into
account in this paper. Furthermore, the vertex and box parts entering
in $\Delta r$ or $\Delta \hat{r}$ are different from the ones present
in
\dr. Moreover the calculation of the reducible contributions in the
SM
is more complicated because two coupling constants, $g$ and
$g^\prime$,
have to be taken into account.

To conclude, it is fair to say that the subleading two-loop $\mt$
effects
can be larger than what is ``nai\"vely'' expected. Although it is
quite
unlikely that these effects can modify the theoretical predictions of
the
various observables by amounts larger than the foreseeable
experimental
accuracy, it is realistic to assume that their influence can be
comparable
to,
or maybe larger than, the uncertainty in the evaluation of the
hadronic
contribution to the photonic \self.

\section*{Acknowledgements}
The authors want to thank F.~Feruglio, A.~Masiero, S.~Peris,
M.~Porrati,
A.~Santamaria
and A.~Sirlin for valuable discussions.
In particular we thank S.~Peris for bringing to our attention the
work of
\efe{b11bis}.
Two of us (G.D. and S.F.) would like
to thank the Physics Department of New York University for its kind
hospitality during November 1993, when part of this work was carried
out.
This research was supported in part by the National Science
Foundation under
Grant No.\ . One of 	us (S.F.) is currently supported by
an
ICSC World Laboratory Fellowship at the CERN Theory Division.
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\end{thebibliography}

%%%%%%%%%%%%% Figure Captions Page %%%%%%%%%%%%%%%
\newpage
\section*{Figure captions}
\begin{description}
\item[Fig.~1:] Two-loop diagrams contributing to  $O(\gmuq)$  to
             the vector boson propagators. In the figure wavy lines
             represent vector bosons, dashed lines indicate scalars
             while solid lines are fermions.
\item[Fig.~2:] Relevant two-loop diagrams contributing to $\Pi_{\Phi
\Phi}$
             and $\Pi_{\Phi_2 \Phi_2}$.
\item[Fig.~3:] Some of the two-loop diagrams involving the Higgs
boson
	     contributing
             to $O(\gmud)$ to the vector boson propagators.
\item[Fig.~4:] One-loop graphs relevant to charged- or
neutral-current
             interactions. The shadowed blob in figs.~4c and 4d
indicates
             schematically the sum of graphs in which the virtual
gauge
             bosons are attached in all possible ways to the external
lines.
\item[Fig.~5:] Two-loop graphs relevant to charged- or
neutral-current
             interactions. The meaning of the shadowed blob is as in
fig.~4.
%\item[Fig.~6:]
\end{description}
\clearpage
%%%%%%%%%%%%%%%% Figures %%%%%%%%%%%%%%%%
\newpage
\vfil
\par\centerline{\psfig{figure=fig1.ps,height=10cm}}
\vspace{0.4 true cm}
\centerline{\bf Fig.~1}\par
%\vspace{0.8 true cm}
\vfill
\par\centerline{\psfig{figure=fig2.ps,height=7cm}}
\vspace{0.4 true cm}
\centerline{\bf Fig.~2}\par
\vfil
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\vfil
\par\centerline{\psfig{figure=fig3.ps,height=9cm}}
\vspace{0.4 true cm}
\centerline{\bf Fig.~3}\par
%\vspace{0.8 true cm}
\vfill
\par\centerline{\psfig{figure=fig4.ps,height=7.5cm}}
\vspace{0.4 true cm}
\centerline{\bf Fig.~4}\par
\vfil
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\par\centerline{\psfig{figure=fig5.ps,height=19cm}}
\vspace{0.4 true cm}
\centerline{\bf Fig.~5}\par

%%%%%%%%%%%%%%%% Table %%%%%%%%%%%%%%%%
\renewcommand{\arraystretch}{1.2}
\begin{table}[h]
\centering
\caption[TableI]{Values of $\delta \rho^{(2)}$, for $N_c=3,\,I_3=-1$
and
                 in units $N_c\,x_t^2$, as a function of $\mh /\mt$
                 for some values of $zt \equiv \mz^2 /\mt^2$}

\begin{tabular}{|c|c|c|c|c|}
\hline
$\mh/\mt$ &  \multicolumn{4}{c|}{ $ \delta\rho^{(2)}$ } \\
\cline{2-5}
  {} & $zt = 0$ & \phantom{$zt$}0.2\phantom{$zt$}  &
  \phantom{$zt$}0.25\phantom{$zt$}  & \phantom{$zt$}0.3\phantom{$zt$}
\\
        \hline
  0.10    & \phantom{$-$}1.18   &    $-$8.28    &   $-$9.56    &
$-$10.7 \\
  0.20    & \phantom{$-$}0.30   &    $-$8.96    &   $-$10.2    &
$-$11.3 \\
  0.30    &         $-$0.46  &    $-$9.51    &   $-$10.8    &
$-$11.8 \\
  0.40    &         $-$1.13  &    $-$10.0    &   $-$11.2    &
$-$12.2 \\
  0.50    &         $-$1.72  &    $-$10.4    &   $-$11.6    &
$-$12.6 \\
  0.60    &         $-$2.25  &    $-$10.7    &   $-$11.9    &
$-$12.9 \\
  0.70    &         $-$2.74  &    $-$11.1    &   $-$12.2    &
$-$13.2 \\
  0.80    &         $-$3.18  &    $-$11.4    &   $-$12.5    &
$-$13.4 \\
  0.90    &         $-$3.58  &    $-$11.6    &   $-$12.7    &
$-$13.7 \\
  1.00    &         $-$3.96  &    $-$11.9    &   $-$13.0    &
$-$13.9 \\
  1.10    &         $-$4.30  &    $-$12.1    &   $-$13.2    &
$-$14.1 \\
  1.20    &         $-$4.62  &    $-$12.4    &   $-$13.4    &
$-$14.3 \\
  1.30    &         $-$4.91  &    $-$12.6    &   $-$13.6    &
$-$14.5 \\
  1.40    &         $-$5.19  &    $-$12.8    &   $-$13.8    &
$-$14.6 \\
  1.50    &         $-$5.44  &    $-$12.9    &   $-$13.9    &
$-$14.8 \\
  1.60    &         $-$5.68  &    $-$13.1    &   $-$14.1    &
$-$14.9 \\
  1.70    &         $-$5.90  &    $-$13.3    &   $-$14.2    &
$-$15.1 \\
  1.80    &         $-$6.11  &    $-$13.4    &   $-$14.4    &
$-$15.2 \\
  1.90    &         $-$6.30  &    $-$13.6    &   $-$14.5    &
$-$15.3 \\
  2.00    &         $-$6.48  &    $-$13.7    &   $-$14.6    &
$-$15.4 \\
  2.50    &         $-$7.24  &    $-$14.2    &   $-$15.1    &
$-$15.9 \\
  3.00    &         $-$7.78  &    $-$14.6    &   $-$15.5    &
$-$16.2 \\
  3.50    &         $-$8.17  &    $-$14.9    &   $-$15.7    &
$-$16.4 \\
  4.00    &         $-$8.44  &    $-$15.1    &   $-$15.9    &
$-$16.6 \\
  4.50    &         $-$8.61  &    $-$15.2    &   $-$16.0    &
$-$16.6 \\
  5.00    &         $-$8.72  &    $-$15.2    &   $-$16.0    &
$-$16.6 \\
  5.50    &         $-$8.76  &    $-$15.2    &   $-$16.0    &
$-$16.6 \\
  6.00    &         $-$8.76  &    $-$15.2    &   $-$15.9    &
$-$16.6 \\
\hline
\end{tabular}
\end{table}
\end{document}

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        xpts txtwid 2 div sub
          yminpts fnthgt sub delpts extndfr mul sub M2
        x 20 string cvs S
        } if

      /labseptrk labseptrk delpts sub def
      } for

    /labseptrk -1 def
    0 1 ysteps
      {
      /yn exch def
      /ypts yminpts delpts yn mul add def
      /y    ymin    delta    yn mul add def

      /dolabel labseptrk 0 lt def

      graylevel setgray
      dolabel { xminpts delpts extndfr mul sub } { xminpts } ifelse
      ypts M2
      xmaxpts ypts L2
      ST

      dolabel
        {
        /labseptrk labsep def
        0 setgray
        xminpts ypts M2
        y 20 string cvs false CP
        PBX
        exch 4 -1 roll sub /txtwid exch def
        sub neg /txthgt exch def
        NP
        xminpts txtwid sub fnthgt sub delpts extndfr mul sub
          ypts txthgt 2 div sub M2
        y 20 string cvs S
        } if

      /labseptrk labseptrk delpts sub def
      } for

    0 setgray

    end
  } def

%%EndProlog
%%Page: 1 1
%%PageBoundingBox: 113.144 267.84 498.856 622.925
%%PageOrientation: Portrait
save

NP
0.756 SLW
1 SLC
180 576 21.6 180 0 arcn
ST
NP
0.756 SLW
1 SLC
180 576 21.6 360 180 arcn
ST
NP
0.756 SLW
1 SLC
[ 4.536 3.024 ] 0 SD
180 597.6 M2
180 554.4 L2
ST
[] 0 SD
NP
0.756 SLW
1 SLC
115.2 576 M2
116 573.944 116.8 573.944 117.6 576 C2
118.4 578.056 119.2 578.056 120 576 C2
120.8 573.944 121.6 573.944 122.4 576 C2
123.2 578.056 124 578.056 124.8 576 C2
125.6 573.944 126.4 573.944 127.2 576 C2
128 578.056 128.8 578.056 129.6 576 C2
130.4 573.944 131.2 573.944 132 576 C2
132.8 578.056 133.6 578.056 134.4 576 C2
135.2 573.944 136 573.944 136.8 576 C2
137.6 578.056 138.4 578.056 139.2 576 C2
140 573.944 140.8 573.944 141.6 576 C2
142.4 578.056 143.2 578.056 144 576 C2
144.8 573.944 145.6 573.944 146.4 576 C2
147.2 578.056 148 578.056 148.8 576 C2
149.6 573.944 150.4 573.944 151.2 576 C2
152 578.056 152.8 578.056 153.6 576 C2
154.4 573.944 155.2 573.944 156 576 C2
156.8 578.056 157.6 578.056 158.4 576 C2
ST
NP
0.756 SLW
1 SLC
201.6 576 M2
202.4 573.944 203.2 573.944 204 576 C2
204.8 578.056 205.6 578.056 206.4 576 C2
207.2 573.944 208 573.944 208.8 576 C2
209.6 578.056 210.4 578.056 211.2 576 C2
212 573.944 212.8 573.944 213.6 576 C2
214.4 578.056 215.2 578.056 216 576 C2
216.8 573.944 217.6 573.944 218.4 576 C2
219.2 578.056 220 578.056 220.8 576 C2
221.6 573.944 222.4 573.944 223.2 576 C2
224 578.056 224.8 578.056 225.6 576 C2
226.4 573.944 227.2 573.944 228 576 C2
228.8 578.056 229.6 578.056 230.4 576 C2
231.2 573.944 232 573.944 232.8 576 C2
233.6 578.056 234.4 578.056 235.2 576 C2
236 573.944 236.8 573.944 237.6 576 C2
238.4 578.056 239.2 578.056 240 576 C2
240.8 573.944 241.6 573.944 242.4 576 C2
243.2 578.056 244 578.056 244.8 576 C2
ST
1.134 SLW
{vtx_dot} 158.4 576 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 180 597.6 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 201.6 576 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 180 554.4 0.108 0 vtx_create
180 545.76 0 txtsta
0 0 M2
CF dup 5 mul setflat
CPT
/Times-Roman 12.96 SF
(\(a\)) true CP
FP PBX
3 -1 roll sub
3 1 roll sub neg
-0.5 mul 2 1 roll -1 mul
NP
4 2 roll M2 RM2
setflat
/Times-Roman 12.96 SF
(\(a\)) S
txtend
gsave
NP
432 576 21.6 21.6 ellipse
clip
NP
grestore
NP
432 576 21.6 21.6 ellipse
0.756 SLW
ST
NP
0.756 SLW
1 SLC
[ 4.536 3.024 ] 0 SD
432 606.034 17.4863 584.886 315.114 arcn
ST
[] 0 SD
NP
0.756 SLW
1 SLC
367.2 576 M2
368 573.944 368.8 573.944 369.6 576 C2
370.4 578.056 371.2 578.056 372 576 C2
372.8 573.944 373.6 573.944 374.4 576 C2
375.2 578.056 376 578.056 376.8 576 C2
377.6 573.944 378.4 573.944 379.2 576 C2
380 578.056 380.8 578.056 381.6 576 C2
382.4 573.944 383.2 573.944 384 576 C2
384.8 578.056 385.6 578.056 386.4 576 C2
387.2 573.944 388 573.944 388.8 576 C2
389.6 578.056 390.4 578.056 391.2 576 C2
392 573.944 392.8 573.944 393.6 576 C2
394.4 578.056 395.2 578.056 396 576 C2
396.8 573.944 397.6 573.944 398.4 576 C2
399.2 578.056 400 578.056 400.8 576 C2
401.6 573.944 402.4 573.944 403.2 576 C2
404 578.056 404.8 578.056 405.6 576 C2
406.4 573.944 407.2 573.944 408 576 C2
408.8 578.056 409.6 578.056 410.4 576 C2
ST
NP
0.756 SLW
1 SLC
453.6 576 M2
454.4 573.944 455.2 573.944 456 576 C2
456.8 578.056 457.6 578.056 458.4 576 C2
459.2 573.944 460 573.944 460.8 576 C2
461.6 578.056 462.4 578.056 463.2 576 C2
464 573.944 464.8 573.944 465.6 576 C2
466.4 578.056 467.2 578.056 468 576 C2
468.8 573.944 469.6 573.944 470.4 576 C2
471.2 578.056 472 578.056 472.8 576 C2
473.6 573.944 474.4 573.944 475.2 576 C2
476 578.056 476.8 578.056 477.6 576 C2
478.4 573.944 479.2 573.944 480 576 C2
480.8 578.056 481.6 578.056 482.4 576 C2
483.2 573.944 484 573.944 484.8 576 C2
485.6 578.056 486.4 578.056 487.2 576 C2
488 573.944 488.8 573.944 489.6 576 C2
490.4 578.056 491.2 578.056 492 576 C2
492.8 573.944 493.6 573.944 494.4 576 C2
495.2 578.056 496 578.056 496.8 576 C2
ST
1.134 SLW
{vtx_dot} 410.4 576 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 453.6 576 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 419.611 593.694 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 444.389 593.694 0.108 0 vtx_create
432 550.08 0 txtsta
0 0 M2
CF dup 5 mul setflat
CPT
/Times-Roman 12.96 SF
(\(b\)) true CP
FP PBX
3 -1 roll sub
3 1 roll sub neg
-0.5 mul 2 1 roll -1 mul
NP
4 2 roll M2 RM2
setflat
/Times-Roman 12.96 SF
(\(b\)) S
txtend
NP
0.756 SLW
1 SLC
158.4 432 M2
180 453.6 L2
ST
NP
0.756 SLW
1 SLC
158.4 432 M2
180 410.4 L2
ST
NP
0.756 SLW
1 SLC
[ 4.536 3.024 ] 0 SD
180 453.6 M2
201.6 432 L2
ST
[] 0 SD
NP
0.756 SLW
1 SLC
[ 4.536 3.024 ] 0 SD
201.6 432 M2
180 410.4 L2
ST
[] 0 SD
NP
0.756 SLW
1 SLC
180 453.6 M2
180 410.4 L2
ST
NP
0.756 SLW
1 SLC
115.2 432 M2
116 429.944 116.8 429.944 117.6 432 C2
118.4 434.056 119.2 434.056 120 432 C2
120.8 429.944 121.6 429.944 122.4 432 C2
123.2 434.056 124 434.056 124.8 432 C2
125.6 429.944 126.4 429.944 127.2 432 C2
128 434.056 128.8 434.056 129.6 432 C2
130.4 429.944 131.2 429.944 132 432 C2
132.8 434.056 133.6 434.056 134.4 432 C2
135.2 429.944 136 429.944 136.8 432 C2
137.6 434.056 138.4 434.056 139.2 432 C2
140 429.944 140.8 429.944 141.6 432 C2
142.4 434.056 143.2 434.056 144 432 C2
144.8 429.944 145.6 429.944 146.4 432 C2
147.2 434.056 148 434.056 148.8 432 C2
149.6 429.944 150.4 429.944 151.2 432 C2
152 434.056 152.8 434.056 153.6 432 C2
154.4 429.944 155.2 429.944 156 432 C2
156.8 434.056 157.6 434.056 158.4 432 C2
ST
NP
0.756 SLW
1 SLC
201.6 432 M2
202.4 429.944 203.2 429.944 204 432 C2
204.8 434.056 205.6 434.056 206.4 432 C2
207.2 429.944 208 429.944 208.8 432 C2
209.6 434.056 210.4 434.056 211.2 432 C2
212 429.944 212.8 429.944 213.6 432 C2
214.4 434.056 215.2 434.056 216 432 C2
216.8 429.944 217.6 429.944 218.4 432 C2
219.2 434.056 220 434.056 220.8 432 C2
221.6 429.944 222.4 429.944 223.2 432 C2
224 434.056 224.8 434.056 225.6 432 C2
226.4 429.944 227.2 429.944 228 432 C2
228.8 434.056 229.6 434.056 230.4 432 C2
231.2 429.944 232 429.944 232.8 432 C2
233.6 434.056 234.4 434.056 235.2 432 C2
236 429.944 236.8 429.944 237.6 432 C2
238.4 434.056 239.2 434.056 240 432 C2
240.8 429.944 241.6 429.944 242.4 432 C2
243.2 434.056 244 434.056 244.8 432 C2
ST
1.134 SLW
{vtx_dot} 158.4 432 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 180 453.6 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 201.6 432 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 180 410.4 0.108 0 vtx_create
180 401.76 0 txtsta
0 0 M2
CF dup 5 mul setflat
CPT
/Times-Roman 12.96 SF
(\(c\)) true CP
FP PBX
3 -1 roll sub
3 1 roll sub neg
-0.5 mul 2 1 roll -1 mul
NP
4 2 roll M2 RM2
setflat
/Times-Roman 12.96 SF
(\(c\)) S
txtend
gsave
NP
432 453.6 10.8 10.8 ellipse
clip
NP
grestore
NP
432 453.6 10.8 10.8 ellipse
0.756 SLW
ST
NP
0.756 SLW
1 SLC
[ 4.536 3.024 ] 0 SD
433.704 433.848 23.377 184.534 122.336 arcn
ST
[] 0 SD
NP
0.756 SLW
1 SLC
[ 4.536 3.024 ] 0 SD
430.296 433.848 23.377 417.664 355.466 arcn
ST
[] 0 SD
NP
0.756 SLW
1 SLC
[ 4.536 3.024 ] 0 SD
432 432 21.6 360 180 arcn
ST
[] 0 SD
NP
0.756 SLW
1 SLC
367.2 432 M2
368 429.944 368.8 429.944 369.6 432 C2
370.4 434.056 371.2 434.056 372 432 C2
372.8 429.944 373.6 429.944 374.4 432 C2
375.2 434.056 376 434.056 376.8 432 C2
377.6 429.944 378.4 429.944 379.2 432 C2
380 434.056 380.8 434.056 381.6 432 C2
382.4 429.944 383.2 429.944 384 432 C2
384.8 434.056 385.6 434.056 386.4 432 C2
387.2 429.944 388 429.944 388.8 432 C2
389.6 434.056 390.4 434.056 391.2 432 C2
392 429.944 392.8 429.944 393.6 432 C2
394.4 434.056 395.2 434.056 396 432 C2
396.8 429.944 397.6 429.944 398.4 432 C2
399.2 434.056 400 434.056 400.8 432 C2
401.6 429.944 402.4 429.944 403.2 432 C2
404 434.056 404.8 434.056 405.6 432 C2
406.4 429.944 407.2 429.944 408 432 C2
408.8 434.056 409.6 434.056 410.4 432 C2
ST
NP
0.756 SLW
1 SLC
453.6 432 M2
454.4 429.944 455.2 429.944 456 432 C2
456.8 434.056 457.6 434.056 458.4 432 C2
459.2 429.944 460 429.944 460.8 432 C2
461.6 434.056 462.4 434.056 463.2 432 C2
464 429.944 464.8 429.944 465.6 432 C2
466.4 434.056 467.2 434.056 468 432 C2
468.8 429.944 469.6 429.944 470.4 432 C2
471.2 434.056 472 434.056 472.8 432 C2
473.6 429.944 474.4 429.944 475.2 432 C2
476 434.056 476.8 434.056 477.6 432 C2
478.4 429.944 479.2 429.944 480 432 C2
480.8 434.056 481.6 434.056 482.4 432 C2
483.2 429.944 484 429.944 484.8 432 C2
485.6 434.056 486.4 434.056 487.2 432 C2
488 429.944 488.8 429.944 489.6 432 C2
490.4 434.056 491.2 434.056 492 432 C2
492.8 429.944 493.6 429.944 494.4 432 C2
495.2 434.056 496 434.056 496.8 432 C2
ST
1.134 SLW
{vtx_dot} 421.2 453.6 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 442.8 453.6 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 410.4 432 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 453.6 432 0.108 0 vtx_create
432 401.76 0 txtsta
0 0 M2
CF dup 5 mul setflat
CPT
/Times-Roman 12.96 SF
(\(d\)) true CP
FP PBX
3 -1 roll sub
3 1 roll sub neg
-0.5 mul 2 1 roll -1 mul
NP
4 2 roll M2 RM2
setflat
/Times-Roman 12.96 SF
(\(d\)) S
txtend
gsave
NP
306 378 10.8 10.8 ellipse
clip
NP
grestore
NP
306 378 10.8 10.8 ellipse
0.756 SLW
ST
NP
0.756 SLW
1 SLC
241.2 324 M2
242 321.944 242.8 321.944 243.6 324 C2
244.4 326.056 245.2 326.056 246 324 C2
246.8 321.944 247.6 321.944 248.4 324 C2
249.2 326.056 250 326.056 250.8 324 C2
251.6 321.944 252.4 321.944 253.2 324 C2
254 326.056 254.8 326.056 255.6 324 C2
256.4 321.944 257.2 321.944 258 324 C2
258.8 326.056 259.6 326.056 260.4 324 C2
261.2 321.944 262 321.944 262.8 324 C2
263.6 326.056 264.4 326.056 265.2 324 C2
266 321.944 266.8 321.944 267.6 324 C2
268.4 326.056 269.2 326.056 270 324 C2
270.8 321.944 271.6 321.944 272.4 324 C2
273.2 326.056 274 326.056 274.8 324 C2
275.6 321.944 276.4 321.944 277.2 324 C2
278 326.056 278.8 326.056 279.6 324 C2
280.4 321.944 281.2 321.944 282 324 C2
282.8 326.056 283.6 326.056 284.4 324 C2
285.2 321.944 286 321.944 286.8 324 C2
287.6 326.056 288.4 326.056 289.2 324 C2
290 321.944 290.8 321.944 291.6 324 C2
292.4 326.056 293.2 326.056 294 324 C2
294.8 321.944 295.6 321.944 296.4 324 C2
297.2 326.056 298 326.056 298.8 324 C2
299.6 321.944 300.4 321.944 301.2 324 C2
302 326.056 302.8 326.056 303.6 324 C2
304.4 321.944 305.2 321.944 306 324 C2
ST
NP
0.756 SLW
1 SLC
306 324 M2
306.8 321.944 307.6 321.944 308.4 324 C2
309.2 326.056 310 326.056 310.8 324 C2
311.6 321.944 312.4 321.944 313.2 324 C2
314 326.056 314.8 326.056 315.6 324 C2
316.4 321.944 317.2 321.944 318 324 C2
318.8 326.056 319.6 326.056 320.4 324 C2
321.2 321.944 322 321.944 322.8 324 C2
323.6 326.056 324.4 326.056 325.2 324 C2
326 321.944 326.8 321.944 327.6 324 C2
328.4 326.056 329.2 326.056 330 324 C2
330.8 321.944 331.6 321.944 332.4 324 C2
333.2 326.056 334 326.056 334.8 324 C2
335.6 321.944 336.4 321.944 337.2 324 C2
338 326.056 338.8 326.056 339.6 324 C2
340.4 321.944 341.2 321.944 342 324 C2
342.8 326.056 343.6 326.056 344.4 324 C2
345.2 321.944 346 321.944 346.8 324 C2
347.6 326.056 348.4 326.056 349.2 324 C2
350 321.944 350.8 321.944 351.6 324 C2
352.4 326.056 353.2 326.056 354 324 C2
354.8 321.944 355.6 321.944 356.4 324 C2
357.2 326.056 358 326.056 358.8 324 C2
359.6 321.944 360.4 321.944 361.2 324 C2
362 326.056 362.8 326.056 363.6 324 C2
364.4 321.944 365.2 321.944 366 324 C2
366.8 326.056 367.6 326.056 368.4 324 C2
369.2 321.944 370 321.944 370.8 324 C2
ST
NP
0.756 SLW
1 SLC
[ 4.536 3.024 ] 0 SD
337.935 358.467 46.9875 227.184 155.436 arcn
ST
[] 0 SD
NP
0.756 SLW
1 SLC
[ 4.536 3.024 ] 0 SD
274.065 358.467 46.9875 312.816 384.564 arc
ST
[] 0 SD
1.134 SLW
{vtx_dot} 306 324 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 295.2 378 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 316.8 378 0.108 0 vtx_create
306 293.76 0 txtsta
0 0 M2
CF dup 5 mul setflat
CPT
/Times-Roman 12.96 SF
(\(e\)) true CP
FP PBX
3 -1 roll sub
3 1 roll sub neg
-0.5 mul 2 1 roll -1 mul
NP
4 2 roll M2 RM2
setflat
/Times-Roman 12.96 SF
(\(e\)) S
txtend
restore
showpage
%%EOF


%%%%%%%%%%%%%%%%%%%% File fig2.ps %%%%%%%%%%%%%%%%%%%%%%%%%%%%
%!
%%Creator: FeynDiagram 1.21  by Bill Dimm
%%BoundingBox: 115.2 375.84 496.8 622.925
%%LanguageLevel: 1
%%Pages: 1
%%EndComments
%%BeginProlog
% @(#) abbrev.ps 1.9@(#)

/CP /charpath load def
/CF /currentflat load def
/CPT /currentpoint load def
/C2 /curveto load def
/FP /flattenpath load def
/L2 /lineto load def
/M2 /moveto load def
/NP /newpath load def
/PBX /pathbbox load def
/RM2 /rmoveto load def
/SD /setdash load def
/SLC /setlinecap load def
/SLW /setlinewidth load def
/S /show load def
/ST /stroke load def

% @(#) vertex.ps 1.9@(#)

/vtx_dict 20 dict def
vtx_dict /vtx_mtrxstor matrix put

/vtx_create
  {
  vtx_dict begin

  /ang exch def
  /rad exch def
  /y exch def
  /x exch def
  /vtx_proc exch def

  /vtx_oldmatrix vtx_mtrxstor currentmatrix def

  x y translate
  ang rotate

  vtx_proc

  vtx_oldmatrix setmatrix
  end
  } bind def


/vtx_dot
  {
  NP
  rad 0 M2
  0 0 rad 0 360 arc
  0 setgray
  fill
  } bind def


/vtx_box
  {
  NP
  rad rad M2
  rad rad neg L2
  rad neg rad neg L2
  rad neg rad L2
  closepath
  0 setgray
  fill
  } def

/vtx_cross
  {
  NP
  -1 2 sqrt div rad mul
  -1 2 sqrt div rad mul
  M2
  1 2 sqrt div rad mul
  1 2 sqrt div rad mul
  L2

  -1 2 sqrt div rad mul
  1 2 sqrt div rad mul
  M2
  1 2 sqrt div rad mul
  -1 2 sqrt div rad mul
  L2

  ST
  } def

/vtx_circlecross
  {
  /xshrfct .63 def

    % blank out whatever is underneath
  NP
  rad 0 M2
  0 0 rad 0 360 arc
  1 setgray
  fill

  NP
  -1 2 sqrt div rad mul xshrfct mul
  -1 2 sqrt div rad mul xshrfct mul
  M2
  1 2 sqrt div rad mul xshrfct mul
  1 2 sqrt div rad mul xshrfct mul
  L2

  -1 2 sqrt div rad mul xshrfct mul
  1 2 sqrt div rad mul xshrfct mul
  M2
  1 2 sqrt div rad mul xshrfct mul
  -1 2 sqrt div rad mul xshrfct mul
  L2

  rad 0 M2
  0 0 rad 0 360 arc

  0 setgray
  ST
  } def

% @(#) arrow.ps 1.9@(#)

/tan
  {
  dup
  sin exch cos div
  } bind def

/arrow_dict 12 dict def
arrow_dict /arrow_mtrxstor matrix put

/arrow
  {
  arrow_dict
    begin

    /alpha exch def
    /y2 exch def
    /x2 exch def
    /y1 exch def
    /x1 exch def

    /arrow_oldmtrx arrow_mtrxstor currentmatrix def

    /dx x2 x1 sub def
    /dy y2 y1 sub def

    /len dx dx mul dy dy mul add sqrt def

    /theta 1 alpha tan atan def
    /rad len alpha tan mul
      theta sin alpha tan theta cos -1 add mul add div def

    x1 y1 translate
    dy dx atan rotate

    len 0 M2
    rad neg 0 rad theta 360 theta sub arcn
    closepath

    arrow_oldmtrx setmatrix
    end
  } bind def

% @(#) fill.ps 1.9@(#)

/fillbox_dict 20 dict def
fillbox_dict /fillbox_mtrxstor matrix put

% fillbox_create is to be used for filling a box which is AT LEAST AS
% LARGE
% as box given by xmin,ymin xmax,ymax - use clipping to strip off
% what
% you don't want

/fillbox_create
  {
  fillbox_dict begin

  /ang exch def
  /ymax exch def
  /xmax exch def
  /ymin exch def
  /xmin exch def
  /fillbox_proc exch def  % the fillbox_proc fetches rest of stack

  /fillbox_oldmtrx fillbox_mtrxstor currentmatrix def

  xmin xmax add 2 div
  ymin ymax add 2 div
  translate

  ang rotate

  % rad tells how big a circle which encompasses the box must be
  /rad
    xmax xmin sub xmax xmin sub mul
    ymax ymin sub ymax ymin sub mul
    add sqrt 2 div
    def

  fillbox_proc

  fillbox_oldmtrx setmatrix
  end
  } def


/fillbox_lines
  {
  /incr exch def

  rad neg incr rad
    {
    /yv exch def
    rad neg yv M2
    rad yv L2
    ST
    } for
  } bind def


/fillbox_dots
  {
  /dtrad exch def
  /incr exch def

  /shx 0 def

  rad neg incr rad
    {
    /yval exch def
    rad neg shx sub incr rad
      {
      /xval exch def
      xval dtrad add yval M2
      xval yval dtrad 0 360 arc
      fill
      } for
    shx 0 eq { /shx incr 2 div def } { /shx 0 def } ifelse
    } for
  } bind def

% @(#) ellipse.ps 1.9@(#)

/ellipse_dict 6 dict def
ellipse_dict /ellipse_mtrxstor matrix put

/ellipse
  {
  ellipse_dict begin

  /yrad exch def
  /xrad exch def
  /y exch def
  /x exch def

  /ellipse_oldmtrx ellipse_mtrxstor currentmatrix def

  x y translate
  xrad yrad scale
  0 0 1 0 360 arc

  ellipse_oldmtrx setmatrix

  end
  } def

% @(#) text.ps 1.9@(#)

/text_dict 20 dict def
text_dict /text_mtrxstor matrix put

/SF* {  exch findfont exch scalefont setfont } bind def

systemdict /selectfont known
  { /SF /selectfont load def }
  { /SF /SF* load def }
  ifelse

/txtsta
  {
  text_dict begin

  /angle exch def
  /y exch def
  /x exch def

  /text_oldmtrx text_mtrxstor currentmatrix def

  x y translate
  angle rotate
  } bind def


/txtend
  {
  text_oldmtrx setmatrix
  end
  } bind def

% @(#) max.ps 1.9@(#)

/max { 2 copy lt {exch} if pop } bind def

% @(#) grid.ps 1.9@(#)

/grid_dict 40 dict def
grid_dict /grid_mtrxstor matrix put

/grid
  {
  grid_dict
    begin

    /ysteps    exch def
    /xsteps    exch def
    /delpts  exch def
    /yminpts   exch def
    /xminpts   exch def
    /delta     exch def
    /ymin      exch def
    /xmin      exch def

    0 SLW

    /xmaxpts xminpts xsteps delpts mul add def
    /ymaxpts yminpts ysteps delpts mul add def

    /fnthgt 8 def
    /Times-Roman findfont fnthgt scalefont setfont
    /labsep fnthgt 2 mul def
    /extndfr 1 2 div def
    /graylevel 0.2 def

    /labseptrk -1 def
    0 1 xsteps
      {
      /xn exch def
      /xpts xminpts delpts xn mul add def
      /x    xmin    delta    xn mul add def

      /dolabel labseptrk 0 lt def

      graylevel setgray
      xpts
      dolabel { yminpts delpts extndfr mul sub } { yminpts } ifelse
      M2
      xpts ymaxpts L2
      ST

      dolabel
        {
        /labseptrk labsep def
        0 setgray
        xpts yminpts M2
        x 20 string cvs false CP
        PBX
        pop exch pop exch sub /txtwid exch def
        NP
        xpts txtwid 2 div sub
          yminpts fnthgt sub delpts extndfr mul sub M2
        x 20 string cvs S
        } if

      /labseptrk labseptrk delpts sub def
      } for

    /labseptrk -1 def
    0 1 ysteps
      {
      /yn exch def
      /ypts yminpts delpts yn mul add def
      /y    ymin    delta    yn mul add def

      /dolabel labseptrk 0 lt def

      graylevel setgray
      dolabel { xminpts delpts extndfr mul sub } { xminpts } ifelse
      ypts M2
      xmaxpts ypts L2
      ST

      dolabel
        {
        /labseptrk labsep def
        0 setgray
        xminpts ypts M2
        y 20 string cvs false CP
        PBX
        exch 4 -1 roll sub /txtwid exch def
        sub neg /txthgt exch def
        NP
        xminpts txtwid sub fnthgt sub delpts extndfr mul sub
          ypts txthgt 2 div sub M2
        y 20 string cvs S
        } if

      /labseptrk labseptrk delpts sub def
      } for

    0 setgray

    end
  } def

%%EndProlog
%%Page: 1 1
%%PageBoundingBox: 115.2 375.84 496.8 622.925
%%PageOrientation: Portrait
save

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NP
0.756 SLW
1 SLC
180 576 21.6 360 180 arcn
ST
NP
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1 SLC
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180 597.6 M2
180 554.4 L2
ST
[] 0 SD
NP
0.756 SLW
1 SLC
[ 4.536 3.024 ] 0 SD
115.2 576 M2
158.4 576 L2
ST
[] 0 SD
NP
0.756 SLW
1 SLC
[ 4.536 3.024 ] 0 SD
201.6 576 M2
244.8 576 L2
ST
[] 0 SD
1.134 SLW
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1.134 SLW
{vtx_dot} 201.6 576 0.108 0 vtx_create
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0 0 M2
CF dup 5 mul setflat
CPT
/Times-Roman 12.96 SF
(\(a\)) true CP
FP PBX
3 -1 roll sub
3 1 roll sub neg
-0.5 mul 2 1 roll -1 mul
NP
4 2 roll M2 RM2
setflat
/Times-Roman 12.96 SF
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gsave
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432 576 21.6 21.6 ellipse
clip
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grestore
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NP
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432 606.034 17.4863 584.886 315.114 arcn
ST
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NP
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1 SLC
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367.2 576 M2
410.4 576 L2
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[] 0 SD
NP
0.756 SLW
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[ 4.536 3.024 ] 0 SD
453.6 576 M2
496.8 576 L2
ST
[] 0 SD
1.134 SLW
{vtx_dot} 410.4 576 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 453.6 576 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 419.611 593.694 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 444.389 593.694 0.108 0 vtx_create
432 550.08 0 txtsta
0 0 M2
CF dup 5 mul setflat
CPT
/Times-Roman 12.96 SF
(\(b\)) true CP
FP PBX
3 -1 roll sub
3 1 roll sub neg
-0.5 mul 2 1 roll -1 mul
NP
4 2 roll M2 RM2
setflat
/Times-Roman 12.96 SF
(\(b\)) S
txtend
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0.756 SLW
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158.4 432 M2
180 453.6 L2
ST
NP
0.756 SLW
1 SLC
158.4 432 M2
180 410.4 L2
ST
NP
0.756 SLW
1 SLC
[ 4.536 3.024 ] 0 SD
180 453.6 M2
201.6 432 L2
ST
[] 0 SD
NP
0.756 SLW
1 SLC
[ 4.536 3.024 ] 0 SD
201.6 432 M2
180 410.4 L2
ST
[] 0 SD
NP
0.756 SLW
1 SLC
180 453.6 M2
180 410.4 L2
ST
NP
0.756 SLW
1 SLC
[ 4.536 3.024 ] 0 SD
115.2 432 M2
158.4 432 L2
ST
[] 0 SD
NP
0.756 SLW
1 SLC
[ 4.536 3.024 ] 0 SD
201.6 432 M2
244.8 432 L2
ST
[] 0 SD
1.134 SLW
{vtx_dot} 158.4 432 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 180 453.6 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 201.6 432 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 180 410.4 0.108 0 vtx_create
180 401.76 0 txtsta
0 0 M2
CF dup 5 mul setflat
CPT
/Times-Roman 12.96 SF
(\(c\)) true CP
FP PBX
3 -1 roll sub
3 1 roll sub neg
-0.5 mul 2 1 roll -1 mul
NP
4 2 roll M2 RM2
setflat
/Times-Roman 12.96 SF
(\(c\)) S
txtend
gsave
NP
432 453.6 10.8 10.8 ellipse
clip
NP
grestore
NP
432 453.6 10.8 10.8 ellipse
0.756 SLW
ST
NP
0.756 SLW
1 SLC
[ 4.536 3.024 ] 0 SD
433.704 433.848 23.377 184.534 122.336 arcn
ST
[] 0 SD
NP
0.756 SLW
1 SLC
[ 4.536 3.024 ] 0 SD
430.296 433.848 23.377 417.664 355.466 arcn
ST
[] 0 SD
NP
0.756 SLW
1 SLC
[ 4.536 3.024 ] 0 SD
432 432 21.6 360 180 arcn
ST
[] 0 SD
NP
0.756 SLW
1 SLC
[ 4.536 3.024 ] 0 SD
367.2 432 M2
410.4 432 L2
ST
[] 0 SD
NP
0.756 SLW
1 SLC
[ 4.536 3.024 ] 0 SD
453.6 432 M2
496.8 432 L2
ST
[] 0 SD
1.134 SLW
{vtx_dot} 421.2 453.6 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 442.8 453.6 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 410.4 432 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 453.6 432 0.108 0 vtx_create
432 401.76 0 txtsta
0 0 M2
CF dup 5 mul setflat
CPT
/Times-Roman 12.96 SF
(\(d\)) true CP
FP PBX
3 -1 roll sub
3 1 roll sub neg
-0.5 mul 2 1 roll -1 mul
NP
4 2 roll M2 RM2
setflat
/Times-Roman 12.96 SF
(\(d\)) S
txtend
restore
showpage
%%EOF

%%%%%%%%%%%%%%%%%%%% File fig3.ps %%%%%%%%%%%%%%%%%%%%%%%%%%%%
%!
%%Creator: FeynDiagram 1.21  by Bill Dimm
%%BoundingBox: 113.144 267.84 498.856 608.4
%%LanguageLevel: 1
%%Pages: 1
%%EndComments
%%BeginProlog
% @(#) abbrev.ps 1.9@(#)

/CP /charpath load def
/CF /currentflat load def
/CPT /currentpoint load def
/C2 /curveto load def
/FP /flattenpath load def
/L2 /lineto load def
/M2 /moveto load def
/NP /newpath load def
/PBX /pathbbox load def
/RM2 /rmoveto load def
/SD /setdash load def
/SLC /setlinecap load def
/SLW /setlinewidth load def
/S /show load def
/ST /stroke load def

% @(#) vertex.ps 1.9@(#)

/vtx_dict 20 dict def
vtx_dict /vtx_mtrxstor matrix put

/vtx_create
  {
  vtx_dict begin

  /ang exch def
  /rad exch def
  /y exch def
  /x exch def
  /vtx_proc exch def

  /vtx_oldmatrix vtx_mtrxstor currentmatrix def

  x y translate
  ang rotate

  vtx_proc

  vtx_oldmatrix setmatrix
  end
  } bind def


/vtx_dot
  {
  NP
  rad 0 M2
  0 0 rad 0 360 arc
  0 setgray
  fill
  } bind def


/vtx_box
  {
  NP
  rad rad M2
  rad rad neg L2
  rad neg rad neg L2
  rad neg rad L2
  closepath
  0 setgray
  fill
  } def

/vtx_cross
  {
  NP
  -1 2 sqrt div rad mul
  -1 2 sqrt div rad mul
  M2
  1 2 sqrt div rad mul
  1 2 sqrt div rad mul
  L2

  -1 2 sqrt div rad mul
  1 2 sqrt div rad mul
  M2
  1 2 sqrt div rad mul
  -1 2 sqrt div rad mul
  L2

  ST
  } def

/vtx_circlecross
  {
  /xshrfct .63 def

    % blank out whatever is underneath
  NP
  rad 0 M2
  0 0 rad 0 360 arc
  1 setgray
  fill

  NP
  -1 2 sqrt div rad mul xshrfct mul
  -1 2 sqrt div rad mul xshrfct mul
  M2
  1 2 sqrt div rad mul xshrfct mul
  1 2 sqrt div rad mul xshrfct mul
  L2

  -1 2 sqrt div rad mul xshrfct mul
  1 2 sqrt div rad mul xshrfct mul
  M2
  1 2 sqrt div rad mul xshrfct mul
  -1 2 sqrt div rad mul xshrfct mul
  L2

  rad 0 M2
  0 0 rad 0 360 arc

  0 setgray
  ST
  } def

% @(#) arrow.ps 1.9@(#)

/tan
  {
  dup
  sin exch cos div
  } bind def

/arrow_dict 12 dict def
arrow_dict /arrow_mtrxstor matrix put

/arrow
  {
  arrow_dict
    begin

    /alpha exch def
    /y2 exch def
    /x2 exch def
    /y1 exch def
    /x1 exch def

    /arrow_oldmtrx arrow_mtrxstor currentmatrix def

    /dx x2 x1 sub def
    /dy y2 y1 sub def

    /len dx dx mul dy dy mul add sqrt def

    /theta 1 alpha tan atan def
    /rad len alpha tan mul
      theta sin alpha tan theta cos -1 add mul add div def

    x1 y1 translate
    dy dx atan rotate

    len 0 M2
    rad neg 0 rad theta 360 theta sub arcn
    closepath

    arrow_oldmtrx setmatrix
    end
  } bind def

% @(#) fill.ps 1.9@(#)

/fillbox_dict 20 dict def
fillbox_dict /fillbox_mtrxstor matrix put

% fillbox_create is to be used for filling a box which is AT LEAST AS
% LARGE
% as box given by xmin,ymin xmax,ymax - use clipping to strip off
% what
% you don't want

/fillbox_create
  {
  fillbox_dict begin

  /ang exch def
  /ymax exch def
  /xmax exch def
  /ymin exch def
  /xmin exch def
  /fillbox_proc exch def  % the fillbox_proc fetches rest of stack

  /fillbox_oldmtrx fillbox_mtrxstor currentmatrix def

  xmin xmax add 2 div
  ymin ymax add 2 div
  translate

  ang rotate

  % rad tells how big a circle which encompasses the box must be
  /rad
    xmax xmin sub xmax xmin sub mul
    ymax ymin sub ymax ymin sub mul
    add sqrt 2 div
    def

  fillbox_proc

  fillbox_oldmtrx setmatrix
  end
  } def


/fillbox_lines
  {
  /incr exch def

  rad neg incr rad
    {
    /yv exch def
    rad neg yv M2
    rad yv L2
    ST
    } for
  } bind def


/fillbox_dots
  {
  /dtrad exch def
  /incr exch def

  /shx 0 def

  rad neg incr rad
    {
    /yval exch def
    rad neg shx sub incr rad
      {
      /xval exch def
      xval dtrad add yval M2
      xval yval dtrad 0 360 arc
      fill
      } for
    shx 0 eq { /shx incr 2 div def } { /shx 0 def } ifelse
    } for
  } bind def

% @(#) ellipse.ps 1.9@(#)

/ellipse_dict 6 dict def
ellipse_dict /ellipse_mtrxstor matrix put

/ellipse
  {
  ellipse_dict begin

  /yrad exch def
  /xrad exch def
  /y exch def
  /x exch def

  /ellipse_oldmtrx ellipse_mtrxstor currentmatrix def

  x y translate
  xrad yrad scale
  0 0 1 0 360 arc

  ellipse_oldmtrx setmatrix

  end
  } def

% @(#) text.ps 1.9@(#)

/text_dict 20 dict def
text_dict /text_mtrxstor matrix put

/SF* {  exch findfont exch scalefont setfont } bind def

systemdict /selectfont known
  { /SF /selectfont load def }
  { /SF /SF* load def }
  ifelse

/txtsta
  {
  text_dict begin

  /angle exch def
  /y exch def
  /x exch def

  /text_oldmtrx text_mtrxstor currentmatrix def

  x y translate
  angle rotate
  } bind def


/txtend
  {
  text_oldmtrx setmatrix
  end
  } bind def

% @(#) max.ps 1.9@(#)

/max { 2 copy lt {exch} if pop } bind def

% @(#) grid.ps 1.9@(#)

/grid_dict 40 dict def
grid_dict /grid_mtrxstor matrix put

/grid
  {
  grid_dict
    begin

    /ysteps    exch def
    /xsteps    exch def
    /delpts  exch def
    /yminpts   exch def
    /xminpts   exch def
    /delta     exch def
    /ymin      exch def
    /xmin      exch def

    0 SLW

    /xmaxpts xminpts xsteps delpts mul add def
    /ymaxpts yminpts ysteps delpts mul add def

    /fnthgt 8 def
    /Times-Roman findfont fnthgt scalefont setfont
    /labsep fnthgt 2 mul def
    /extndfr 1 2 div def
    /graylevel 0.2 def

    /labseptrk -1 def
    0 1 xsteps
      {
      /xn exch def
      /xpts xminpts delpts xn mul add def
      /x    xmin    delta    xn mul add def

      /dolabel labseptrk 0 lt def

      graylevel setgray
      xpts
      dolabel { yminpts delpts extndfr mul sub } { yminpts } ifelse
      M2
      xpts ymaxpts L2
      ST

      dolabel
        {
        /labseptrk labsep def
        0 setgray
        xpts yminpts M2
        x 20 string cvs false CP
        PBX
        pop exch pop exch sub /txtwid exch def
        NP
        xpts txtwid 2 div sub
          yminpts fnthgt sub delpts extndfr mul sub M2
        x 20 string cvs S
        } if

      /labseptrk labseptrk delpts sub def
      } for

    /labseptrk -1 def
    0 1 ysteps
      {
      /yn exch def
      /ypts yminpts delpts yn mul add def
      /y    ymin    delta    yn mul add def

      /dolabel labseptrk 0 lt def

      graylevel setgray
      dolabel { xminpts delpts extndfr mul sub } { xminpts } ifelse
      ypts M2
      xmaxpts ypts L2
      ST

      dolabel
        {
        /labseptrk labsep def
        0 setgray
        xminpts ypts M2
        y 20 string cvs false CP
        PBX
        exch 4 -1 roll sub /txtwid exch def
        sub neg /txthgt exch def
        NP
        xminpts txtwid sub fnthgt sub delpts extndfr mul sub
          ypts txthgt 2 div sub M2
        y 20 string cvs S
        } if

      /labseptrk labseptrk delpts sub def
      } for

    0 setgray

    end
  } def

%%EndProlog
%%Page: 1 1
%%PageBoundingBox: 113.144 267.84 498.856 608.4
%%PageOrientation: Portrait
save

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0.756 SLW
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0.756 SLW
1 SLC
[ 4.536 3.024 ] 0 SD
180 597.6 M2
201.6 576 L2
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[] 0 SD
NP
0.756 SLW
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201.6 576 M2
199.592 576.9 199.038 576.346 199.938 574.338 C2
200.839 572.331 200.285 571.777 198.277 572.677 C2
196.269 573.577 195.715 573.023 196.615 571.015 C2
197.516 569.008 196.962 568.454 194.954 569.354 C2
192.946 570.254 192.392 569.7 193.292 567.692 C2
194.192 565.684 193.639 565.131 191.631 566.031 C2
189.623 566.931 189.069 566.377 189.969 564.369 C2
190.869 562.361 190.316 561.808 188.308 562.708 C2
186.3 563.608 185.746 563.054 186.646 561.046 C2
187.546 559.038 186.992 558.484 184.985 559.385 C2
182.977 560.285 182.423 559.731 183.323 557.723 C2
184.223 555.715 183.669 555.161 181.662 556.062 C2
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0.756 SLW
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116 573.944 116.8 573.944 117.6 576 C2
118.4 578.056 119.2 578.056 120 576 C2
120.8 573.944 121.6 573.944 122.4 576 C2
123.2 578.056 124 578.056 124.8 576 C2
125.6 573.944 126.4 573.944 127.2 576 C2
128 578.056 128.8 578.056 129.6 576 C2
130.4 573.944 131.2 573.944 132 576 C2
132.8 578.056 133.6 578.056 134.4 576 C2
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137.6 578.056 138.4 578.056 139.2 576 C2
140 573.944 140.8 573.944 141.6 576 C2
142.4 578.056 143.2 578.056 144 576 C2
144.8 573.944 145.6 573.944 146.4 576 C2
147.2 578.056 148 578.056 148.8 576 C2
149.6 573.944 150.4 573.944 151.2 576 C2
152 578.056 152.8 578.056 153.6 576 C2
154.4 573.944 155.2 573.944 156 576 C2
156.8 578.056 157.6 578.056 158.4 576 C2
ST
NP
0.756 SLW
1 SLC
201.6 576 M2
202.4 573.944 203.2 573.944 204 576 C2
204.8 578.056 205.6 578.056 206.4 576 C2
207.2 573.944 208 573.944 208.8 576 C2
209.6 578.056 210.4 578.056 211.2 576 C2
212 573.944 212.8 573.944 213.6 576 C2
214.4 578.056 215.2 578.056 216 576 C2
216.8 573.944 217.6 573.944 218.4 576 C2
219.2 578.056 220 578.056 220.8 576 C2
221.6 573.944 222.4 573.944 223.2 576 C2
224 578.056 224.8 578.056 225.6 576 C2
226.4 573.944 227.2 573.944 228 576 C2
228.8 578.056 229.6 578.056 230.4 576 C2
231.2 573.944 232 573.944 232.8 576 C2
233.6 578.056 234.4 578.056 235.2 576 C2
236 573.944 236.8 573.944 237.6 576 C2
238.4 578.056 239.2 578.056 240 576 C2
240.8 573.944 241.6 573.944 242.4 576 C2
243.2 578.056 244 578.056 244.8 576 C2
ST
1.134 SLW
{vtx_dot} 158.4 576 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 180 597.6 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 201.6 576 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 180 554.4 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 158.4 576 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 180 597.6 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 201.6 576 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 180 554.4 0.108 0 vtx_create
180 545.76 0 txtsta
0 0 M2
CF dup 5 mul setflat
CPT
/Times-Roman 12.96 SF
(\(a\)) true CP
FP PBX
3 -1 roll sub
3 1 roll sub neg
-0.5 mul 2 1 roll -1 mul
NP
4 2 roll M2 RM2
setflat
/Times-Roman 12.96 SF
(\(a\)) S
txtend
gsave
NP
432 597.6 10.8 10.8 ellipse
clip
NP
grestore
NP
432 597.6 10.8 10.8 ellipse
0.756 SLW
ST
NP
0.756 SLW
1 SLC
[ 4.536 3.024 ] 0 SD
433.704 577.848 23.377 184.534 122.336 arcn
ST
[] 0 SD
NP
0.756 SLW
1 SLC
[ 4.536 3.024 ] 0 SD
430.296 577.848 23.377 417.664 355.466 arcn
ST
[] 0 SD
NP
0.756 SLW
1 SLC
453.6 576 M2
451.53 575.269 451.489 574.539 453.441 573.507 C2
455.392 572.474 455.244 571.602 453.061 571.272 C2
450.878 570.942 450.675 570.239 452.348 568.798 C2
454.021 567.357 453.682 566.54 451.481 566.704 C2
449.279 566.868 448.925 566.228 450.236 564.451 C2
451.546 562.674 451.034 561.952 448.924 562.602 C2
446.814 563.252 446.327 562.707 447.208 560.683 C2
448.09 558.659 447.431 558.069 445.518 559.172 C2
443.605 560.275 443.009 559.852 443.419 557.683 C2
443.828 555.513 443.054 555.085 441.435 556.586 C2
439.815 558.087 439.14 557.807 439.056 555.601 C2
438.973 553.395 438.123 553.15 436.878 554.973 C2
435.633 556.797 434.913 556.675 434.34 554.542 C2
433.768 552.41 432.885 552.36 432.077 554.415 C2
431.269 556.47 430.539 556.511 429.507 554.559 C2
428.474 552.608 427.602 552.756 427.272 554.939 C2
426.942 557.122 426.239 557.325 424.798 555.652 C2
423.357 553.979 422.54 554.318 422.704 556.519 C2
422.868 558.721 422.228 559.075 420.451 557.764 C2
418.674 556.454 417.952 556.966 418.602 559.076 C2
419.252 561.186 418.707 561.673 416.683 560.792 C2
414.659 559.91 414.069 560.569 415.172 562.482 C2
416.275 564.395 415.852 564.991 413.683 564.581 C2
411.513 564.172 411.085 564.946 412.586 566.565 C2
414.087 568.185 413.807 568.86 411.601 568.944 C2
409.395 569.027 409.15 569.877 410.973 571.122 C2
412.797 572.367 412.675 573.087 410.542 573.66 C2
408.41 574.232 408.36 575.115 410.4 576 C2
ST
NP
0.756 SLW
1 SLC
367.2 576 M2
368 573.944 368.8 573.944 369.6 576 C2
370.4 578.056 371.2 578.056 372 576 C2
372.8 573.944 373.6 573.944 374.4 576 C2
375.2 578.056 376 578.056 376.8 576 C2
377.6 573.944 378.4 573.944 379.2 576 C2
380 578.056 380.8 578.056 381.6 576 C2
382.4 573.944 383.2 573.944 384 576 C2
384.8 578.056 385.6 578.056 386.4 576 C2
387.2 573.944 388 573.944 388.8 576 C2
389.6 578.056 390.4 578.056 391.2 576 C2
392 573.944 392.8 573.944 393.6 576 C2
394.4 578.056 395.2 578.056 396 576 C2
396.8 573.944 397.6 573.944 398.4 576 C2
399.2 578.056 400 578.056 400.8 576 C2
401.6 573.944 402.4 573.944 403.2 576 C2
404 578.056 404.8 578.056 405.6 576 C2
406.4 573.944 407.2 573.944 408 576 C2
408.8 578.056 409.6 578.056 410.4 576 C2
ST
NP
0.756 SLW
1 SLC
453.6 576 M2
454.4 573.944 455.2 573.944 456 576 C2
456.8 578.056 457.6 578.056 458.4 576 C2
459.2 573.944 460 573.944 460.8 576 C2
461.6 578.056 462.4 578.056 463.2 576 C2
464 573.944 464.8 573.944 465.6 576 C2
466.4 578.056 467.2 578.056 468 576 C2
468.8 573.944 469.6 573.944 470.4 576 C2
471.2 578.056 472 578.056 472.8 576 C2
473.6 573.944 474.4 573.944 475.2 576 C2
476 578.056 476.8 578.056 477.6 576 C2
478.4 573.944 479.2 573.944 480 576 C2
480.8 578.056 481.6 578.056 482.4 576 C2
483.2 573.944 484 573.944 484.8 576 C2
485.6 578.056 486.4 578.056 487.2 576 C2
488 573.944 488.8 573.944 489.6 576 C2
490.4 578.056 491.2 578.056 492 576 C2
492.8 573.944 493.6 573.944 494.4 576 C2
495.2 578.056 496 578.056 496.8 576 C2
ST
1.134 SLW
{vtx_dot} 421.2 597.6 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 442.8 597.6 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 410.4 576 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 453.6 576 0.108 0 vtx_create
432 545.76 0 txtsta
0 0 M2
CF dup 5 mul setflat
CPT
/Times-Roman 12.96 SF
(\(b\)) true CP
FP PBX
3 -1 roll sub
3 1 roll sub neg
-0.5 mul 2 1 roll -1 mul
NP
4 2 roll M2 RM2
setflat
/Times-Roman 12.96 SF
(\(b\)) S
txtend
gsave
NP
180 453.6 10.8 10.8 ellipse
clip
NP
grestore
NP
180 453.6 10.8 10.8 ellipse
0.756 SLW
ST
NP
0.756 SLW
1 SLC
[ 4.536 3.024 ] 0 SD
181.704 433.848 23.377 184.534 122.336 arcn
ST
[] 0 SD
NP
0.756 SLW
1 SLC
190.8 453.6 M2
190.344 451.438 190.973 450.991 192.915 452.071 C2
194.857 453.151 195.545 452.539 194.698 450.484 C2
193.851 448.43 194.368 447.858 196.497 448.494 C2
198.626 449.13 199.166 448.384 197.896 446.561 C2
196.627 444.737 197.009 444.067 199.225 444.229 C2
201.441 444.392 201.807 443.548 200.175 442.04 C2
198.542 440.533 198.77 439.796 200.97 439.477 C2
203.169 439.159 203.345 438.255 201.426 437.135 C2
199.506 436.015 199.571 435.246 201.65 434.461 C2
203.729 433.676 203.706 432.756 201.6 432 C2
ST
NP
0.756 SLW
1 SLC
201.6 432 M2
199.53 431.269 199.489 430.539 201.441 429.507 C2
203.392 428.474 203.244 427.602 201.061 427.272 C2
198.878 426.942 198.675 426.239 200.348 424.798 C2
202.021 423.357 201.682 422.54 199.481 422.704 C2
197.279 422.868 196.925 422.228 198.236 420.451 C2
199.546 418.674 199.034 417.952 196.924 418.602 C2
194.814 419.252 194.327 418.707 195.208 416.683 C2
196.09 414.659 195.431 414.069 193.518 415.172 C2
191.605 416.275 191.009 415.852 191.419 413.683 C2
191.828 411.513 191.054 411.085 189.435 412.586 C2
187.815 414.087 187.14 413.807 187.056 411.601 C2
186.973 409.395 186.123 409.15 184.878 410.973 C2
183.633 412.797 182.913 412.675 182.34 410.542 C2
181.768 408.41 180.885 408.36 180.077 410.415 C2
179.269 412.47 178.539 412.511 177.507 410.559 C2
176.474 408.608 175.602 408.756 175.272 410.939 C2
174.942 413.122 174.239 413.325 172.798 411.652 C2
171.357 409.979 170.54 410.318 170.704 412.519 C2
170.868 414.721 170.228 415.075 168.451 413.764 C2
166.674 412.454 165.952 412.966 166.602 415.076 C2
167.252 417.186 166.707 417.673 164.683 416.792 C2
162.659 415.91 162.069 416.569 163.172 418.482 C2
164.275 420.395 163.852 420.991 161.683 420.581 C2
159.513 420.172 159.085 420.946 160.586 422.565 C2
162.087 424.185 161.807 424.86 159.601 424.944 C2
157.395 425.027 157.15 425.877 158.973 427.122 C2
160.797 428.367 160.675 429.087 158.542 429.66 C2
156.41 430.232 156.36 431.115 158.4 432 C2
ST
NP
0.756 SLW
1 SLC
115.2 432 M2
116 429.944 116.8 429.944 117.6 432 C2
118.4 434.056 119.2 434.056 120 432 C2
120.8 429.944 121.6 429.944 122.4 432 C2
123.2 434.056 124 434.056 124.8 432 C2
125.6 429.944 126.4 429.944 127.2 432 C2
128 434.056 128.8 434.056 129.6 432 C2
130.4 429.944 131.2 429.944 132 432 C2
132.8 434.056 133.6 434.056 134.4 432 C2
135.2 429.944 136 429.944 136.8 432 C2
137.6 434.056 138.4 434.056 139.2 432 C2
140 429.944 140.8 429.944 141.6 432 C2
142.4 434.056 143.2 434.056 144 432 C2
144.8 429.944 145.6 429.944 146.4 432 C2
147.2 434.056 148 434.056 148.8 432 C2
149.6 429.944 150.4 429.944 151.2 432 C2
152 434.056 152.8 434.056 153.6 432 C2
154.4 429.944 155.2 429.944 156 432 C2
156.8 434.056 157.6 434.056 158.4 432 C2
ST
NP
0.756 SLW
1 SLC
201.6 432 M2
202.4 429.944 203.2 429.944 204 432 C2
204.8 434.056 205.6 434.056 206.4 432 C2
207.2 429.944 208 429.944 208.8 432 C2
209.6 434.056 210.4 434.056 211.2 432 C2
212 429.944 212.8 429.944 213.6 432 C2
214.4 434.056 215.2 434.056 216 432 C2
216.8 429.944 217.6 429.944 218.4 432 C2
219.2 434.056 220 434.056 220.8 432 C2
221.6 429.944 222.4 429.944 223.2 432 C2
224 434.056 224.8 434.056 225.6 432 C2
226.4 429.944 227.2 429.944 228 432 C2
228.8 434.056 229.6 434.056 230.4 432 C2
231.2 429.944 232 429.944 232.8 432 C2
233.6 434.056 234.4 434.056 235.2 432 C2
236 429.944 236.8 429.944 237.6 432 C2
238.4 434.056 239.2 434.056 240 432 C2
240.8 429.944 241.6 429.944 242.4 432 C2
243.2 434.056 244 434.056 244.8 432 C2
ST
1.134 SLW
{vtx_dot} 169.2 453.6 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 190.8 453.6 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 158.4 432 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 201.6 432 0.108 0 vtx_create
180 401.76 0 txtsta
0 0 M2
CF dup 5 mul setflat
CPT
/Times-Roman 12.96 SF
(\(c\)) true CP
FP PBX
3 -1 roll sub
3 1 roll sub neg
-0.5 mul 2 1 roll -1 mul
NP
4 2 roll M2 RM2
setflat
/Times-Roman 12.96 SF
(\(c\)) S
txtend
gsave
NP
432 453.6 10.8 10.8 ellipse
clip
NP
grestore
NP
432 453.6 10.8 10.8 ellipse
0.756 SLW
ST
NP
0.756 SLW
1 SLC
[ 4.536 3.024 ] 0 SD
433.704 433.848 23.377 184.534 122.336 arcn
ST
[] 0 SD
NP
0.756 SLW
1 SLC
442.8 453.6 M2
442.344 451.438 442.973 450.991 444.915 452.071 C2
446.857 453.151 447.545 452.539 446.698 450.484 C2
445.851 448.43 446.368 447.858 448.497 448.494 C2
450.626 449.13 451.166 448.384 449.896 446.561 C2
448.627 444.737 449.009 444.067 451.225 444.229 C2
453.441 444.392 453.807 443.548 452.175 442.04 C2
450.542 440.533 450.77 439.796 452.97 439.477 C2
455.169 439.159 455.345 438.255 453.426 437.135 C2
451.506 436.015 451.571 435.246 453.65 434.461 C2
455.729 433.676 455.706 432.756 453.6 432 C2
ST
NP
0.756 SLW
1 SLC
[ 4.536 3.024 ] 0 SD
432 432 21.6 360 180 arcn
ST
[] 0 SD
NP
0.756 SLW
1 SLC
367.2 432 M2
368 429.944 368.8 429.944 369.6 432 C2
370.4 434.056 371.2 434.056 372 432 C2
372.8 429.944 373.6 429.944 374.4 432 C2
375.2 434.056 376 434.056 376.8 432 C2
377.6 429.944 378.4 429.944 379.2 432 C2
380 434.056 380.8 434.056 381.6 432 C2
382.4 429.944 383.2 429.944 384 432 C2
384.8 434.056 385.6 434.056 386.4 432 C2
387.2 429.944 388 429.944 388.8 432 C2
389.6 434.056 390.4 434.056 391.2 432 C2
392 429.944 392.8 429.944 393.6 432 C2
394.4 434.056 395.2 434.056 396 432 C2
396.8 429.944 397.6 429.944 398.4 432 C2
399.2 434.056 400 434.056 400.8 432 C2
401.6 429.944 402.4 429.944 403.2 432 C2
404 434.056 404.8 434.056 405.6 432 C2
406.4 429.944 407.2 429.944 408 432 C2
408.8 434.056 409.6 434.056 410.4 432 C2
ST
NP
0.756 SLW
1 SLC
453.6 432 M2
454.4 429.944 455.2 429.944 456 432 C2
456.8 434.056 457.6 434.056 458.4 432 C2
459.2 429.944 460 429.944 460.8 432 C2
461.6 434.056 462.4 434.056 463.2 432 C2
464 429.944 464.8 429.944 465.6 432 C2
466.4 434.056 467.2 434.056 468 432 C2
468.8 429.944 469.6 429.944 470.4 432 C2
471.2 434.056 472 434.056 472.8 432 C2
473.6 429.944 474.4 429.944 475.2 432 C2
476 434.056 476.8 434.056 477.6 432 C2
478.4 429.944 479.2 429.944 480 432 C2
480.8 434.056 481.6 434.056 482.4 432 C2
483.2 429.944 484 429.944 484.8 432 C2
485.6 434.056 486.4 434.056 487.2 432 C2
488 429.944 488.8 429.944 489.6 432 C2
490.4 434.056 491.2 434.056 492 432 C2
492.8 429.944 493.6 429.944 494.4 432 C2
495.2 434.056 496 434.056 496.8 432 C2
ST
1.134 SLW
{vtx_dot} 421.2 453.6 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 442.8 453.6 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 410.4 432 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 453.6 432 0.108 0 vtx_create
432 401.76 0 txtsta
0 0 M2
CF dup 5 mul setflat
CPT
/Times-Roman 12.96 SF
(\(d\)) true CP
FP PBX
3 -1 roll sub
3 1 roll sub neg
-0.5 mul 2 1 roll -1 mul
NP
4 2 roll M2 RM2
setflat
/Times-Roman 12.96 SF
(\(d\)) S
txtend
gsave
NP
306 345.6 10.8 10.8 ellipse
clip
NP
grestore
NP
306 345.6 10.8 10.8 ellipse
0.756 SLW
ST
NP
0.756 SLW
1 SLC
284.4 324 M2
286.403 324.933 286.384 325.704 284.354 326.61 C2
282.325 327.515 282.402 328.432 284.554 328.988 C2
286.706 329.543 286.853 330.3 285.067 331.621 C2
283.28 332.943 283.553 333.822 285.774 333.901 C2
287.995 333.979 288.302 334.687 286.842 336.362 C2
285.382 338.038 285.838 338.837 288.023 338.436 C2
290.209 338.034 290.661 338.659 289.596 340.609 C2
288.532 342.56 289.149 343.243 291.197 342.38 C2
293.244 341.516 293.821 342.029 293.201 344.163 C2
292.582 346.298 293.332 346.831 295.2 345.6 C2
ST
NP
0.756 SLW
1 SLC
316.8 345.6 M2
316.344 343.438 316.973 342.991 318.915 344.071 C2
320.857 345.151 321.545 344.539 320.698 342.484 C2
319.851 340.43 320.368 339.858 322.497 340.494 C2
324.626 341.13 325.166 340.384 323.896 338.561 C2
322.627 336.737 323.009 336.067 325.225 336.229 C2
327.441 336.392 327.807 335.548 326.175 334.04 C2
324.542 332.533 324.77 331.796 326.97 331.477 C2
329.169 331.159 329.345 330.255 327.426 329.135 C2
325.506 328.015 325.571 327.246 327.65 326.461 C2
329.729 325.676 329.706 324.756 327.6 324 C2
ST
NP
0.756 SLW
1 SLC
[ 4.536 3.024 ] 0 SD
306 324 21.6 360 180 arcn
ST
[] 0 SD
NP
0.756 SLW
1 SLC
241.2 324 M2
242 321.944 242.8 321.944 243.6 324 C2
244.4 326.056 245.2 326.056 246 324 C2
246.8 321.944 247.6 321.944 248.4 324 C2
249.2 326.056 250 326.056 250.8 324 C2
251.6 321.944 252.4 321.944 253.2 324 C2
254 326.056 254.8 326.056 255.6 324 C2
256.4 321.944 257.2 321.944 258 324 C2
258.8 326.056 259.6 326.056 260.4 324 C2
261.2 321.944 262 321.944 262.8 324 C2
263.6 326.056 264.4 326.056 265.2 324 C2
266 321.944 266.8 321.944 267.6 324 C2
268.4 326.056 269.2 326.056 270 324 C2
270.8 321.944 271.6 321.944 272.4 324 C2
273.2 326.056 274 326.056 274.8 324 C2
275.6 321.944 276.4 321.944 277.2 324 C2
278 326.056 278.8 326.056 279.6 324 C2
280.4 321.944 281.2 321.944 282 324 C2
282.8 326.056 283.6 326.056 284.4 324 C2
ST
NP
0.756 SLW
1 SLC
327.6 324 M2
328.4 321.944 329.2 321.944 330 324 C2
330.8 326.056 331.6 326.056 332.4 324 C2
333.2 321.944 334 321.944 334.8 324 C2
335.6 326.056 336.4 326.056 337.2 324 C2
338 321.944 338.8 321.944 339.6 324 C2
340.4 326.056 341.2 326.056 342 324 C2
342.8 321.944 343.6 321.944 344.4 324 C2
345.2 326.056 346 326.056 346.8 324 C2
347.6 321.944 348.4 321.944 349.2 324 C2
350 326.056 350.8 326.056 351.6 324 C2
352.4 321.944 353.2 321.944 354 324 C2
354.8 326.056 355.6 326.056 356.4 324 C2
357.2 321.944 358 321.944 358.8 324 C2
359.6 326.056 360.4 326.056 361.2 324 C2
362 321.944 362.8 321.944 363.6 324 C2
364.4 326.056 365.2 326.056 366 324 C2
366.8 321.944 367.6 321.944 368.4 324 C2
369.2 326.056 370 326.056 370.8 324 C2
ST
1.134 SLW
{vtx_dot} 295.2 345.6 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 316.8 345.6 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 284.4 324 0.108 0 vtx_create
1.134 SLW
{vtx_dot} 327.6 324 0.108 0 vtx_create
306 293.76 0 txtsta
0 0 M2
CF dup 5 mul setflat
CPT
/Times-Roman 12.96 SF
(\(e\)) true CP
FP PBX
3 -1 roll sub
3 1 roll sub neg
-0.5 mul 2 1 roll -1 mul
NP
4 2 roll M2 RM2
setflat
/Times-Roman 12.96 SF
(\(e\)) S
txtend
restore
showpage
%%EOF

%%%%%%%%%%%%%%%%%%%% File fig4.ps %%%%%%%%%%%%%%%%%%%%%%%%%%%%
%!
%%Creator: FeynDiagram 1.21  by Bill Dimm
%%BoundingBox: 28.8 234.72 403.2 460.8
%%LanguageLevel: 1
%%Pages: 1
%%EndComments
%%BeginProlog
% @(#) abbrev.ps 1.9@(#)

/CP /charpath load def
/CF /currentflat load def
/CPT /currentpoint load def
/C2 /curveto load def
/FP /flattenpath load def
/L2 /lineto load def
/M2 /moveto load def
/NP /newpath load def
/PBX /pathbbox load def
/RM2 /rmoveto load def
/SD /setdash load def
/SLC /setlinecap load def
/SLW /setlinewidth load def
/S /show load def
/ST /stroke load def

% @(#) vertex.ps 1.9@(#)

/vtx_dict 20 dict def
vtx_dict /vtx_mtrxstor matrix put

/vtx_create
  {
  vtx_dict begin

  /ang exch def
  /rad exch def
  /y exch def
  /x exch def
  /vtx_proc exch def

  /vtx_oldmatrix vtx_mtrxstor currentmatrix def

  x y translate
  ang rotate

  vtx_proc

  vtx_oldmatrix setmatrix
  end
  } bind def


/vtx_dot
  {
  NP
  rad 0 M2
  0 0 rad 0 360 arc
  0 setgray
  fill
  } bind def


/vtx_box
  {
  NP
  rad rad M2
  rad rad neg L2
  rad neg rad neg L2
  rad neg rad L2
  closepath
  0 setgray
  fill
  } def

/vtx_cross
  {
  NP
  -1 2 sqrt div rad mul
  -1 2 sqrt div rad mul
  M2
  1 2 sqrt div rad mul
  1 2 sqrt div rad mul
  L2

  -1 2 sqrt div rad mul
  1 2 sqrt div rad mul
  M2
  1 2 sqrt div rad mul
  -1 2 sqrt div rad mul
  L2

  ST
  } def

/vtx_circlecross
  {
  /xshrfct .63 def

    % blank out whatever is underneath
  NP
  rad 0 M2
  0 0 rad 0 360 arc
  1 setgray
  fill

  NP
  -1 2 sqrt div rad mul xshrfct mul
  -1 2 sqrt div rad mul xshrfct mul
  M2
  1 2 sqrt div rad mul xshrfct mul
  1 2 sqrt div rad mul xshrfct mul
  L2

  -1 2 sqrt div rad mul xshrfct mul
  1 2 sqrt div rad mul xshrfct mul
  M2
  1 2 sqrt div rad mul xshrfct mul
  -1 2 sqrt div rad mul xshrfct mul
  L2

  rad 0 M2
  0 0 rad 0 360 arc

  0 setgray
  ST
  } def

% @(#) arrow.ps 1.9@(#)

/tan
  {
  dup
  sin exch cos div
  } bind def

/arrow_dict 12 dict def
arrow_dict /arrow_mtrxstor matrix put

/arrow
  {
  arrow_dict
    begin

    /alpha exch def
    /y2 exch def
    /x2 exch def
    /y1 exch def
    /x1 exch def

    /arrow_oldmtrx arrow_mtrxstor currentmatrix def

    /dx x2 x1 sub def
    /dy y2 y1 sub def

    /len dx dx mul dy dy mul add sqrt def

    /theta 1 alpha tan atan def
    /rad len alpha tan mul
      theta sin alpha tan theta cos -1 add mul add div def

    x1 y1 translate
    dy dx atan rotate

    len 0 M2
    rad neg 0 rad theta 360 theta sub arcn
    closepath

    arrow_oldmtrx setmatrix
    end
  } bind def

% @(#) fill.ps 1.9@(#)

/fillbox_dict 20 dict def
fillbox_dict /fillbox_mtrxstor matrix put

% fillbox_create is to be used for filling a box which is AT LEAST AS
% LARGE
% as box given by xmin,ymin xmax,ymax - use clipping to strip off
% what
% you don't want

/fillbox_create
  {
  fillbox_dict begin

  /ang exch def
  /ymax exch def
  /xmax exch def
  /ymin exch def
  /xmin exch def
  /fillbox_proc exch def  % the fillbox_proc fetches rest of stack

  /fillbox_oldmtrx fillbox_mtrxstor currentmatrix def

  xmin xmax add 2 div
  ymin ymax add 2 div
  translate

  ang rotate

  % rad tells how big a circle which encompasses the box must be
  /rad
    xmax xmin sub xmax xmin sub mul
    ymax ymin sub ymax ymin sub mul
    add sqrt 2 div
    def

  fillbox_proc

  fillbox_oldmtrx setmatrix
  end
  } def


/fillbox_lines
  {
  /incr exch def

  rad neg incr rad
    {
    /yv exch def
    rad neg yv M2
    rad yv L2
    ST
    } for
  } bind def


/fillbox_dots
  {
  /dtrad exch def
  /incr exch def

  /shx 0 def

  rad neg incr rad
    {
    /yval exch def
    rad neg shx sub incr rad
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showpage
%%EOF

%%%%%%%%%%%%%%%%%%%% File fig5.ps %%%%%%%%%%%%%%%%%%%%%%%%%%%%
%!
%%Creator: FeynDiagram 1.21  by Bill Dimm
%%BoundingBox: 14.4 119.52 417.6 748.8
%%LanguageLevel: 1
%%Pages: 1
%%EndComments
%%BeginProlog
% @(#) abbrev.ps 1.9@(#)

/CP /charpath load def
/CF /currentflat load def
/CPT /currentpoint load def
/C2 /curveto load def
/FP /flattenpath load def
/L2 /lineto load def
/M2 /moveto load def
/NP /newpath load def
/PBX /pathbbox load def
/RM2 /rmoveto load def
/SD /setdash load def
/SLC /setlinecap load def
/SLW /setlinewidth load def
/S /show load def
/ST /stroke load def

% @(#) vertex.ps 1.9@(#)

/vtx_dict 20 dict def
vtx_dict /vtx_mtrxstor matrix put

/vtx_create
  {
  vtx_dict begin

  /ang exch def
  /rad exch def
  /y exch def
  /x exch def
  /vtx_proc exch def

  /vtx_oldmatrix vtx_mtrxstor currentmatrix def

  x y translate
  ang rotate

  vtx_proc

  vtx_oldmatrix setmatrix
  end
  } bind def


/vtx_dot
  {
  NP
  rad 0 M2
  0 0 rad 0 360 arc
  0 setgray
  fill
  } bind def


/vtx_box
  {
  NP
  rad rad M2
  rad rad neg L2
  rad neg rad neg L2
  rad neg rad L2
  closepath
  0 setgray
  fill
  } def

/vtx_cross
  {
  NP
  -1 2 sqrt div rad mul
  -1 2 sqrt div rad mul
  M2
  1 2 sqrt div rad mul
  1 2 sqrt div rad mul
  L2

  -1 2 sqrt div rad mul
  1 2 sqrt div rad mul
  M2
  1 2 sqrt div rad mul
  -1 2 sqrt div rad mul
  L2

  ST
  } def

/vtx_circlecross
  {
  /xshrfct .63 def

    % blank out whatever is underneath
  NP
  rad 0 M2
  0 0 rad 0 360 arc
  1 setgray
  fill

  NP
  -1 2 sqrt div rad mul xshrfct mul
  -1 2 sqrt div rad mul xshrfct mul
  M2
  1 2 sqrt div rad mul xshrfct mul
  1 2 sqrt div rad mul xshrfct mul
  L2

  -1 2 sqrt div rad mul xshrfct mul
  1 2 sqrt div rad mul xshrfct mul
  M2
  1 2 sqrt div rad mul xshrfct mul
  -1 2 sqrt div rad mul xshrfct mul
  L2

  rad 0 M2
  0 0 rad 0 360 arc

  0 setgray
  ST
  } def

% @(#) arrow.ps 1.9@(#)

/tan
  {
  dup
  sin exch cos div
  } bind def

/arrow_dict 12 dict def
arrow_dict /arrow_mtrxstor matrix put

/arrow
  {
  arrow_dict
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    /alpha exch def
    /y2 exch def
    /x2 exch def
    /y1 exch def
    /x1 exch def

    /arrow_oldmtrx arrow_mtrxstor currentmatrix def

    /dx x2 x1 sub def
    /dy y2 y1 sub def

    /len dx dx mul dy dy mul add sqrt def

    /theta 1 alpha tan atan def
    /rad len alpha tan mul
      theta sin alpha tan theta cos -1 add mul add div def

    x1 y1 translate
    dy dx atan rotate

    len 0 M2
    rad neg 0 rad theta 360 theta sub arcn
    closepath

    arrow_oldmtrx setmatrix
    end
  } bind def

% @(#) fill.ps 1.9@(#)

/fillbox_dict 20 dict def
fillbox_dict /fillbox_mtrxstor matrix put

% fillbox_create is to be used for filling a box which is AT LEAST AS
% LARGE
% as box given by xmin,ymin xmax,ymax - use clipping to strip off
% what
% you don't want

/fillbox_create
  {
  fillbox_dict begin

  /ang exch def
  /ymax exch def
  /xmax exch def
  /ymin exch def
  /xmin exch def
  /fillbox_proc exch def  % the fillbox_proc fetches rest of stack

  /fillbox_oldmtrx fillbox_mtrxstor currentmatrix def

  xmin xmax add 2 div
  ymin ymax add 2 div
  translate

  ang rotate

  % rad tells how big a circle which encompasses the box must be
  /rad
    xmax xmin sub xmax xmin sub mul
    ymax ymin sub ymax ymin sub mul
    add sqrt 2 div
    def

  fillbox_proc

  fillbox_oldmtrx setmatrix
  end
  } def


/fillbox_lines
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  /incr exch def

  rad neg incr rad
    {
    /yv exch def
    rad neg yv M2
    rad yv L2
    ST
    } for
  } bind def


/fillbox_dots
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  /dtrad exch def
  /incr exch def

  /shx 0 def

  rad neg incr rad
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    /yval exch def
    rad neg shx sub incr rad
      {
      /xval exch def
      xval dtrad add yval M2
      xval yval dtrad 0 360 arc
      fill
      } for
    shx 0 eq { /shx incr 2 div def } { /shx 0 def } ifelse
    } for
  } bind def

% @(#) ellipse.ps 1.9@(#)

/ellipse_dict 6 dict def
ellipse_dict /ellipse_mtrxstor matrix put

/ellipse
  {
  ellipse_dict begin

  /yrad exch def
  /xrad exch def
  /y exch def
  /x exch def

  /ellipse_oldmtrx ellipse_mtrxstor currentmatrix def

  x y translate
  xrad yrad scale
  0 0 1 0 360 arc

  ellipse_oldmtrx setmatrix

  end
  } def

% @(#) text.ps 1.9@(#)

/text_dict 20 dict def
text_dict /text_mtrxstor matrix put

/SF* {  exch findfont exch scalefont setfont } bind def

systemdict /selectfont known
  { /SF /selectfont load def }
  { /SF /SF* load def }
  ifelse

/txtsta
  {
  text_dict begin

  /angle exch def
  /y exch def
  /x exch def

  /text_oldmtrx text_mtrxstor currentmatrix def

  x y translate
  angle rotate
  } bind def


/txtend
  {
  text_oldmtrx setmatrix
  end
  } bind def

% @(#) max.ps 1.9@(#)

/max { 2 copy lt {exch} if pop } bind def

% @(#) grid.ps 1.9@(#)

/grid_dict 40 dict def
grid_dict /grid_mtrxstor matrix put

/grid
  {
  grid_dict
    begin

    /ysteps    exch def
    /xsteps    exch def
    /delpts  exch def
    /yminpts   exch def
    /xminpts   exch def
    /delta     exch def
    /ymin      exch def
    /xmin      exch def

    0 SLW

    /xmaxpts xminpts xsteps delpts mul add def
    /ymaxpts yminpts ysteps delpts mul add def

    /fnthgt 8 def
    /Times-Roman findfont fnthgt scalefont setfont
    /labsep fnthgt 2 mul def
    /extndfr 1 2 div def
    /graylevel 0.2 def

    /labseptrk -1 def
    0 1 xsteps
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      /xn exch def
      /xpts xminpts delpts xn mul add def
      /x    xmin    delta    xn mul add def

      /dolabel labseptrk 0 lt def

      graylevel setgray
      xpts
      dolabel { yminpts delpts extndfr mul sub } { yminpts } ifelse
      M2
      xpts ymaxpts L2
      ST

      dolabel
        {
        /labseptrk labsep def
        0 setgray
        xpts yminpts M2
        x 20 string cvs false CP
        PBX
        pop exch pop exch sub /txtwid exch def
        NP
        xpts txtwid 2 div sub
          yminpts fnthgt sub delpts extndfr mul sub M2
        x 20 string cvs S
        } if

      /labseptrk labseptrk delpts sub def
      } for

    /labseptrk -1 def
    0 1 ysteps
      {
      /yn exch def
      /ypts yminpts delpts yn mul add def
      /y    ymin    delta    yn mul add def

      /dolabel labseptrk 0 lt def

      graylevel setgray
      dolabel { xminpts delpts extndfr mul sub } { xminpts } ifelse
      ypts M2
      xmaxpts ypts L2
      ST

      dolabel
        {
        /labseptrk labsep def
        0 setgray
        xminpts ypts M2
        y 20 string cvs false CP
        PBX
        exch 4 -1 roll sub /txtwid exch def
        sub neg /txthgt exch def
        NP
        xminpts txtwid sub fnthgt sub delpts extndfr mul sub
          ypts txthgt 2 div sub M2
        y 20 string cvs S
        } if

      /labseptrk labseptrk delpts sub def
      } for

    0 setgray

    end
  } def

%%EndProlog
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271.479 333.823 272.007 333.973 272.882 332.79 C2
273.758 331.606 274.265 331.736 274.466 333.194 C2
274.666 334.652 275.202 334.773 276.01 333.543 C2
276.818 332.313 277.332 332.414 277.613 333.859 C2
277.894 335.303 278.436 335.394 279.174 334.121 C2
279.913 332.848 280.431 332.92 280.793 334.347 C2
281.154 335.774 281.699 335.835 282.366 334.523 C2
283.033 333.211 283.554 333.254 283.994 334.659 C2
284.434 336.064 284.982 336.094 285.575 334.747 C2
286.168 333.399 286.691 333.414 287.208 334.792 C2
287.726 336.17 288.274 336.17 288.792 334.792 C2
289.309 333.414 289.832 333.399 290.425 334.747 C2
291.018 336.094 291.566 336.064 292.006 334.659 C2
292.446 333.254 292.967 333.211 293.634 334.523 C2
294.301 335.835 294.846 335.774 295.207 334.347 C2
295.569 332.92 296.087 332.848 296.826 334.121 C2
297.564 335.394 298.106 335.303 298.387 333.859 C2
298.668 332.414 299.182 332.313 299.99 333.543 C2
300.798 334.773 301.334 334.652 301.534 333.194 C2
301.735 331.736 302.242 331.606 303.118 332.79 C2
303.993 333.973 304.521 333.823 304.64 332.356 C2
304.76 330.889 305.259 330.731 306.198 331.864 C2
307.138 332.998 307.657 332.818 307.695 331.346 C2
307.732 329.875 308.222 329.69 309.223 330.769 C2
310.224 331.849 310.732 331.64 310.688 330.169 C2
310.644 328.698 311.123 328.486 312.172 329.516 C2
ST
NP
0.504 SLW
1 SLC
263.828 318.484 M2
263.756 317.012 264.258 316.79 265.288 317.841 C2
266.319 318.892 266.803 318.693 266.8 317.221 C2
266.797 315.749 267.311 315.556 268.282 316.662 C2
269.252 317.769 269.747 317.597 269.826 316.128 C2
269.904 314.658 270.428 314.493 271.336 315.651 C2
272.243 316.81 272.747 316.667 272.907 315.203 C2
273.067 313.74 273.599 313.604 274.441 314.812 C2
275.283 316.019 275.794 315.904 276.035 314.452 C2
276.276 313 276.814 312.894 277.588 314.146 C2
278.362 315.398 278.878 315.311 279.199 313.875 C2
279.521 312.439 280.064 312.363 280.767 313.656 C2
281.47 314.949 281.991 314.891 282.391 313.475 C2
282.792 312.058 283.339 312.013 283.969 313.343 C2
284.599 314.673 285.122 314.644 285.6 313.252 C2
286.079 311.86 286.628 311.845 287.183 313.208 C2
287.738 314.571 288.262 314.571 288.817 313.208 C2
289.372 311.845 289.921 311.86 290.4 313.252 C2
290.878 314.644 291.401 314.673 292.031 313.343 C2
292.661 312.013 293.208 312.058 293.609 313.475 C2
294.009 314.891 294.53 314.949 295.233 313.656 C2
295.936 312.363 296.479 312.439 296.801 313.875 C2
297.122 315.311 297.638 315.398 298.412 314.146 C2
299.186 312.894 299.724 313 299.965 314.452 C2
300.206 315.904 300.717 316.019 301.559 314.812 C2
302.401 313.604 302.933 313.74 303.093 315.203 C2
303.253 316.667 303.757 316.81 304.664 315.651 C2
305.572 314.493 306.096 314.658 306.174 316.128 C2
306.253 317.597 306.748 317.769 307.718 316.662 C2
308.689 315.556 309.203 315.749 309.2 317.221 C2
309.197 318.693 309.681 318.892 310.712 317.841 C2
311.742 316.79 312.244 317.012 312.172 318.484 C2
ST
NP
0.504 SLW
1 SLC
244.8 352.8 M2
254.572 329.516 L2
ST
NP
0.504 SLW
1 SLC
244.8 295.2 M2
254.572 318.484 L2
ST
NP
0.504 SLW
1 SLC
331.2 352.8 M2
321.428 329.516 L2
ST
NP
0.504 SLW
1 SLC
331.2 295.2 M2
321.428 318.484 L2
ST
1.512 SLW
{vtx_cross} 288 334.8 5.0568 0 vtx_create
216 280.8 0 txtsta
0 0 M2
CF dup 5 mul setflat
CPT
/Times-Roman 8.64 SF
(\(c\)) true CP
FP PBX
3 -1 roll sub
3 1 roll sub neg
-0.5 mul 2 1 roll -1 mul
NP
4 2 roll M2 RM2
setflat
/Times-Roman 8.64 SF
(\(c\)) S
txtend
NP
0.504 SLW
1 SLC
187.2 180 M2
187.733 178.629 188.267 178.629 188.8 180 C2
189.333 181.371 189.867 181.371 190.4 180 C2
190.933 178.629 191.467 178.629 192 180 C2
192.533 181.371 193.067 181.371 193.6 180 C2
194.133 178.629 194.667 178.629 195.2 180 C2
195.733 181.371 196.267 181.371 196.8 180 C2
197.333 178.629 197.867 178.629 198.4 180 C2
198.933 181.371 199.467 181.371 200 180 C2
200.533 178.629 201.067 178.629 201.6 180 C2
202.133 181.371 202.667 181.371 203.2 180 C2
203.733 178.629 204.267 178.629 204.8 180 C2
205.333 181.371 205.867 181.371 206.4 180 C2
206.933 178.629 207.467 178.629 208 180 C2
208.533 181.371 209.067 181.371 209.6 180 C2
210.133 178.629 210.667 178.629 211.2 180 C2
211.733 181.371 212.267 181.371 212.8 180 C2
213.333 178.629 213.867 178.629 214.4 180 C2
214.933 181.371 215.467 181.371 216 180 C2
216.533 178.629 217.067 178.629 217.6 180 C2
218.133 181.371 218.667 181.371 219.2 180 C2
219.733 178.629 220.267 178.629 220.8 180 C2
221.333 181.371 221.867 181.371 222.4 180 C2
222.933 178.629 223.467 178.629 224 180 C2
224.533 181.371 225.067 181.371 225.6 180 C2
226.133 178.629 226.667 178.629 227.2 180 C2
227.733 181.371 228.267 181.371 228.8 180 C2
229.333 178.629 229.867 178.629 230.4 180 C2
230.933 181.371 231.467 181.371 232 180 C2
232.533 178.629 233.067 178.629 233.6 180 C2
234.133 181.371 234.667 181.371 235.2 180 C2
235.733 178.629 236.267 178.629 236.8 180 C2
237.333 181.371 237.867 181.371 238.4 180 C2
238.933 178.629 239.467 178.629 240 180 C2
240.533 181.371 241.067 181.371 241.6 180 C2
242.133 178.629 242.667 178.629 243.2 180 C2
243.733 181.371 244.267 181.371 244.8 180 C2
ST
NP
0.504 SLW
1 SLC
172.8 208.8 M2
187.2 180 L2
ST
NP
0.504 SLW
1 SLC
172.8 151.2 M2
187.2 180 L2
ST
NP
0.504 SLW
1 SLC
259.2 208.8 M2
244.8 180 L2
ST
NP
0.504 SLW
1 SLC
259.2 151.2 M2
244.8 180 L2
ST
1.512 SLW
{vtx_cross} 187.2 180 5.0568 0 vtx_create
1.512 SLW
{vtx_cross} 244.8 180 5.0568 0 vtx_create
192 192 0 txtsta
0 0 M2
CF dup 5 mul setflat
CPT
/Times-Roman 8.64 SF
() true CP
/Symbol 8.64 SF
(\144) true CP
/Times-Roman 8.64 SF
(g) true CP
FP PBX
3 -1 roll sub
3 1 roll sub neg
-0.5 mul 2 1 roll -0 mul
NP
4 2 roll M2 RM2
setflat
/Times-Roman 8.64 SF
() S
/Symbol 8.64 SF
(\144) S
/Times-Roman 8.64 SF
(g) S
txtend
240 192 0 txtsta
0 0 M2
CF dup 5 mul setflat
CPT
/Times-Roman 8.64 SF
() true CP
/Symbol 8.64 SF
(\144) true CP
/Times-Roman 8.64 SF
(g) true CP
FP PBX
3 -1 roll sub
3 1 roll sub neg
-0.5 mul 2 1 roll -0 mul
NP
4 2 roll M2 RM2
setflat
/Times-Roman 8.64 SF
() S
/Symbol 8.64 SF
(\144) S
/Times-Roman 8.64 SF
(g) S
txtend
216 136.8 0 txtsta
0 0 M2
CF dup 5 mul setflat
CPT
/Times-Roman 8.64 SF
(\(d\)) true CP
FP PBX
3 -1 roll sub
3 1 roll sub neg
-0.5 mul 2 1 roll -1 mul
NP
4 2 roll M2 RM2
setflat
/Times-Roman 8.64 SF
(\(d\)) S
txtend
restore
showpage
%%EOF

%%%%%%%%%%%%%%%%%%% End of figures %%%%%%%%%%%%%%%%

