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\title{
\hfill {\large IFJ 1653/PH}\\
\vspace{1 cm}
 {\bf Implications of scaling violations of
       $F_2$ at HERA for perturbative QCD}
\\[30pt]
\author{
A.J.Askew\thanks{On leave from Department of Physics,
University of Durham, England}\ ,
K.Golec\--Biernat, J.Kwieci\'nski, A.D.Martin$^*$
\\[3pt]
and
\\[3pt]
P.J.Sutton\thanks{On leave from Department of Physics,
University of Manchester, England}
\\[10pt]
{\it Department of Theoretical Physics,}
\\[3pt]
{\it Henryk Niewodniczanski Institute of Nuclear Physics,}
\\[3pt]
{\it ul. Radzikowskiego 152, 31-342 Krak\'ow, Poland }
\\[10pt]
       }
%\date
     }

\maketitle

\begin{abstract}
  We critically examine the QCD predictions for the $Q^2$
dependence of the electron-proton deep-inelastic structure
function $F_2(x,Q^2)$ in the small $x$ region, which is
being probed at HERA.  The standard results based on
next-to-leading order Altarelli-Parisi evolution are compared
with those that follow from the BFKL equation, which corresponds
to the resummation of the leading log$(1/x)$ terms.  The effects of parton
screening are also quantified.  The theoretical predictions
are confronted with each other, and with existing data from HERA.
\end{abstract}

\thispagestyle{empty}
\newpage
\setcounter{page}{1}
\vspace{1 cm}


   The first measurements of the proton structure function
$F_2(x,Q^2)$ at small $x$ have been made by the H1\cite{H1}
and ZEUS\cite{ZEUS} collaborations at HERA.  A striking increase
of $F_2$ with decreasing $x$ is observed which is consistent
with the expectations of perturbative QCD at small $x$ as
embodied in the BFKL equation\cite{BFKL}.  This equation
effectively performs a leading $\alpha_s{\rm log}(1/x)$
resummation of soft gluon emissions, which results in a
small $x$ behaviour $F_2 \sim x^{-\lambda}$ with $\lambda \sim
0.5$.

   The data at $Q^2=$ 15 and 30 GeV$^2$ are shown in Fig.1,
together with a representative set of predictions and extrapolations,
whose distinguishing features we elucidate below.  These curves fall into
two general categories.  The first, category (A), is phenomenological
and is based on parametric forms extrapolated to small $x$ with
$Q^2$ behaviour governed by the next-to-leading order Altarelli-Parisi
equations.  The parameters are determined by global fits to data
at larger $x$ (examples are the curves in Fig. 1 labelled
 MRS(D$_-^{\prime}$)\cite{MRSD}, MRS(H)\cite{MRSH} and,
to some extent, also GRV\cite{GRV}, but see below).
The second approach, denoted (B), is, in principle,
more fundamental.  Here perturbative QCD is used in the form of the
BFKL equation to evolve to small $x$ from known behaviour at larger
$x$ (e.g. AKMS\cite{AKMS}).  In other words in approach (A) the
small $x$ behaviour is input in the parametric forms used for the
parton distributions at some scale $Q^2=Q^2_0$, whereas in (B)
an $x^{-\lambda}$ behaviour at small $x$ is generated dynamically
with a determined value of $\lambda $. Of course in the phenomenological
approach, (A), it is possible to input a BFKL-motivated small $x$ behaviour
into the starting distributions (e.g. MRS(D$_-^{\prime}$) and
MRS(H) have $xg, xq_{\rm sea} \sim x^{-\lambda}$ with $\lambda
= 0.5$ and 0.3 respectively).  Since the $x^{-\lambda}$ behaviour,
for these values of $\lambda$, is stable to evolution in $Q^2$
we may anticipate that it will be difficult to distinguish approaches
(A) and (B).  However the $Q^2$ behaviour (or scaling violations) of $F_2$
is, in principle, different in the two approaches.

The Altarelli-Parisi $Q^2$ evolution is controlled by the
anomalous dimensions of the splitting functions (and by the
coefficient functions) which have been computed perturbatively
up to next-to-leading order.  On the other hand the
BFKL approach, at small $x$, corresponds to an infinite order resummation
of these quantities, keeping only leading log$(1/x)$ terms.
Summing the leading log$(1/x)$ terms, besides generating an
$x^{-\lambda}$ behaviour, gives its own characteristic $Q^2$
dependence.  One of our main purposes
is to study whether or not the
BFKL behaviour, which is more theoretically valid at small $x$,
can be distinguished from the approximate
Altarelli-Parisi parametric forms which neglect the log($1/x$) resummation.

  If we were to assume that Altarelli-Parisi evolution is valid at
small $x$ then
\begin{equation}
\frac{\partial F_2(x,Q^2)}{\partial {\rm log}Q^2} \ \simeq \ 2\sum_q e_q^2
\frac{\alpha _s(Q^2)}{2\pi}\int^1_x\frac{dy}{y}\frac{x}{y}
P_{qg}\left(\frac{x}{y} \right)
yg(y,Q^2)\ +\ ...\ ,
\end{equation}
and hence the $Q^2$ behaviour of $F_2$ can be varied by simply exploiting
the freedom in the gluon distribution at small $x$.  However the
situation is much more constrained when the BFKL equation is used to
determine the (unintegrated) gluon distribution $f(x,k^2_T)$.
Then $F_2$ may be calculated\cite{AKMS} using the $k_T$-factorization theorem
\cite{CATANI}
\begin{equation}
F_2(x,Q^2)\ =\ \int^1_x\frac{dx^{\prime}}{x^{\prime}}\int\frac{dk^2_T}
{k^4_T}f\left( \frac{x}{x^{\prime}},k^2_T \right)F^{(0)}_2(x^{\prime},
k^2_T,Q^2)
\end{equation}
where $x/x^{\prime}$ and $k_T$ are the longitudinal momentum
fraction and transverse momentum that are carried by
the gluon which dissociates into the $q\bar{q}$ pair, see Fig.\ 2.
$F^{(0)}_2$ is the quark box (and crossed box) amplitude for gluon-virtual
photon fusion\cite{AKMS}.

    In order to gain insight into the  different
possible $Q^2$ dependences of $F_2$
it is useful to introduce the moment function of the (unintegrated)
gluon distribution
\begin{equation}
f(n,k^2_T) = \int^1_0 dx x^{n-2} f(x,k^2_T).
\end{equation}
The evolution
of the moment function is given by the renormalization group
equation
\begin{equation}
f(n,k^2_T)\ =\ f(n,k^2_0)\ {\rm exp}\left[\int^{k^2_T}_{k^2_0}
\frac {dk^{\prime 2}}{k^{\prime 2}}\gamma (n,\alpha {_s}(k^{\prime2}_T))
\right]
\end{equation}
where the anomalous dimension $\gamma (n,\alpha {_s})$ is known.
{}From eq.(3)
we see that the behaviour at small $x$ is controlled by the leading
singularity of $f(n,k^2_T)$ in the $n$ plane.  In the leading
log$(1/x)$ approximation $\gamma (n,\alpha {_s})$ is just a function of
the single variable $\alpha {_s}(k^2_T)/(n-1)$ and is determined by
the BFKL kernel.  Its value is such that\cite{JAR}
\be
1\ -\ \frac{3\alpha _s(k^2_T)}{\pi (n-1)}{\tilde K}(\gamma )\ =\ 0
\ee
is satisfied, with
\be
{\tilde K}(\gamma )\ =\ 2\Psi (1)-\Psi(\gamma )-\Psi(1-\gamma ),
\ee
where $\Psi $ is the logarithmic derivative of the Euler gamma function.

For fixed $\alpha _s$  the leading singularity of $f(n,k^2_T)$
is a square root branch point at $n=1+\lambda _L$
where $\lambda _{L} =3\alpha _s{\tilde K}({1 \over 2})/\pi$
=$12\alpha _s\rm {log}2/\pi $.
Comparing with eq.(5) we find that
$\gamma (1+\lambda _{L},\alpha _s)={1\over 2}$.
Thus, from eq.(4), it directly follows that
\be
f(x,k^2_T)\sim(k^2_T)^{1\over 2} x^{-\lambda _{L}}.
\ee
Since $F^{(0)}_2/k^2_T$ in eq.(2) is simply a function of $k^2_T/Q^2$, this
leading behaviour feeds through into $F_2$ to give
\be
F_2(x,Q^2)\sim (Q^2)^{1\over 2} x^{-\lambda _{L}},
\ee
where in (7) and (8) we have omitted slowly varying logarithmic
factors.

   Formula (4) is valid for running $\alpha _s$, provided
$n$ remains to the right of the branch point throughout the
region of integration, that is provided
$n>1+12\alpha _s(k_0^2){\rm log}2/\pi$.
(For smaller values of $n$ the $k^2_T$ dependence of $f(n,k^2_T)$
is more involved\cite{JK}.)  For running $\alpha _s$ the small $x$
behaviour of $f(x,k^2_T)$ is controlled by the leading $pole$
singularity of $f(n,k_T^2)$ which occurs at $n=1+\bar{\lambda}$,
where now $\bar {\lambda}$ has to be calculated numerically\cite{KMS}.
A value of $\bar {\lambda}\approx 0.5$ is found, with rather
little sensitivity to the treatment of the infrared region of
the BFKL equation\cite{AKMS2}.
The $k^2_T$ dependence of $f$ (and hence the $Q^2$ dependence of
$F_2$) is determined by the residue $\beta$ of this pole.
Using eq.(4) we have
\be
f\sim {\beta (k^2_T)}x^{-\bar {\lambda}}
\ee
where
\be
\beta (k^2_T)\sim {\rm exp}\left[ \int^{k^2_T}_{k^2_0}
\frac {dk^{\prime 2}_T}{k^{\prime 2}_T} \gamma (1+{\bar {\lambda}},
\alpha_s(k^{\prime 2}_T))\right].
\ee
{}From the above discussion we see that this form is valid provided
$k^2_T\ge k^2_0\ge \kappa ^2(\bar {\lambda})$, where
$\kappa ^2(\bar {\lambda})$ satisfies the implicit equation
$\bar {\lambda}=12\alpha _{s}(\kappa ^2(\bar {\lambda})){\rm log}2/\pi$.
Similarly, provided that $Q^2\ge \kappa ^2(\bar {\lambda})$,
we have
\be
F_2(x,Q^2)\sim {\beta (Q^2)}x^{-\bar {\lambda}},
\ee
up to slight modifications which result from known $Q^2$ effects embedded in
$F^{(0)}_2$.  Note that for
$Q^2\stackrel{>}{\sim} \kappa ^2(\bar {\lambda})$ we should again get
an approximate $(Q^2)^{1\over 2}$ behaviour of $F_2(x,Q^2)$,
although it may (at moderately small values of $x$) be modified
by the non-leading contributions.
  Here we are also interested in $Q^2<\kappa^2(\bar{
\lambda})$ and then the form of $\beta$ is more involved\cite{JK}.

  In the leading log$(1/x)$ approximation
the anomalous dimension, $\gamma (n,\alpha _{s})$,
is a power series in $\alpha _{s}/(n-1)$.  For the BFKL approach
$\gamma (n,\alpha _{s})$ contains the sum of all these terms.
If only the first term were retained then the $Q^2$ behaviour
would correspond to Altarelli-Parisi
evolution from a singular $x^{-\bar {\lambda}}$ gluon starting
distribution with only $g\to gg$ transitions included and
with the splitting function $P_{gg}(z)$ approximated by its
singular $1/z$ term.

   It is useful to compare the $Q^2$ dependence of $F_2$ which
results from the theoretically
motivated BFKL approach, (B), with that of the
Altarelli-Parisi $Q^2$ evolution of approach (A).
For Altarelli-Parisi evolution the $Q^2$ behaviour of $F_2$ depends on the
small $x$ behaviour of the parton starting distributions.
If we assume that the starting distributions are non-singular at small $x$
(i.e.
$xg(x,Q_0^2)$ and $xq_{\rm sea}(x,Q_0^2)$ approach a constant
limit for $x\rightarrow 0$), then the leading term, which drives
both the $Q^2$ and $x$ dependence at small $x$, is of the double
logarithmic form
\be
F_2(x,Q^2) \sim {\rm exp}\left[2{\lbrace \xi(Q_0^2,Q^2){\rm log}(1/x)\rbrace
}^{1\over 2}\right],
\ee
where
\be
\xi(Q_0^2,Q^2)=\int_{Q_0^2}^{Q^2}{dq^2\over q^2}{3\alpha_s(q^2)\over \pi}.
\ee
{}From (12) we see that, as $x$ decreases, $F_2$ increases faster
than any power of log$(1/x)$ but slower than any power of $x$.


If, on the other hand,
the starting gluon and sea quark distributions are assumed to have
singular behaviour in the small $x$ limit i.e.
\be
 xg(x,Q_0^2),xq_{\rm sea}(x,Q^2_0) \sim x^{-\lambda}
\ee
with $\lambda > 0$,
then the  structure function $F_2(x,Q^2)$ behaves as
\be
F_2(x,Q^2) \sim x^{-\lambda}h(Q^2)
\ee
where the function $h(Q^2)$ is determined by the corresponding anomalous
dimensions of the moments of the (singlet) parton distributions
 at $n=1+\lambda$, as well as by the coefficient functions.

  We emphasize again that, in contrast to the BFKL
approach, for (next-to-leading
order) Altarelli-Parisi evolution the relevant quantities which
determine $h(Q^2)$ are computed from the first (two) terms in the
perturbative expansion in $\alpha _s$.  Thus terms are neglected, which
may in principle be important at small $x$, corresponding
to the infinite sum of powers of $\alpha _s/(n-1)$ in $\gamma $
(and in the coefficient function).


  Note that in both cases (i.e. eqs.(12) and (15))
Altarelli-Parisi evolution gives a slope of the structure function,
${\partial F_2(x,Q^2)/\partial {\rm log}(Q^2)}$,
which increases with decreasing $x$.
The MRS(D$_-^{\prime}$)\cite{MRSD} and MRS(H)\cite{MRSH} extrapolations
are examples of (15), with $\lambda = 0.5$ and 0.3 respectively.
On the other hand, the behaviour of $F_2$ obtained from the GRV\cite{GRV}
partons is an example of (12).  In the GRV model the partons
are generated from a valence-like input at a very low scale,
$Q^2_0=0.3{\rm GeV}^2$ (and then the valence is matched to MRS at
much higher $Q^2$). Due to the long evolution length,
$\xi (Q_0^2,Q^2)$, in reaching the $Q^2$ values corresponding to the small
$x$ HERA data the GRV prediction tends to the double
logarithmic form of (12).
The GRV model is probably best regarded as a phenomenological
way of obtaining steep distributions at a conventional input scale,
say 4GeV$^2$, since the steepness is mainly generated in the very
low $Q^2$ region where perturbative QCD is unreliable\cite{JRF}.
  Note, however,
that the steepness is specified by the evolution and is not a free
parameter.  In fact, in the region of the HERA data, the GRV
form mimics an $x^{-\lambda}$ behaviour with $\lambda \sim 0.4$,
although for smaller $x$ it is less steep.

   To summarize, we have discussed four different ways of generating
a steep $x$ behaviour of $F_2(x,Q^2)$ at small $x$, each with its own
characteristic $Q^2$ dependence: the BFKL fixed and running
$\alpha _s$ forms, (8) and (11), the Altarelli-Parisi double
leading logarithmic form with a long $Q^2$ evolution, (12),
and finally Altarelli-Parisi evolution from a steep
$x^{-\lambda}$ input, (15).  Examples of such forms are,
respectively, the fixed and running $\alpha _s$
AKMS predictions\cite{AKMS,AKMS2}, and the GRV\cite{GRV}
and MRS(H)\cite{MRSH} extrapolations.
Their $Q^2$ dependences are compared
with each other in Fig. 3
at given values of small $x$ in the HERA regime.  For reference the
MRS(D$_-^{\prime}$)\cite{MRSD} extrapolation is also shown.
The theoretical curves are calculated either from eq.(2) (where $f$
is the complete numerical solution of the BFKL equation
obtained as described in ref.\cite{AKMS2}) or from the
full next-to-leading order Altarelli-Parisi evolution.
We also show, in Fig. 3, H1\cite{H1Q} and ZEUS\cite{ZEUS}
measurements of $F_2$ made during the 1992 HERA run, corresponding
to an integrated luminosity of 25nb$^{-1}$.  Only the
statistical errors of the data are shown.
Measurements will be made with much higher luminosity, and
at smaller $x$ values, in the future.


  Several features
of this plot are noteworthy.   First, if we compare the data with
the ``$x^{-\lambda}$ dependences'' of the Altarelli-Parisi forms
of MRS(D$_-^{\prime}$), GRV and MRS(H) (which have respectively
$\lambda $ =0.5, ``$\approx $0.4'', and 0.3), then we see that
MRS(D$_-^{\prime }$) and GRV are disfavoured. So we are left with
MRS(H), which, in fact, was devised simply to reproduce\footnote
{See also the partons of the CTEQ collaboration which have
$\lambda $=0.27\cite{CTEQ2}.} the HERA data of refs.\cite
{H1,ZEUS}.

    Second, we see that the AKMS prediction (which
pre-dated the HERA data) is, like MRS(H), in good agreement
with the $x$ and $Q^2$ dependence of the data.  In principle,
it is an absolute perturbative QCD prediction of $F_2(x,Q^2)$
at small $x$ in terms of the known behaviour at larger $x$,
but, in practice, the overall normalization depends on the
treatment of the infrared region of the BFKL
equation\cite{AKMS,AKMS2}.  We can therefore normalise
the BFKL-based predictions so as to approximately describe the data at
$x=0.0027$ by adjusting a parameter which is introduced\cite{AKMS2}
in the description of the infrared region.  For the
running $\alpha _s$  AKMS calculation, this is achieved if
the infrared parameter
$k^2_a\approx 2{\rm GeV}^2$ (with $k^2_c=1{\rm GeV}^2$), in the
notation of ref.\cite{AKMS2}.  Strictly speaking, within the
genuine leading log$(1/x)$ approximation the coupling
$\alpha _s$ should be kept fixed\footnote {The use of
running $\alpha _s$ has the advantage that then the BFKL
equation reduces to the Altarelli-Parisi equation in the
double leading logarithm approximation when the transverse
momenta of the gluons become strongly ordered.}.
We therefore also solved the BFKL equation
with fixed $\alpha _s$, choosing a value $\alpha _s$=0.25 so
as to have a satisfactory normalization.  The resulting $Q^2$
dependence of $F_2(x,Q^2)$ turned out to be almost identical to that
calculated from the solution of the BFKL equation with
running $\alpha _s$.   For clarity, we therefore
have omitted the fixed $\alpha _s$ curve from Fig. 3.
Also a
background (or non-BFKL) contribution to $F_2$
has to be included in the
AKMS calculation\footnote {To be precise, we take
$F_2({\rm background})=F_2(x_0=0.1,Q^2)(x/x_0)^{-0.08}$
\cite{AKMS2}; a form which is motivated by
``soft'' Pomeron Regge behaviour\cite{DL}.  Other
reasonable choices of the background do not change our conclusions.}; this
explains why MRS(H), with $\lambda $=0.3, and AKMS, with
$\bar {\lambda} \approx 0.5$, both give equally good
descriptions of the HERA data. However, by the smallest $x$ value shown
we see that the BFKL-based AKMS predictions for $F_2$ begin to lie
significantly above those for MRS(H), due to this difference
in $\lambda $.

   A third feature of Fig. 3 is the stronger $Q^2$ dependence
of the AKMS predictions as compared with the MRS and GRV
extrapolations which are based on Altarelli-Parisi evolution.
This we had anticipated, with a growth approaching
$(Q^2)^{1\over 2}$ for BFKL as compared with the
approximately linear log$Q^2$
behaviour characteristic of Altarelli-Parisi evolution.
In reality, at the smallest $x$ value shown we find that the
AKMS growth is reduced to about $(Q^2)^{1\over 3}$, due to the fact
that $F_2({\rm background})$ is still significant.  Although
we see that the BFKL and Altarelli-Parisi $Q^2$ behaviours
are quite distinctive, to actually distinguish between them will
clearly be an experimental challenge, particularly since
 $Q^2\stackrel {<}{\sim}15{\rm GeV}^2$ is
the kinematic reach of HERA at the lowest $x$ value shown.
Recall that  the BFKL and Altarelli-Parisi equations effectively
resum the leading log$(1/x)$ and log$(Q^2)$ contributions respectively.
Thus the BFKL equation is appropriate in the small $x$ region
where
$\alpha _s{\rm log}(1/x)\sim 1$ yet
$\alpha _s{\rm log}(Q^2/Q_0^2)\ll 1$, where $Q_0^2$ is some
(sufficiently large) reference scale.  If the latter were
also $\sim 1$ then both log$(1/x)$ and log$(Q^2/Q_0^2)$
have to be treated on an equal footing\cite{GLR}, as is done, for
instance, in the unified equation proposed by Marchesini
et al.\cite{MARCH}.  For this reason we
restrict our study of small $x$ via the BFKL equation
to the region
$5\stackrel{<}{\sim}Q^2\stackrel{<}{\sim}50{\rm GeV}^2$.
As it happens, the very small $x$ HERA data lie well within this
limited $Q^2$ interval.

   So far we have neglected the effects of parton
shadowing.  If, as is conventionally expected, the gluons
are spread reasonably uniformly across the proton then
we anticipate that the effects will be small in the HERA
regime\cite{AKMS2}.  For illustration we have therefore
shown the effects of (speculative) ``hot-spot''
shadowing, corresponding to concentrations of gluons
in small hot-spots of transverse area
$\pi R^2$ inside the proton with, say, $R=2{\rm GeV}^{-1}$.
In this case, to normalise the predictions at $x$=0.0027,
we need to take the infrared parameter $k^2_a\approx 1.5{\rm GeV}^2$.
With decreasing $x$, we see from Fig. 3, that this shadowed
AKMS prediction increases more slowly than the
unshadowed one, but that it keeps the characteristic
``BFKL $Q^2$ curvature''.

     To conclude, we have performed a detailed analysis of
the $Q^2$ dependence of the structure function $F_2(x,Q^2)$
in the small $x$ region which is being probed at HERA.
We have found that the theoretically-motivated BFKL-based
predictions do indeed lead, in the HERA small $x$ regime,
to a more pronounced
curvature of $F_2(x,Q^2)$ than those based on next-to-leading
order Altarelli-Parisi evolution. The difference is illustrated
in Fig. 3 by the comparison of the AKMS curve with that for
MRS(H).  From the figure we see that data at the smallest
possible $x$ values will be the most revealing.  The
measurements shown are from the 1992 run, but data with
much higher luminosity, and at smaller $x$, will become
available in the near future.  Clearly the experimental
identification of the characteristic BFKL $Q^2$ behaviour will pose
a difficult, though hopefully not an impossible, task.

\vspace{1cm}

\noindent {\large\bf Acknowledgements}

We thank Dick Roberts and James Stirling for valuable discussions.
Two of us (JK, ADM) thank the European Community for
Fellowships and three of us (AJA, KGB, PJS) thank the Polish
KBN - British Council collaborative research programme for partial support.
This work has also been supported in part by Polish KBN grant
no. 2 0198 91 01 and by the UK Science and
Engineering Research Council.


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\bibitem{GLR} L.V. Gribov, E.M. Levin and M.G. Ryskin, Phys. Rep. {\bf 100}
(1983) 1
\bibitem{MARCH} G. Marchesini, Proc. of Workshop ``QCD at 200TeV'',
Erice, 1990, eds. L. Cifarelli and Yu.L. Dokshitzer, Plenum Press,
1992, p.183, and references therein

\end{thebibliography}


\newpage

\noindent{\Large \bf Figure Captions}
\begin{itemize}
\item[Fig.\ 1:]
 The measurements of $F_2(x,Q^2)$ at $Q^2 = 15$ and 30 GeV$^2$ by
the H1\cite{H1} and ZEUS\cite{ZEUS} collaborations shown
 by closed and open data points respectively, with the statistical
and systematic errors added in quadrature; the H1 and ZEUS data
have a global normalization uncertainty
of $\pm $8\% and $\pm $7\% respectively. The continuous,
dotted and dashed curves respectively correspond
to the values of $F_2$ obtained from MRS(H)\cite{MRSH},
GRV\cite{GRV} and MRS(D$_-^{\prime }$)\cite{MRSD} partons.
The curves that are shown as a sequence of small squares
(triangles) correspond to the unshadowed (strong
or ``hot-spot'' shadowing) AKMS
predictions obtained by computing
$F_2=f\otimes F_2^{(0)}+F_2({\rm background})$ as in
ref.\cite{AKMS2} and as described in the text.


\item[Fig.\ 2:]
Diagrammatic display of the $k_T$-factorization formula (2),
which is symbolically of the form $F_2=f\otimes F_2^{(0)}$, where $f$
denotes the gluon ladder and $F_2^{(0)}$ the quark box (and crossed box)
amplitude.

\item[Fig.\ 3:]
The $Q^2$ dependence of $F_2(x,Q^2)$ at small $x$ (note
the shifts of scale between
the plots at the different $x$ values, which have
been introduced for clarity). The curves are as in Fig. 1.
Also shown are the measurements of the 1992 HERA
run obtained by the ZEUS
collaboration\cite{ZEUS} (open points) and, by the H1
collaboration\cite{H1Q} using their ``electron''
analysis (closed points). Only statistical errors of the data are shown.
The ZEUS points shown on the $x$=0.00098 curves are measured at an
average $x$=0.00085.  A challenge for future experiments is to distinguish
between curves like AKMS and MRS(H), both of which give a satisfactory
description of the existing data.

\end{itemize}

\end{document}

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m21 614 1012 m21 631 1000 m21 649 988 m21 667 978 m21 685
 968 m21 702 956 m21 720 946
m21 738 936 m21 756 926 m21 774 916 m21 791 907 m21
 809 897 m21 827 888
m21 845 879 m21 863 871 m21 880 862 m21 898 854 m21 916 846
 m21 934 837 m21 951 829 m21 969 821 m21 987 814 m21 1005 806 m21 1023 800 m21
 1040 792 m21 1058 785 m21 1076 779 m21 1094 772 m21 1112 766 m21 1129 760 m21
 1147 753 m21 1165 746 m21 1183 741 m21 1200 735 m21 1218 729 m21 1236 722 m21
 1254 717 m21 1272 712 m21 1289 706 m21 1307 700 m21 1325 695 m21 1343 691 m21
 1360 686 m21 1378 681 m21 1396 676 m21 1414 671 m21 1432 667 m21 1449 662 m21
 1467 657 m21 1485 653 m21 1503 649 m21 1521 644 m21 1538 640 m21 1556 636 m21
 1574 632 m21 1592 628 m21 1609 624 m21 1627 620 m21 1645 616 m21 1663 612 m21
 1681 609 m21 1698 605 m21 1716 601 m21 1734 597 m21 1752 594 m21 1770 590 m21
 1787 587 m21 1805 583 m21 1823 580 m21 1841 577 m21 1858 573 m21 1876 570 m21
 1894 567 m21 1912 563 m21 1930 560 m21 1947 557 m21 1965 554 m21 1983 551 m21
 240 1091 m22 258 1084 m22 276 1074 m22 293 1065 m22 311 1055 m22 329 1049 m22
 347 1042 m22 365 1032 m22 382 1025 m22 400 1015 m22 418 1007 m22 436 1000 m22
 453 992 m22 471 984
m22 489 978 m22 507 971 m22 525 962 m22 542 955 m22 560 948
 m22 578 941 m22 596 934
m22 614 927 m22 631 920 m22 649 913 m22 667 907 m22 685
 900 m22 702 893 m22 720 887
m22 738 880 m22 756 874 m22 774 868 m22 791 861 m22
 809 855 m22 827 849 m22 845 843
m22 863 837 m22 880 831 m22 898 826 m22 916 820
 m22 934 813 m22 951 808 m22 969 802 m22 987 797 m22 1005 792 m22 1023 787 m22
 1040 782 m22 1058 776 m22 1076 771 m22 1094 766 m22 1112 761 m22 1129 757 m22
 1147 751 m22 1165 746 m22 1183 742 m22 1200 737 m22 1218 732 m22 1236 727 m22
 1254 723 m22 1272 719 m22 1289 713 m22 1307 708 m22 1325 705 m22 1343 700 m22
 1360 696 m22 1378 692 m22 1396 687 m22 1414 683 m22 1432 680 m22 1449 675 m22
 1467 671 m22 1485 667 m22 1503 663 m22 1521 659 m22 1538 655 m22 1556 651 m22
 1574 647 m22 1592 643 m22 1609 640 m22 1627 636 m22 1645 632 m22 1663 628 m22
 1681 625 m22 1698 621 m22 1716 617 m22 1734 613 m22 1752 610 m22 1770 606 m22
 1787 602 m22 1805 599 m22 1823 595 m22 1841 592 m22 1858 588 m22 1876 585 m22
 1894 581 m22 1912 577 m22 1930 574 m22 1947 570 m22 1965 567 m22 1983 564 m22
 /w 31 def /w2 {w 2 div} def /w3 {w 3 div} def 914 1096
m20 914 851 m 914 1080 l
 s 914 1111 m 914 1340 l s 1096 803 m20 1096 664 m 1096 788 l s 1096 819 m 1096
 943 l s 1236 727 m20 1236 622 m 1236 712 l s 1236 743 m 1236 832 l s 1445 676
 m20 1445 597 m 1445 661 l s 1445 692 m 1445 756 l s 1667 642 m20 1667 578 m
 1667 626 l s 1667 657 m 1667 705 l s 1890 546 m20 1890 499 m 1890 531 l s 1890
 562 m 1890 594 l s s s 777 921
m24 777 749 m 777 905 l s 777 937 m 777 1092 l s
 1049 832 m24 1049 759 m 1049 816 l s 1049 848 m 1049 905 l s 1316 689 m24 1316
 645 m 1316 674 l s 1316 705
m 1316 734 l s 1725 556 m24 1725 489 m 1725 540 l s
 1725 571 m 1725 622 l s 3 lw 259 1206 m 308 1171 l 323 1161 l 378 1124 l 426
 1093 l 468 1066 l 507 1043 l 542 1023 l 574 1004 l 603 988 l 631 973 l 657 960
 l 681 947 l 703 936 l 725 925 l 745 915
l 764 906 l 783 897 l 800 889 l 817 881
 l 833 874 l 848 867 l 863 861
l 877 854 l 891 848 l 905 843 l 917 837 l 930 832
 l 942 827 l 954 822 l 965 818 l 976 813 l 987 809 l 997 805 l 1007 801 l 1017
 797 l 1027 794 l 1036 790 l 1045 786 l 1054 783 l 1063 780 l 1072 777 l 1080
 774 l 1088 771 l 1096 768 l 1104 765 l 1112 762 l 1119 760 l 1127 757 l 1134
 754 l 1141 752 l s 1141 752 m 1148 750 l 1155 747 l 1162 745 l 1169 743 l 1175
 740 l 1182 738 l 1188 736 l 1194 734 l 1200 732 l 1206 730 l 1212 728 l 1218
 726 l 1224 725 l 1229 723 l 1235 721 l 1241 719 l 1246 717 l 1251 716 l 1257
 714 l 1262 713 l 1267 711 l 1272 709 l 1277 708 l 1282 706 l 1287 705 l 1292
 703 l 1297 702 l 1301 701 l 1306 699 l 1310 698 l 1315 696 l 1320 695 l 1324
 694 l 1328 693 l 1333 691 l 1337 690 l 1341 689 l 1345 688 l 1350 686 l 1354
 685 l 1358 684 l 1362 683 l 1366 682 l 1370 681 l 1374 680 l 1377 679 l 1381
 678 l 1385 677 l 1389 676 l s 1389 676 m 1392 675 l 1396 673 l 1400 673 l 1403
 672 l 1407 671 l 1411 670 l 1414 669 l 1418 668 l 1421 667 l 1424 666 l 1428
 665 l 1431 664 l 1434 663 l 1438 662 l 1441 662 l 1444 661 l 1447 660 l 1451
 659 l 1454 658 l 1457 657 l 1460 657 l 1463 656 l 1466 655 l 1469 654 l 1472
 653 l 1475 653 l 1478 652 l 1481 651 l 1484 650 l 1487 650 l 1490 649 l 1493
 648 l 1496 648 l 1498 647 l 1501 646 l 1504 646 l 1507 645 l 1509 644 l 1512
 644 l 1515 643 l 1518 642 l 1520 642 l 1523 641 l 1526 640 l 1528 640 l 1531
 639 l 1533 638 l 1536 638 l 1538 637 l s 1538 637 m 1541 637 l 1543 636 l 1546
 635 l 1548 635 l 1551 634 l 1553 634 l 1556 633 l 1558 633 l 1561 632 l 1563
 631 l 1565 631 l 1568 630 l 1570 630 l 1572 629 l 1575 629 l 1577 628 l 1579
 628 l 1581 627 l 1584 627 l 1586 626 l 1588 626 l 1590 625 l 1593 625 l 1595
 624 l 1597 624 l 1599 623 l 1601 623 l 1603 622 l 1606 622 l 1608 621 l 1610
 621 l 1612 620 l 1614 620 l 1616 619 l 1618 619 l 1620 619 l 1622 618 l 1624
 618 l 1626 617 l 1628 617 l 1630 616 l 1632 616 l 1634 615 l 1636 615 l 1638
 615 l 1640 614 l 1642 614 l 1644 613 l 1646 613 l s 1646 613 m 1648 613 l 1650
 612 l 1652 612 l 1654 611 l 1655 611 l 1657 611 l 1659 610 l 1661 610 l 1663
 609 l 1665 609 l 1666 609 l 1668 608 l 1670 608 l 1672 607 l 1674 607 l 1675
 607 l 1677 606 l 1679 606 l 1681 606 l 1683 605 l 1684 605 l 1686 605 l 1688
 604 l 1689 604 l 1691 604 l 1693 603 l 1695 603 l 1696 602 l 1698 602 l 1700
 602 l 1701 601 l 1703 601 l 1705 601 l 1706 600 l 1708 600 l 1709 600 l 1711
 599 l 1713 599 l 1714 599 l 1716 599 l 1717 598 l 1719 598 l 1721 598 l 1722
 597 l 1724 597 l 1725 597 l 1727 596 l 1728 596 l 1730 596 l s 1730 596 m 1732
 595 l 1733 595 l 1735 595 l 1736 595 l 1738 594 l 1739 594 l 1741 594 l 1742
 593 l 1744 593 l 1745 593 l 1747 593 l 1748 592 l 1750 592 l 1751 592 l 1752
 591 l 1754 591 l 1755 591 l 1757 591 l 1758 590 l 1760 590 l 1761 590 l 1763
 589 l 1764 589 l 1765 589 l 1767 589 l 1768 588 l 1770 588 l 1771 588 l 1772
 588 l 1774 587 l 1775 587 l 1776 587 l 1778 587 l 1779 586 l 1781 586 l 1782
 586 l 1783 586 l 1785 585 l 1786 585 l 1787 585 l 1789 585 l 1790 584 l 1791
 584 l 1792 584 l 1794 584 l 1795 583 l 1796 583 l 1798 583 l 1799 583 l s 1799
 583 m 1800 582 l 1802 582 l 1803 582 l 1804 582 l 1805 582 l 1807 581 l 1808
 581 l 1809 581 l 1810 581 l 1812 580 l 1813 580 l 1814 580 l 1815 580 l 1817
 580 l 1818 579 l 1819 579 l 1820 579 l 1822 579 l 1823 578 l 1824 578 l 1825
 578 l 1826 578 l 1828 578 l 1829 577 l 1830 577 l 1831 577 l 1832 577 l 1833
 577 l 1835 576 l 1836 576 l 1837 576 l 1838 576 l 1839 575 l 1841 575 l 1842
 575 l 1843 575 l 1844 575 l 1845 574 l 1846 574 l 1847 574 l 1849 574 l 1850
 574 l 1851 574 l 1852 573 l 1853 573 l 1854 573 l 1855 573 l 1856 573 l 1857
 572 l s 1857 572 m 1859 572 l 1860 572 l 1861 572 l 1862 572 l 1863 571 l 1864
 571 l 1865 571 l 1866 571 l 1867 571 l 1868 571 l 1869 570 l 1871 570 l 1872
 570 l 1873 570 l 1874 570 l 1875 569 l 1876 569 l 1877 569 l 1878 569 l 1879
 569 l 1880 569 l 1881 568 l 1882 568 l 1883 568 l 1884 568 l 1885 568 l 1886
 568 l 1887 567 l 1888 567 l 1889 567 l 1890 567 l 1891 567 l 1892 567 l 1893
 566 l 1894 566 l 1895 566 l 1896 566 l 1897 566 l 1898 566 l 1899 565 l 1900
 565 l 1901 565 l 1902 565 l 1903 565 l 1904 565 l 1905 564 l 1906 564 l 1907
 564 l 1908 564 l s 1908 564 m 1909 564 l 1910 564 l 1911 563 l 1912 563 l 1913
 563 l 1914 563 l 1915 563 l 1916 563 l 1917 563 l 1918 562 l 1919 562 l 1920
 562 l 1921 562 l 1922 562 l 1923 562 l 1924 561 l 1925 561 l 1926 561 l 1927
 561 l 1928 561 l 1929 561 l 1930 560 l 1931 560 l 1932 560 l 1933 560 l 1934
 560 l 1935 560 l 1936 559 l 1937 559 l 1938 559 l 1939 559 l 1940 559 l 1941
 559 l 1942 559 l 1943 559 l 1944 558 l 1945 558 l 1946 558 l 1947 558 l 1948
 558 l 1949 558 l 1950 557 l 1951 557 l 1952 557 l 1953 557 l s 1953 557 m 1954
 557 l 1955 557 l 1956 556 l 1957 556 l 1958 556 l 1959 556 l 1960 556 l 1961
 556 l 1962 556 l 1963 555 l 1964 555 l 1965 555 l 1966 555 l 1967 555 l 1968
 555 l 1969 554 l 1970 554 l 1971 554 l 1972 554 l 1973 554 l 1974 554 l 1975
 554 l 1976 553 l 1977 553 l 1978 553 l 1979 553 l 1980 553 l 1981 553 l 1982
 553 l 1983 552 l 1984 552 l 1985 552 l 1986 552 l 1987 552 l 1988 552 l 1989
 552 l 1989 551 l 1990 551 l 1991 551 l 1992 551 l 1993 551 l s 1993 551 m 1994
 551 l 1995 551 l 1996 551 l 1996 550 l 1997 550 l 1998 550 l 1999 550 l 2000
 550 l s [12 12] 0 sd 354 1461 m 359 1454 l 359 1454
m 378 1430 l 415 1383 l 426
 1369 l 468 1319 l 507 1275 l 542 1237 l 574 1204 l 603 1175 l 631 1148 l 657
 1124 l 681 1103 l 703 1083 l 725 1064 l 745 1048 l 764 1032 l 783 1017 l 800
 1004 l 817 991 l 833 979 l 848 968 l 863 958 l 877 948 l 891 938 l 905 929 l
 917 921 l 930 913 l 942 905 l 954 897
l 965 890 l 976 884 l 987 877 l 997 871 l
 1007 865 l 1017 859 l 1027 854 l 1036 848 l 1045 843 l 1054 838 l 1063 834 l
 1072 829 l 1080 824 l 1088 820 l 1096 816 l 1104 812 l 1112 808 l 1119 804 l
 1127 800 l 1134 797 l 1141 793 l 1148 790 l s 1148 790 m 1155 787 l 1162 783 l
 1169 780 l 1175 777 l 1182 774 l 1188 771 l 1194 768 l 1200 766 l 1206 763 l
 1212 760 l 1218 758 l 1224 755 l 1229 753 l 1235 750 l 1241 748 l 1246 746 l
 1251 744 l 1257 741 l 1262 739 l 1267 737 l 1272 735 l 1277 733 l 1282 731 l
 1287 729 l 1292 727 l 1297 725 l 1301 723 l 1306 722 l 1310 720 l 1315 718 l
 1320 716 l 1324 715 l 1328 713 l 1333 711 l 1337 710 l 1341 708 l 1345 707 l
 1350 705 l 1354 704 l 1358 702 l 1362 701 l 1366 699 l 1370 698 l 1374 697 l
 1377 695 l 1381 694 l 1385 693 l 1389 691 l 1392 690 l s 1392 690 m 1396 689 l
 1400 687 l 1403 686 l 1407 685 l 1411 684 l 1414 683 l 1418 681 l 1421 680 l
 1424 679 l 1428 678 l 1431 677 l 1434 676 l 1438 675 l 1441 674 l 1444 673 l
 1447 672 l 1451 671 l 1454 670 l 1457 669 l 1460 668 l 1463 667 l 1466 666 l
 1469 665 l 1472 664 l 1475 663 l 1478 662 l 1481 661 l 1484 660 l 1487 659 l
 1490 659 l 1493 658 l 1496 657 l 1498 656 l 1501 655 l 1504 654 l 1507 654 l
 1509 653 l 1512 652 l 1515 651 l 1518 650 l 1520 650 l 1523 649 l 1526 648 l
 1528 647 l 1531 647 l 1533 646 l 1536 645 l 1538 644 l 1541 644 l s 1541 644 m
 1543 643 l 1546 642 l 1548 642 l 1551 641 l 1553 640 l 1556 640 l 1558 639 l
 1561 638 l 1563 638 l 1565 637 l 1568 636 l 1570 636 l 1572 635 l 1575 634 l
 1577 634 l 1579 633 l 1581 633 l 1584 632 l 1586 631 l 1588 631 l 1590 630 l
 1593 630 l 1595 629 l 1597 629 l 1599 628 l 1601 627 l 1603 627 l 1606 626 l
 1608 626 l 1610 625 l 1612 625 l 1614 624 l 1616 624 l 1618 623 l 1620 623 l
 1622 622 l 1624 622 l 1626 621 l 1628 621 l 1630 620 l 1632 620 l 1634 619 l
 1636 619 l 1638 618 l 1640 618 l 1642 617 l 1644 617 l 1646 616 l 1648 616 l s
 1648 616 m 1650 615 l 1652 615 l 1654 614 l 1655 614 l 1657 613 l 1659 613 l
 1661 613 l 1663 612 l 1665 612 l 1666 611 l 1668 611 l 1670 610 l 1672 610 l
 1674 610 l 1675 609 l 1677 609 l 1679 608 l 1681 608 l 1683 608 l 1684 607 l
 1686 607 l 1688 606 l 1689 606 l 1691 606 l 1693 605 l 1695 605 l 1696 604 l
 1698 604 l 1700 604 l 1701 603 l 1703 603 l 1705 602 l 1706 602 l 1708 602 l
 1709 601 l 1711 601 l 1713 601 l 1714 600 l 1716 600 l 1717 600 l 1719 599 l
 1721 599 l 1722 599 l 1724 598 l 1725 598 l 1727 598 l 1728 597 l 1730 597 l
 1732 597 l s 1732 597 m 1733 596 l 1735 596 l 1736 596 l 1738 595 l 1739 595 l
 1741 595 l 1742 594 l 1744 594 l 1745 594 l 1747 593 l 1748 593 l 1750 593 l
 1751 592 l 1752 592 l 1754 592 l 1755 591 l 1757 591 l 1758 591 l 1760 591 l
 1761 590 l 1763 590 l 1764 590 l 1765 589 l 1767 589 l 1768 589 l 1770 589 l
 1771 588 l 1772 588 l 1774 588 l 1775 587 l 1776 587 l 1778 587 l 1779 587 l
 1781 586 l 1782 586 l 1783 586 l 1785 585 l 1786 585 l 1787 585 l 1789 585 l
 1790 584 l 1791 584 l 1792 584 l 1794 584 l 1795 583 l 1796 583 l 1798 583 l
 1799 583 l 1800 582 l s 1800 582 m 1802 582 l 1803 582 l 1804 582 l 1805 581 l
 1807 581 l 1808 581 l 1809 581 l 1810 580 l 1812 580 l 1813 580 l 1814 580 l
 1815 579 l 1817 579 l 1818 579 l 1819 579 l 1820 579 l 1822 578 l 1823 578 l
 1824 578 l 1825 578 l 1826 577 l 1828 577 l 1829 577 l 1830 577 l 1831 576 l
 1832 576 l 1833 576 l 1835 576 l 1836 576 l 1837 575 l 1838 575 l 1839 575 l
 1841 575 l 1842 574 l 1843 574 l 1844 574 l 1845 574 l 1846 574 l 1847 573 l
 1849 573 l 1850 573 l 1851 573 l 1852 573 l 1853 572 l 1854 572 l 1855 572 l
 1856 572 l 1857 572 l 1859 571 l s 1859 571 m 1860 571 l 1861 571 l 1862 571 l
 1863 571 l 1864 570 l 1865 570 l 1866 570 l 1867 570 l 1868 570 l 1869 569 l
 1871 569 l 1872 569 l 1873 569 l 1874 569 l 1875 569 l 1876 568 l 1877 568 l
 1878 568 l 1879 568 l 1880 568 l 1881 567 l 1882 567 l 1883 567 l 1884 567 l
 1885 567 l 1886 567 l 1887 566 l 1888 566 l 1889 566 l 1890 566 l 1891 566 l
 1892 565 l 1893 565 l 1894 565 l 1895 565 l 1896 565 l 1897 565 l 1898 564 l
 1899 564 l 1900 564 l 1901 564 l 1902 564 l 1903 564 l 1904 563 l 1905 563 l
 1906 563 l 1907 563 l 1908 563 l 1909 563 l s 1909 563 m 1910 562 l 1911 562 l
 1912 562 l 1913 562 l 1914 562 l 1915 562 l 1916 562 l 1917 561 l 1918 561 l
 1919 561 l 1920 561 l 1921 561 l 1922 561 l 1923 560 l 1924 560 l 1925 560 l
 1926 560 l 1927 560 l 1928 560 l 1929 559 l 1930 559 l 1931 559 l 1932 559 l
 1933 559 l 1934 559 l 1935 558 l 1936 558 l 1937 558 l 1938 558 l 1939 558 l
 1940 558 l 1941 557 l 1942 557 l 1943 557 l 1944 557 l 1945 557 l 1946 557 l
 1947 556 l 1948 556 l 1949 556 l 1950 556 l 1951 556 l 1952 556 l 1953 556 l
 1954 555 l s 1954 555 m 1955 555 l 1956 555 l 1957 555 l 1958 555 l 1959 555 l
 1960 554 l 1961 554 l 1962 554 l 1963 554 l 1964 554 l 1965 554 l 1966 554 l
 1967 553 l 1968 553 l 1969 553 l 1970 553 l 1971 553 l 1972 553 l 1973 553 l
 1973 552 l 1974 552 l 1975 552 l 1976 552 l 1977 552 l 1978 552 l 1979 552 l
 1980 551 l 1981 551 l 1982 551 l 1983 551 l 1984 551 l 1985 551 l 1986 551 l
 1986 550 l 1987 550 l 1988 550 l 1989 550 l 1990 550 l 1991 550 l 1992 550 l
 1993 550 l 1993 549 l 1994 549 l s 1994 549 m 1995 549 l 1996 549 l 1997 549 l
 1998 549 l 1999 549 l 2000 549 l 2000 548 l s [4 8] 0 sd 259 1348 m 311 1303 l
 323 1294 l 378 1249 l 426 1212 l 468 1180 l 507 1152 l 542 1127 l 574 1105 l
 603 1085 l 631 1067 l 657 1050
l 681 1035 l 703 1021 l 725 1008 l 745 996 l 764
 984 l 783 974 l 800 964
l 817 954 l 833 945 l 848 937 l 863 929 l 877 921 l 891
 913 l 905 906 l 917 900
l 930 893 l 942 887 l 954 881 l 965 876 l 976 870 l 987
 865 l 997 860 l 1007 855
l 1017 850 l 1027 845 l 1036 841 l 1045 837 l 1054 833
 l 1063 828 l 1072 825 l 1080 821 l 1088 817 l 1096 813 l 1104 810 l 1112 807 l
 1119 803 l 1127 800 l 1134 797 l 1141 794 l s 1141 794 m 1148 791 l 1155 788 l
 1162 785 l 1169 782 l 1175 779 l 1182 777 l 1188 774 l 1194 772 l 1200 769 l
 1206 767 l 1212 764 l 1218 762 l 1224 760 l 1229 757 l 1235 755 l 1241 753 l
 1246 751 l 1251 749 l 1257 747 l 1262 745 l 1267 743 l 1272 741 l 1277 739 l
 1282 737 l 1287 735 l 1292 733 l 1297 731 l 1301 730 l 1306 728 l 1310 726 l
 1315 725 l 1320 723 l 1324 721 l 1328 720 l 1333 718 l 1337 717 l 1341 715 l
 1345 714 l 1350 712 l 1354 711 l 1358 709 l 1362 708 l 1366 706 l 1370 705 l
 1374 704 l 1377 702 l 1381 701 l 1385 700 l 1389 699 l s 1389 699 m 1392 697 l
 1396 696 l 1400 695 l 1403 694 l 1407 692 l 1411 691 l 1414 690 l 1418 689 l
 1421 688 l 1424 687 l 1428 685 l 1431 684 l 1434 683 l 1438 682 l 1441 681 l
 1444 680 l 1447 679 l 1451 678 l 1454 677 l 1457 676 l 1460 675 l 1463 674 l
 1466 673 l 1469 672 l 1472 671 l 1475 670 l 1478 669 l 1481 668 l 1484 667 l
 1487 667 l 1490 666 l 1493 665 l 1496 664 l 1498 663 l 1501 662 l 1504 661 l
 1507 660 l 1509 660 l 1512 659 l 1515 658 l 1518 657 l 1520 656 l 1523 656 l
 1526 655 l 1528 654 l 1531 653 l 1533 652 l 1536 652 l 1538 651 l s 1538 651 m
 1541 650 l 1543 649 l 1546 649 l 1548 648 l 1551 647 l 1553 647 l 1556 646 l
 1558 645 l 1561 645 l 1563 644 l 1565 643 l 1568 642 l 1570 642 l 1572 641 l
 1575 641 l 1577 640 l 1579 639 l 1581 639 l 1584 638 l 1586 637 l 1588 637 l
 1590 636 l 1593 635 l 1595 635 l 1597 634 l 1599 634 l 1601 633 l 1603 632 l
 1606 632 l 1608 631 l 1610 631 l 1612 630 l 1614 630 l 1616 629 l 1618 628 l
 1620 628 l 1622 627 l 1624 627 l 1626 626 l 1628 626 l 1630 625 l 1632 625 l
 1634 624 l 1636 624 l 1638 623 l 1640 623 l 1642 622 l 1644 622 l 1646 621 l s
 1646 621 m 1648 621 l 1650 620 l 1652 620 l 1654 619 l 1655 619 l 1657 618 l
 1659 618 l 1661 617 l 1663 617 l 1665 616 l 1666 616 l 1668 615 l 1670 615 l
 1672 614 l 1674 614 l 1675 613 l 1677 613 l 1679 613 l 1681 612 l 1683 612 l
 1684 611 l 1686 611 l 1688 610 l 1689 610 l 1691 610 l 1693 609 l 1695 609 l
 1696 608 l 1698 608 l 1700 607 l 1701 607 l 1703 607 l 1705 606 l 1706 606 l
 1708 605 l 1709 605 l 1711 605 l 1713 604 l 1714 604 l 1716 603 l 1717 603 l
 1719 603 l 1721 602 l 1722 602 l 1724 602 l 1725 601 l 1727 601 l 1728 600 l
 1730 600 l s 1730 600 m 1732 600 l 1733 599 l 1735 599 l 1736 599 l 1738 598 l
 1739 598 l 1741 598 l 1742 597 l 1744 597 l 1745 596 l 1747 596 l 1748 596 l
 1750 595 l 1751 595 l 1752 595 l 1754 594 l 1755 594 l 1757 594 l 1758 593 l
 1760 593 l 1761 593 l 1763 592 l 1764 592 l 1765 592 l 1767 591 l 1768 591 l
 1770 591 l 1771 591 l 1772 590 l 1774 590 l 1775 590 l 1776 589 l 1778 589 l
 1779 589 l 1781 588 l 1782 588 l 1783 588 l 1785 587 l 1786 587 l 1787 587 l
 1789 587 l 1790 586 l 1791 586 l 1792 586 l 1794 585 l 1795 585 l 1796 585 l
 1798 585 l 1799 584 l s 1799 584 m 1800 584 l 1802 584 l 1803 583 l 1804 583 l
 1805 583 l 1807 583 l 1808 582 l 1809 582 l 1810 582 l 1812 582 l 1813 581 l
 1814 581 l 1815 581 l 1817 580 l 1818 580 l 1819 580 l 1820 580 l 1822 579 l
 1823 579 l 1824 579 l 1825 579 l 1826 578 l 1828 578 l 1829 578 l 1830 578 l
 1831 577 l 1832 577 l 1833 577 l 1835 577 l 1836 576 l 1837 576 l 1838 576 l
 1839 576 l 1841 575 l 1842 575 l 1843 575 l 1844 575 l 1845 574 l 1846 574 l
 1847 574 l 1849 574 l 1850 574 l 1851 573 l 1852 573 l 1853 573 l 1854 573 l
 1855 572 l 1856 572 l 1857 572 l s 1857 572 m 1859 572 l 1860 571 l 1861 571 l
 1862 571 l 1863 571 l 1864 571 l 1865 570 l 1866 570 l 1867 570 l 1868 570 l
 1869 569 l 1871 569 l 1872 569 l 1873 569 l 1874 569 l 1875 568 l 1876 568 l
 1877 568 l 1878 568 l 1879 568 l 1880 567 l 1881 567 l 1882 567 l 1883 567 l
 1884 567 l 1885 566 l 1886 566 l 1887 566 l 1888 566 l 1889 566 l 1890 565 l
 1891 565 l 1892 565 l 1893 565 l 1894 565 l 1895 564 l 1896 564 l 1897 564 l
 1898 564 l 1899 564 l 1900 563 l 1901 563 l 1902 563 l 1903 563 l 1904 563 l
 1905 562 l 1906 562 l 1907 562 l 1908 562 l s 1908 562 m 1909 562 l 1910 562 l
 1911 561 l 1912 561 l 1913 561 l 1914 561 l 1915 561 l 1916 560 l 1917 560 l
 1918 560 l 1919 560 l 1920 560 l 1921 560 l 1922 559 l 1923 559 l 1924 559 l
 1925 559 l 1926 559 l 1927 558 l 1928 558 l 1929 558 l 1930 558 l 1931 558 l
 1932 558 l 1933 557 l 1934 557 l 1935 557 l 1936 557 l 1937 557 l 1938 556 l
 1939 556 l 1940 556 l 1941 556 l 1942 556 l 1943 556 l 1944 555 l 1945 555 l
 1946 555 l 1947 555 l 1948 555 l 1949 554 l 1950 554 l 1951 554 l 1952 554 l
 1953 554 l s 1953 554 m 1954 553 l 1955 553 l 1956 553 l 1957 553 l 1958 553 l
 1959 553 l 1960 552 l 1961 552 l 1962 552 l 1963 552 l 1964 552 l 1965 552 l
 1966 551 l 1967 551 l 1968 551 l 1969 551 l 1970 551 l 1971 551 l 1972 550 l
 1973 550 l 1974 550 l 1975 550 l 1976 550 l 1977 550 l 1977 549 l 1978 549 l
 1979 549 l 1980 549 l 1981 549 l 1982 549 l 1983 548 l 1984 548 l 1985 548 l
 1986 548 l 1987 548 l 1988 548 l 1989 547 l 1990 547 l 1991 547 l 1992 547 l
 1993 547 l s 1993 547 m 1994 547 l 1995 546 l 1996 546 l 1997 546 l 1998 546 l
 1999 546 l 2000 546 l s [] 0 sd 1 lw 1977 206 m 1995 184 l s 1979 208 m 1997
 186 l s 1995 206 m 1977 184 l s 1997 208 m 1979 186 l s 43 1284 m 77 1284 l s
 45 1286 m 79 1286 l s 43 1284 m 43 1304 l s 45 1286 m 45 1306 l s 59 1284 m 59
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%%Trailer
%%Pages: 1
gr gr
%%EOF
%%End of figure 1


Here is Figure 2



%! FEYNMAN DRAW
% % A program by David J. Summers to draw
% Feynman diagrams. (c) 1992
% Version 2 (c) 1993
% % Comments and questions to
% D.J.Summers@uk.ac.durham
%
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/Fphotonl {exch dup 3 1 roll 0 ge {{ Fxc Fth cos Frr mul Fxt Fi 180
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%%End of figure 2


Here is Figure 3


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(6)
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(7)
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(8)
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(8)
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(10)
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(20)
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(50)
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 694
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 972 1577
l 997 1586 l 1022 1595 l 1045 1604 l 1068 1612 l 1089 1620 l 1110 1628
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 1788 l 1568 1791 l 1577 1795 l 1587 1798 l 1596 1801 l 1605 1805 l 1614 1808 l
 1623 1811 l 1632 1814 l 1641 1817 l 1650 1820 l 1658 1823 l 1666 1826 l 1675
 1829 l 1683 1832 l 1691 1835 l 1699 1837 l 1706 1840 l 1714 1843 l 1722 1845 l
 1729 1848 l 1737 1851 l 1744 1853 l 1751 1856 l 1759 1858 l 1766 1861 l 1773
 1863 l 1780 1866 l 1786 1868 l 1793 1870 l 1800 1873 l 1807 1875 l 1813 1877 l
 1820 1879 l 1826 1882 l 1832 1884 l 1839 1886 l 1845 1888 l 1851 1890 l 1857
 1892 l 1863 1895 l 1869 1897 l 1875 1899 l 1881 1901 l 1887 1903 l 1892 1905 l
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 1358 l 589 1376 l 635 1393 l 679 1409 l 719 1424 l 757 1438 l 793 1451 l 827
 1464 l 858 1475 l 889 1486 l 918 1497 l 945 1507 l 972 1516 l 997 1525 l 1022
 1534 l 1045 1542 l 1068 1551 l 1089 1558 l 1110 1566 l 1131 1573 l 1150 1580 l
 1169 1587 l 1188 1593 l 1206 1600 l 1223 1606 l 1240 1612 l 1256 1618 l 1272
 1623 l 1288 1629 l 1303 1634 l 1318 1640 l 1333 1645 l 1347 1650 l 1361 1654 l
 1374 1659 l 1387 1664 l 1400 1668 l 1413 1673 l 1425 1677 l 1438 1681 l 1450
 1686 l 1461 1690 l 1473 1694 l 1484 1698 l 1495 1702 l 1506 1705 l 1517 1709 l
 1527 1713 l s 1527 1713 m 1538 1716 l 1548 1720 l 1558 1723 l 1568 1727 l 1577
 1730 l 1587 1733 l 1596 1737 l 1605 1740 l 1614 1743 l 1623 1746 l 1632 1749 l
 1641 1752 l 1650 1755 l 1658 1758 l 1666 1761 l 1675 1764 l 1683 1767 l 1691
 1769 l 1699 1772 l 1706 1775 l 1714 1777 l 1722 1780 l 1729 1783 l 1737 1785 l
 1744 1788 l 1751 1790 l 1759 1793 l 1766 1795 l 1773 1797 l 1780 1800 l 1786
 1802 l 1793 1804 l 1800 1807 l 1807 1809 l 1813 1811 l 1820 1813 l 1826 1816 l
 1832 1818 l 1839 1820 l 1845 1822 l 1851 1824 l 1857 1826 l 1863 1828 l 1869
 1830 l 1875 1832 l 1881 1834 l 1887 1836 l 1892 1838 l 1898 1840 l s 1898 1840
 m 1904 1842 l s [] 0 sd 423 963 m 483 974 l 538 985 l 589 994 l 635 1004 l 679
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 1058 l 945 1064 l 972 1069 l 997 1074 l 1022 1079 l 1045 1083 l 1068 1088 l
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 1114 l 1223 1118 l 1240 1121 l 1256 1124 l 1272 1127 l 1288 1131 l 1303 1133 l
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 1179 l 1548 1181 l 1558 1183 l 1568 1185 l 1577 1187 l 1587 1189 l 1596 1191 l
 1605 1192 l 1614 1194 l 1623 1196 l 1632 1198 l 1641 1199 l 1650 1201 l 1658
 1203 l 1666 1204 l 1675 1206 l 1683 1207 l 1691 1209 l 1699 1210 l 1706 1212 l
 1714 1213 l 1722 1215 l 1729 1216 l 1737 1218 l 1744 1219 l 1751 1221 l 1759
 1222 l 1766 1223 l 1773 1225 l 1780 1226 l 1786 1228 l 1793 1229 l 1800 1230 l
 1807 1231 l 1813 1233 l 1820 1234 l 1826 1235 l 1832 1236 l 1839 1238 l 1845
 1239 l 1851 1240 l 1857 1241 l 1863 1242 l 1869 1244 l 1875 1245 l 1881 1246 l
 1887 1247 l 1892 1248 l 1898 1249 l s 1898 1249 m 1904 1250 l s [12 12] 0 sd
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 1065 l 793 1073 l 827 1082 l 858 1089 l 889 1097 l 918 1104 l 945 1110 l 972
 1117 l 997 1123 l 1022 1128 l 1045 1134 l 1068 1139 l 1089 1144 l 1110 1149 l
 1131 1154 l 1150 1159 l 1169 1163 l 1188 1167 l 1206 1172 l 1223 1176 l 1240
 1180 l 1256 1183 l 1272 1187 l 1288 1191 l 1303 1194 l 1318 1198 l 1333 1201 l
 1347 1204 l 1361 1207 l 1374 1211 l 1387 1214 l 1400 1217 l 1413 1220 l 1425
 1222 l 1438 1225 l 1450 1228 l 1461 1231 l 1473 1233 l 1484 1236 l 1495 1238 l
 1506 1241 l 1517 1243 l 1527 1246 l s 1527 1246 m 1538 1248 l 1548 1250 l 1558
 1252 l 1568 1255 l 1577 1257 l 1587 1259 l 1596 1261 l 1605 1263 l 1614 1265 l
 1623 1267 l 1632 1269 l 1641 1271 l 1650 1273 l 1658 1275 l 1666 1277 l 1675
 1279 l 1683 1280 l 1691 1282 l 1699 1284 l 1706 1286 l 1714 1288 l 1722 1289 l
 1729 1291 l 1737 1293 l 1744 1294 l 1751 1296 l 1759 1297 l 1766 1299 l 1773
 1301 l 1780 1302 l 1786 1304 l 1793 1305 l 1800 1307 l 1807 1308 l 1813 1309 l
 1820 1311 l 1826 1312 l 1832 1314 l 1839 1315 l 1845 1316 l 1851 1318 l 1857
 1319 l 1863 1320 l 1869 1322 l 1875 1323 l 1881 1324 l 1887 1326 l 1892 1327 l
 1898 1328
l s 1898 1328 m 1904 1329 l s [4 8] 0 sd 423 971 m 483 987 l 538 1001
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 858 1082 l 889 1090 l 918 1097 l 945 1104 l 972 1110 l 997 1117 l 1022 1123 l
 1045 1128 l 1068 1134 l 1089 1139 l 1110 1144 l 1131 1149 l 1150 1154 l 1169
 1158 l 1188 1163 l 1206 1167 l 1223 1171 l 1240 1175 l 1256 1179 l 1272 1183 l
 1288 1187 l 1303 1190 l 1318 1194 l 1333 1197 l 1347 1201 l 1361 1204 l 1374
 1207 l 1387 1210 l 1400 1213 l 1413 1216 l 1425 1219 l 1438 1222 l 1450 1225 l
 1461 1228 l 1473 1230 l 1484 1233 l 1495 1236 l 1506 1238 l 1517 1241 l 1527
 1243 l s 1527 1243 m 1538 1245 l 1548 1248 l 1558 1250 l 1568 1252 l 1577 1255
 l 1587 1257 l 1596 1259 l 1605 1261 l 1614 1263 l 1623 1265 l 1632 1267 l 1641
 1269 l 1650 1271 l 1658 1273 l 1666 1275 l 1675 1277 l 1683 1279 l 1691 1281 l
 1699 1282 l 1706 1284 l 1714 1286 l 1722 1288 l 1729 1289 l 1737 1291 l 1744
 1293 l 1751 1294 l 1759 1296 l 1766 1298 l 1773 1299 l 1780 1301 l 1786 1302 l
 1793 1304 l 1800 1305 l 1807 1307 l 1813 1308 l 1820 1310 l 1826 1311 l 1832
 1313 l 1839 1314 l 1845 1315 l 1851 1317 l 1857 1318 l 1863 1319 l 1869 1321 l
 1875 1322 l 1881 1323 l 1887 1325 l 1892 1326 l 1898 1327 l s 1898 1327 m 1904
 1329 l s [] 0 sd 423 643 m 483 651 l 538 658 l 589 664 l 635 670 l 679 675 l
 719 681
l 757 685 l 793 690 l 827 694 l 858 698 l 889 702 l 918 705 l 945 709 l
 972 712
l 997 715 l 1022 718 l 1045 721 l 1068 724 l 1089 727 l 1110 729 l 1131
 732 l 1150 734 l 1169 737 l 1188 739 l 1206 741 l 1223 743 l 1240 745 l 1256
 747 l 1272 749 l 1288 751 l 1303 753 l 1318 755 l 1333 757 l 1347 758 l 1361
 760 l 1374 762 l 1387 763 l 1400 765 l 1413 766 l 1425 768 l 1438 769 l 1450
 771 l 1461 772 l 1473 774 l 1484 775 l 1495 776 l 1506 778 l 1517 779 l 1527
 780 l s 1527 780 m 1538 781 l 1548 783 l 1558 784 l 1568 785 l 1577 786 l 1587
 787 l 1596 788 l 1605 789 l 1614 791 l 1623 792 l 1632 793 l 1641 794 l 1650
 795 l 1658 796 l 1666 797 l 1675 798 l 1683 799 l 1691 800 l 1699 801 l 1706
 802 l 1714 802 l 1722 803 l 1729 804 l 1737 805 l 1744 806 l 1751 807 l 1759
 808 l 1766 809 l 1773 809 l 1780 810 l 1786 811 l 1793 812 l 1800 813 l 1807
 813 l 1813 814 l 1820 815 l 1826 816 l 1832 816 l 1839 817 l 1845 818 l 1851
 819 l 1857 819 l 1863 820 l 1869 821 l 1875 821 l 1881 822 l 1887 823 l 1892
 824 l 1898 824
l s 1898 824 m 1904 825 l s [12 12] 0 sd 423 639 m 483 648 l 538
 656 l 589 664
l 635 671 l 679 677 l 719 683 l 757 688 l 793 693 l 827 698 l 858
 703 l 889 707 l 918 711 l 945 715 l 972 719 l 997 722 l 1022 726 l 1045 729 l
 1068 732 l 1089 735 l 1110 738 l 1131 741 l 1150 743 l 1169 746 l 1188 748 l
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 1495 790 l 1506 791 l 1517 792 l 1527 794 l s 1527 794 m 1538 795 l 1548 796 l
 1558 798 l 1568 799 l 1577 800 l 1587 801 l 1596 803 l 1605 804 l 1614 805 l
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 1683 814 l 1691 815 l 1699 816 l 1706 817 l 1714 818 l 1722 819 l 1729 820 l
 1737 821 l 1744 822 l 1751 823 l 1759 823 l 1766 824 l 1773 825 l 1780 826 l
 1786 827 l 1793 828 l 1800 829 l 1807 830 l 1813 830 l 1820 831 l 1826 832 l
 1832 833 l 1839 834 l 1845 834 l 1851 835 l 1857 836 l 1863 837 l 1869 837 l
 1875 838 l 1881 839 l 1887 840 l 1892 840 l 1898 841 l s 1898 841 m 1904 842 l
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 687 l 757 693
l 793 698 l 827 703 l 858 708 l 889 713 l 918 717 l 945 721 l 972
 725 l 997 729
l 1022 732 l 1045 736 l 1068 739 l 1089 742 l 1110 745 l 1131 748
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 1461 795 l 1473 797 l 1484 798 l 1495 800 l 1506 801 l 1517 803 l 1527 804 l s
 1527 804 m 1538 805 l 1548 807 l 1558 808 l 1568 809 l 1577 811 l 1587 812 l
 1596 813 l 1605 815 l 1614 816 l 1623 817 l 1632 818 l 1641 819 l 1650 820 l
 1658 822 l 1666 823 l 1675 824 l 1683 825 l 1691 826 l 1699 827 l 1706 828 l
 1714 829 l 1722 830 l 1729 831 l 1737 832 l 1744 833 l 1751 834 l 1759 835 l
 1766 836 l 1773 837 l 1780 838 l 1786 838 l 1793 839 l 1800 840 l 1807 841 l
 1813 842 l 1820 843 l 1826 844 l 1832 844 l 1839 845 l 1845 846 l 1851 847 l
 1857 848 l 1863 848 l 1869 849 l 1875 850 l 1881 851 l 1887 851 l 1892 852 l
 1898 853 l s 1898 853 m 1904 854 l s [] 0 sd 1 lw 1810 208 m 1807 207 l 1803
 203 l 1801 200 l 1800 195 l 1800 186 l 1801 181 l 1803 178 l 1807 174 l 1810
 173 l 1817 173 l 1820 174 l 1824 178 l 1825 181 l 1827 186 l 1827 195 l 1825
 200 l 1824 203 l 1820 207 l 1817 208 l 1810 208 l cl s 1812 210 m 1809 209 l
 1805 205 l 1804 202 l 1802 197 l 1802 188 l 1804 183 l 1805 180 l 1809 176 l
 1812 175 l 1819 175 l 1822 176 l 1826 180 l 1827 183 l 1829 188 l 1829 197 l
 1827 202
l 1826 205 l 1822 209 l 1819 210 l 1812 210 l cl s 1815 179 m 1825 169
 l s 1817 182 m 1827 171 l s 1835 213 m 1835 214 l 1836 215 l 1837 216 l 1839
 217 l 1842 217 l 1844 216 l 1845 215 l 1846 214 l 1846 212 l 1845 210 l 1843
 208 l 1835 199 l 1846 199 l s 1837 215 m 1837 216 l 1838 218 l 1839 218 l 1841
 219 l 1844 219 l 1846 218 l 1847 218 l 1848 216 l 1848 214 l 1847 212 l 1845
 210 l 1837 201 l 1849 201 l s 1883 215 m 1852 161 l s 1885 217 m 1854 163 l s
 1917 200 m 1915 203 l 1912 207 l 1908 208 l 1901 208 l 1898 207 l 1895 203 l
 1893 200 l 1891 195 l 1891 186 l 1893 181 l 1895 178 l 1898 174 l 1901 173 l
 1908 173 l 1912 174 l 1915 178 l 1917 181 l 1917 186 l s 1919 202 m 1917 205 l
 1914 209 l 1910 210 l 1904 210 l 1900 209 l 1897 205 l 1895 202 l 1893 197 l
 1893 188 l 1895 183 l 1897 180 l 1900 176 l 1904 175 l 1910 175 l 1914 176 l
 1917 180 l 1919 183 l 1919 188 l s 1908 186 m 1917 186 l s 1910 188 m 1919 188
 l s 1927 186 m 1947 186 l 1947 190 l 1945 193 l 1944 195 l 1940 196 l 1935 196
 l 1932 195 l 1929 191 l 1927 186 l 1927 183 l 1929 178 l 1932 174 l 1935 173 l
 1940 173 l 1944 174 l 1947 178 l s 1929 188 m 1949 188 l 1949 192 l 1948 195 l
 1946 197 l 1943 198 l 1937 198 l 1934 197 l 1931 193 l 1929 188 l 1929 185 l
 1931 180 l 1934 176 l 1937 175 l 1943 175 l 1946 176 l 1949 180 l s 1954 208 m
 1967 173 l s 1956 210 m 1970 175 l s 1981 208 m 1967 173 l s 1983 210 m 1970
 175 l s 1986 213 m 1986 214 l 1987 215 l 1988 216 l 1989 217 l 1993 217 l 1995
 216 l 1995 215 l 1996 214 l 1996 212 l 1995 210 l 1994 208 l 1985 199 l 1997
 199 l s 1988 215 m 1988 216 l 1989 218 l 1990 218 l 1992 219 l 1995 219 l 1997
 218 l 1998 218 l 1998 216 l 1998 214 l 1998 212 l 1996 210 l 1987 201 l 1999
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 2608 m 43 2630 l s 58 2606 m 58 2619 l s 60 2608 m 60 2621 l s 72 2633 m 71
 2633 l 70 2634
l 69 2634 l 68 2636 l 68 2639 l 69 2641 l 70 2642 l 71 2643 l 73
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 71 2637
l 70 2638 l 70 2642 l 71 2643 l 72 2644 l 73 2645 l 75 2645 l 77 2644 l
 79 2642
l 88 2634 l 88 2646 l s 34 2665 m 38 2661 l 43 2658 l 50 2655 l 58 2653
 l 65 2653 l 73 2655 l 80 2658 l 85 2661 l 89 2665 l s 37 2667 m 40 2664 l 45
 2660 l 52 2657 l 60 2655 l 67 2655 l 75 2657 l 82 2660 l 87 2664 l 91 2667 l s
 53 2675 m 77 2694 l s 55 2677 m 79 2696 l s 53 2694 m 77 2675 l s 55 2696 m 79
 2677 l s 75 2709 m 77 2707 l 75 2706 l 73 2707 l 75 2709 l 78 2709 l 82 2707 l
 84 2706
l s 77 2711 m 79 2709 l 77 2708 l 75 2709 l 77 2711 l 81 2711 l 84 2709
 l 86 2708 l s 41 2731 m 43 2728 l 46 2724 l 50 2722 l 55 2721 l 63 2721 l 68
 2722
l 72 2724 l 75 2728 l 77 2731 l 77 2738 l 75 2741 l 72 2744 l 68 2746 l 63
 2748
l 55 2748 l 50 2746 l 46 2744 l 43 2741 l 41 2738 l 41 2731 l cl s 43 2733
 m 45 2730
l 48 2726 l 52 2725 l 57 2723 l 65 2723 l 70 2725 l 74 2726 l 77 2730
 l 79 2733
l 79 2740 l 77 2743 l 74 2747 l 70 2748 l 65 2750 l 57 2750 l 52 2748
 l 48 2747 l 45 2743 l 43 2740 l 43 2733 l cl s 70 2736 m 80 2746 l s 72 2738 m
 82 2748
l s 37 2756 m 36 2756 l 34 2757 l 33 2758 l 32 2760 l 32 2763 l 33 2765
 l 34 2766 l 36 2766 l 37 2766 l 39 2766 l 42 2764 l 50 2755 l 50 2767 l s 39
 2758 m 38 2758
l 36 2759 l 35 2760 l 34 2762 l 34 2765 l 35 2767 l 36 2768 l 38
 2769 l 39 2769 l 41 2768 l 44 2766 l 52 2758 l 52 2769 l s 34 2775 m 38 2778 l
 43 2782
l 50 2785 l 58 2787 l 65 2787 l 73 2785 l 80 2782 l 85 2778 l 89 2775 l
 s 37 2777
m 40 2780 l 45 2784 l 52 2787 l 60 2789 l 67 2789 l 75 2787 l 82 2784
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%%Trailer
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%%EOF
%%End of figure 3

%% end of file

