%Paper: hep-th/9409082
%From: Chi-Sing LAM <lam@inca.physics.mcgill.ca>
%Date: Wed, 14 Sep 1994 15:49:39 -0400 (EDT)

%%%%This is a Latex file in the ichep style. The file ichep.sty
%%%%is appended--search for `ichep.sty' to get it.
%%%%
%%%%The five figures are contained in five  postscript files
%%%%(obtained by searching for  "postscript files"). These
%%%%postscript files have to extracted and printed separately.
%%%%----------------------------------------------------------
\documentstyle{ichep}
\hoffset 0mm
\voffset 23mm
% Use suitable offset values to centre the output on your own system.
\def\.{\!\cdot\!}
\def\tr{{\rm tr}}
\def\p{\partial}
\def\bk#1{\langle#1\rangle}


\begin{document}

\title{String rearrangement of gauge theories}

\author{C.S. Lam \dag }

\affil{Department of Physics, McGill University\\
3600 University St., Montreal, P.Q., Canada H3A 2T8}

%\collab{333}

\abstract{Feynman diagram expressions in ordinary field theories can
be written in a string-like manner. The methods and the advantages for
doing so are briefly discussed.}

\twocolumn[\maketitle]

\fnm{1}{E-mail: lam@physics.mcgill.ca}


\section{Introduction}
There are three reasons why one wants to arrange ordinary Feynman diagrams
in a string-like manner: it simplifies calculations and it gives new insights
into gauge and gravitational theories. Moreover, this is done (graphically) all
the times so we might as well find out exactly what that means. For example,
The tree diagrams for the process
$\pi^+K^0\to \pi^0K^+$
are given by Figs.~1(a) and (b), but  one often
 shows only the quark diagram Fig.~1(c), which can  be considered also as
a string diagram with hadronic strings strung between  $q\bar q$ pairs.
String diagrams are also
used for pure QCD in the large-$N_c$ limit as shown in
Fig.~2.


\begin{figure}
\vspace*{11.5cm}
\caption{Lowest-order diagrams for $\pi^+K^0\to \pi^0K^+$.}
\end{figure}


\begin{figure}
\vspace*{9.5cm}
\caption{A pure-gluon QCD process.}
\end{figure}




Is it possible then to make the string-like diagrams {\it quantitative}
 by writing down for them a set of `Feynman rules'? When one tries to
do that a number of problems is encoutered.
For ordinary Feynman diagrams,
($i$) loop momenta $k_a$ have to be introduced and integrated over;
($ii$) interactions occur at the vertices though particles
propagate freely between them. In particular, loop momenta $k_a$
injected at the vertices can
change the direction of
the combined momentum flow. Similarly, flavour, colour, and spin are altered
at the vertices; ($iii$) gauge invariance
determines the vertex factor
for gauge interactions, for example to be
$\epsilon(p)\cdot(q'+q'')$ in scalar QED, where $p$ is
the momentum of the external photon  and $q',q''$ are the charged
particle momenta; ($iv$)
a sum of many Feynman diagrams is needed to
describle a process in a given order. In contrast, for string diagrams,
($i'$) if we consider only hadronic {\it ground} states so as to freeze
string excitations,  the propagation of a string is described
only by the `proper time' parameter $\tau$, so that scattering amplitudes
are expressed as integrations over
the proper-time parameters $\tau_m$;
($ii'$) the geometrical shape of a string diagram is arbitrary
(reparametrization and conformal invariance), so
vertices cannot be present and
strings propagate freely throughout the diagram. In particular, external
momenta can merge and divide inside a diagram but there are no
additional loop momenta
present to alter their directions. Flavour and colour flow smoothly along the
quark lines
as given respectively by Fig.~1(c)
and Fig.~2(b), and similarly for spin as we shall see later;
($iii'$) conformal invariance
determines external gauge-interaction vertices, for example to be
$\epsilon(p)\cdot[\partial_\tau x(\tau)]\exp[ip\cdot x(\tau)]$ for scalar
QED
where $x^\mu(\tau)$ is the spacetime (operator) coordiantes of the string
at the proper time $\tau$. The vertex factor for this operator is then
$\epsilon(p)\cdot[\partial_\tau(-i\partial/\partial p)]$,
quite different from the corresponding field-theoretic vertex in ($iii$);
($iv'$) (Veneziano) duality [1] is valid. The sum of many Feynman diagrams
is replaced by one or few string diagrams, as in Fig.~1.

In the rest of this note, we will show how to rearrange
($i$)--($iv$) to resemble ($i'$)--($iv'$). The calculational and
conceptual advantages for doing so will also be briefly discussed.

\section{Momentum, flavour, colour, and spin flows}
To convert ($i$) to ($i'$), it is sufficient to introduce a Schwinger
parameter $\alpha_r$ for the denominator of every propagator:
\begin{equation}
(-q_r^2+m_r^2-i\epsilon)^{-1}=i\int_0^\infty d\alpha_r\exp[-i\alpha_r
(m_r^2-q_r^2)]\ .
\end{equation}
Substituting this into the general expression for a T-matrix
amplitude
\begin{equation}
T(p)=\int\left(\prod_ad^4k_a\right)S_0(q,p)\prod_r(-q_r^2+m_r^2-i\epsilon)^{-1}
\ ,
\end{equation}
the loop momentum integrations can be explicitly carried out to yield [2]
\begin{eqnarray}
T(p)\sim&&\int_0^\infty\left(\prod_rd\alpha_r\right)\Delta^{-2}(\alpha)
S(q,p)\nonumber\\
&&\prod_r\exp[-i\alpha_r(m_r^2-q_r^2)]\ .\\
\nonumber
\end{eqnarray}
This is the string-like form where momentum flow
is described by $q_r$, a quantity best thought of as
the current flowing through
the $r$th line of an electric circuit
given by the Feynman diagram, in which
the branch resistances are $\alpha_s$ and the external currents
are $p_i$. Explicit rules are available
to calculate these currents and other quantities in $T(p)$
directly from the Feynman diagram. If a proper time $\tau$ is assigned
to each vertex, and if line $r=(ij)$ connects vertices $j$ to $i$, then
$\alpha_r$ can also be interpreted as the proper-time difference
$|\tau_i-\tau_j|$.

The factor $S_0(q,p)$ in (2) contains all the vertex factors and
numerators of propagators, so it encodes
the flows of flavour, colour, and spin and is given by a sum of products
of these factors. The quantity $S(q,p)$ in (3) is equal to
$S_0+S_1+S_2+\cdots$, and can be obtained from $S_0$ through momentum
contraction [2]. The factors for flavour, colour, and spin flows can all
be read off directly from the appropriate quark, or string-like, diagrams,
and it is important to note that the quark diagrams are {\it different for
different flows}. The general rules and their derivations are given elsewhere
[3], but specific examples can be seen from Figs.~3 and 4 respectively for
colour and spin flows. In each case diagram (a) is the Feynman diagram
and diagram (b) is the equivalent string-like diagram from which
the factors for $S_0$ can be read off to be

\begin{figure}
\vspace*{9cm}
\caption{An example to illustrate colour flow. See eq.~(4).}
\end{figure}

\begin{figure}
\vspace*{10cm}
\caption{An example to illustrate spin flow. See eq.~(5).}
\end{figure}


\begin{eqnarray}
S_0^{(m)colour}=&&T^aT^bT^cT^d \tr(T^eT^fT^gT^h)\ ,\\
S_0^{spin}=&&[p_2q_1]\bk{q_1p_3}\.[p_4p_5]\.
\bk{k_5q_9}[q_9q_3]\bk{q_3p_1}\. \nonumber\\
&&\bk{p_6q_6}[q_6q_5]\bk{q_5q_8}[q_8k_6]\ .\\
\nonumber
\end{eqnarray}
$T^a$ in (5) are the $(S)U(N)$ generators in the fundamental representation.
The superscript $(m)$ indicates that there are many colour
factors for Fig.~3(a), and that in (4) is the one appropriate to
Fig.~3(b). Other colour factors can be obtained from
other string-like diagrams, which in the case of $U(N)$ can be obtained from
3(b) by crossing
the quark lines at the vertices. For $SU(N)$ there are other quark diagrams
in which some of the internal gluon lines are omitted.
The spin-flow (or more correctly helicity-flow) factor $S_0^{spin}$ in (5) is
obtained by assuming
the fermion masses to be zero. In that case the string-like diagram 4(b)
is unique once 4(a) is given, and the direction the fermion lines turn depends
on the helicities of the external particles, indicated in the diagram by a
$+$ or a $-$ sign. More diagrams will be necessary if the internal
particles are massive. The square and angular brackets are the overlap of
the massless Dirac wave functions with definite helicities:
\begin{equation}
[p_ip_j]=\bar u_+(p_i)u_-(p_j),\quad \bk{p_ip_j}=\bar u_-(p_i)u_+(p_j)\ ,
\end{equation}
When internal momenta $q_r$ appear in these brackets, it is understood that
they should first be expanded in terms of the external massless momenta
$p_i$ with only diagonal terms kept. Note that Dirac matrices are completely
absent so the usual four-channel problem is reduced to a one-channel
expression. This is possible because of helicity and chirality conservations
for massless fermions. Note that it would have been impossible to write
(5) before the loop momenta were eliminated in going from
(2) to (3).



The final factor $S_0$ is obtained from
$S_0^{(m)colour}S_0^{spin}\cdots$ by summing over all $m$. The main
advantage in this string-like arrangement is to be able to use the spinor
helicity technique [4], developed originally for {\it tree} diagrams, now  for
arbitrary processes with any number of loops. It also allows
the gauge-invariant colour subamplitudes to be easily separated.

\section{External gauge vertices}
The scalar QED vertex ($iii$) can be replaced by the string-like vertex
($iii'$) by using differential circuit identities [3]. The presence of
$\p/\p\tau$ in the latter expression allows integration-by-parts
to be used, to redistribute the gauge-dependent terms among
different diagrams in order to minimize their appearance and thereby increase
computability. It also makes the Ward-Takahashi identity realized
in a different way, and allows the possibility of formulating gauge
invariance in another way.

\section{Duality}
Veneziano duality ($iv'$) can be simulated by formally summing up a number
of Feynman diagrams into a single integral expression, at least for
QED-like theories [5]. Take for
example the sum of the $6!4!3!$ QED diagrams,
obtained from Fig.~5 by permuting the photon vertices along each of the
three charged lines.



\begin{figure}
\vspace*{5 cm}
\caption{A QED diagram.}
\end{figure}

In the proper-time representation (3),
a proper time [$\tau'_i(1\le i\le 6),\ \tau''_j(1\le j\le 4)$, \
$\tau'''_k(1\le k\le 3)$] is assigned to each vertex, and each Feynman
diagram is given by a fixed ordering of the $\tau'$s, $\tau''$s,
and $\tau'''$s, so for each diagram the proper-time integration region in (3)
is the product of three {\it hyper-triangular regions}. The `dual {\it sum}' of
these $6!4!3!$
diagrams is obtained by summing all these permutations, and is thus
given by a {\it single} integral over the product of three
{\it hypercube} regions, one each for $\tau',\tau''$,  and $\tau'''$.
Moreover, it is gauge invariant. Unfortunately, it is often impossible
to carry out this single integral analytically, but it can be used as
a starting point for gauge-invariant approximations. For example, the
result for the soft-photon eikonal approximation can be obtained very easily
in this way.

It is also worthwhile pointing out an interesting parallel. Each Feynman
diagram with $n$ vertices is a sum of $n!$ time-ordered (old-fashioned)
diagrams. An individual time-ordered diagram is not Lorentz invariant,
but the Feynman diagram is. In our case, the dual sum puts together
Feynman diagrams that have different {\it proper-time} orderings, each
of which is not gauge invariant, but the dual sum is.




\section{References}
\def\i#1{\item{[#1]}}
\def\npb#1{{\it Nucl.~Phys. }{\bf B#1}}
\def\plb#1{{\it Phys.~Lett. }{\bf #1B}}
\def\prl#1{{\it Phys.~Rev.~Lett. }{\bf B#1}}
\def\prd#1{{\it Phys.~Rev. }{\bf D#1}}
\def\pr#1{{\it Phys.~Rep. }{\bf #1}}
\def\ibid#1{{\it ibid.} {\bf #1}}

%\Bibliography
\begin{thebibliography}{9}
\bibitem{1} G. Veneziano, {\it Nuovo Cimento} {\bf 57A} (1968), 190.
\bibitem{2} C.S. Lam and J.P. Lebrun, {\it Nuovo Cimento} {\bf 59A} (1969),
397.
\bibitem{3} C.S. Lam,  \npb{397} (1993), 143; \prd{48} (1993), 873;
{\it Can. J. Phys. } July-August (1994).
\bibitem{4} M.L. Mangano and S.J. Parke, {\it Phys. Rep.}  200 (1991), 301;
R. Gastmans and T.T. Wu, `The Ubiquitous Photon',
International Series of Monographs on
Physics, Vol.~80 (Clarendon Press, Oxford, 1990).
\bibitem{5} Y.J. Feng and C.S. Lam, to be published.

\end{thebibliography}


\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%5 postscript figures%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%fig1.ps
%!PS-Adobe-2.0
%%%%%%BoundingBox: 0 0 468 360

%   DRAWING COMMANDS

/arrowdict 14 dict def
arrowdict begin
 /mtrx matrix def
end

/arrow
 { arrowdict begin
    /headlength exch def
    /halfheadthickness exch 2 div def
    /halfthickness exch 2 div def
    /tipy exch def /tipx exch def
    /taily exch def /tailx exch def
    /dx tipx tailx sub def
    /dy tipy taily sub def
    /arrowlength dx dx mul dy dy mul add
      sqrt def
    /angle dy dx atan def
    /base arrowlength headlength sub def
    /savematrix mtrx currentmatrix def
    tailx taily translate
    angle rotate
    0 halfthickness neg moveto
    base halfthickness neg lineto
    base halfheadthickness neg lineto
    arrowlength 0 lineto
    base halfheadthickness lineto
    base halfthickness lineto
    0 halfthickness lineto
    closepath
    savematrix setmatrix
    end
  }def

/cir{ stroke  0 0 3 -1 roll 0 360 arc stroke} def
/cirf{ stroke 0 0 3 -1 roll 0 360 arc  fill stroke} def
/cp {currentpoint} def
/l {lineto} def
/m{moveto}def
/r {rotate} def
/rl{rlineto} def
/rm {rmoveto} def
/so {currentpoint translate stroke} def
/o {0 0 moveto} def
/t {translate} def
% ---------------------------------------------
%
%     FONT COMMANDS

%  The number `s' must be filled in by hand and /fh replaced by e.g., /fh1

%/fh {/Helvetica findfont s scalefont setfont moveto show} def
%/fhb {/Helvetica-Bold findfont s scalefont setfont moveto show} def
%/fhbo {/Helvetica-BoldOblique findfont s scalefont setfont moveto show} def
%/fho {/Helvetica-Oblique findfont s scalefont setfont moveto show} def
%/ftb {/Times-Bold  findfont s scalefont setfont moveto show} def
%/ftbi {/Times-BoldItalic findfont s scalefont setfont moveto show} def
%/fti {/Times-Italic findfont s scalefont setfont moveto show} def
%/ftr {/Times-Roman findfont s scalefont  setfont moveto show} def
%/fsym {/Symbol findfont s scalefont setfont moveto show} def

%-------------------------------------------------


/cb{/Courier-BoldOblique
 findfont 20 scalefont setfont moveto show} def
/cbs{/Courier-BoldOblique
 findfont 15 scalefont setfont moveto show} def
/cbb{/Courier-BoldOblique
 findfont 30 scalefont setfont moveto show} def

/fb {/Times-Roman findfont 15 scalefont setfont moveto    show
 } def
/fbb {/Times-Roman findfont 18 scalefont setfont moveto    show
 } def
/fs {/Times-Roman findfont 10 scalefont setfont moveto    show
 } def
/gkb{/Symbol findfont 30 scalefont setfont moveto  show} def
/itb{/Times-Italic findfont 20 scalefont setfont moveto show} def
/itm{/Times-Italic findfont 15 scalefont setfont moveto show} def
/its{/Times-Italic findfont 10 scalefont setfont moveto show} def

%----------------------------------------------------
% ADDITIONAL DEFINITIONS

%thick fermion line of length 50 from cp; cp moved to right end of line
/fer{3 setlinewidth  50 0 rlineto cp t stroke 1 setlinewidth
0 0 m} def

/rtl{3 setlinewidth rlineto cp t stroke 1 setlinewidth 0 0 m} def

%arrowhead with base at 0 0 pointing to the left; cp moved to base
/aro{0 0 -10 0 2 10 10 arrow 0 0 t fill stroke 0 0  m} def

%thin arrowhead pointing left at origin
/aros{0 0 -5 0 2 5 10 arrow  0 0 t fill stroke o} def

%dashed photon line from cp to rel coord (x,y); cp moved to end of line.
%usage: x y pho
/pho{[5 5]0 setdash rlineto cp t stroke o []0 setdash} def

%dot at current point
/dot{cp t  4 cirf o} def

%dotted photon lines up and down
/phoup{90 r [5 5]0 setdash 50 0 rlineto
 -50 0 rm cp t stroke o []0 setdash     -90 r} def

/phodown{-90 r [5 5]0 setdash 50 0 rlineto
 -50 0 rm cp t stroke o []0 setdash     90 r} def
%---------------
%definition for Fig. 6 only
/dot1{0 0 3 0 360 arc closepath gsave fill grestore} def
/ca{ newpath 50 0 10 0 360 arc -90 r stroke (A) -5 45 fb 90 r
 0 0 m} def
/cross{-40 0 m 40 0 lineto 0 40 m 0 5 lineto  0 -5 m 0 -40 lineto stroke
0 0 5 90 -90 arcn stroke 0 0 m} def
/fer1{3 setlinewidth -40 0 m 40 0 lineto stroke 1 setlinewidth
stroke
-20 0 -30 0 2 10 10 arrow fill 30 0 20 0 2 10 10 arrow fill
0 0 m} def
/gh{-40 0 m 40 0 lineto [7] 0 setdash stroke []0 setdash
-20 0 -30 0 .5 4 5 arrow fill 30 0 20 0 .5 4 5 arrow fill
0 0 m} def
/quad{round 180 r round 180 r} def
/round{40 0 m 10 0 lineto 0 0 0 10 5 arct  0 40 lineto stroke
0 0 m} def
/triple{
 0 0 m 40 0 lineto stroke
120 r
0 0 m 40 0 lineto stroke
120 r
10 0 t stroke dot1 0 0 m 30 0 lineto
stroke -10 0 t 0 0 m 120 r} def
%--------------------------------------


.35 .35 scale
300 1000 t
o
fer aro fer 45 r fer aro fer -100 0 rm
-90 r fer aro fer -100 0 rm 45 r
-100 0 rm -45 r -100 0 rm fer aro fer
90 r -100 0 rm fer aro fer -45 r
((a)) 250 0 cbb
(\160) 180 80 gkb (\053) 200 100 gkb
(\113) 180 -80 gkb (\060) 200 -60 gkb
(\160) -120 80 gkb (\060) -100 100 gkb
(\113) -120 -80 gkb (\053) -100 -60 gkb
(\113) 40 20 gkb (\052\053) 60 40 gkb

50 -300 t
o
90 r fer aro fer -45 r fer aro fer -100 0 rm
-90 r -100 0 rm fer aro fer -225 r -100 0 rm
-45 r -100 0 rm  fer aro fer  -90 r
fer aro fer -100 0 rm 45 r
((b)) 130 100 cbb

(\160) 20 220 gkb (\053) 40 240 gkb
(\113) 20 0 gkb (\060) 40 20 gkb
(\160) -180 220 gkb (\060) -160 240 gkb
(\113) -180 0 gkb (\053) -160 20 gkb
(\162) -50 110 gkb (\053) -30 130 gkb


-60 -250 t
o
3 setlinewidth
/ql{stroke 0 141.41 100 -45 -135 arcn  -79.29 79.29 rl
 300 0 rm  -79.29 -79.29 rl  cp t stroke o  -70.71 -70.71 translate
0 41.42 t o aro 0 -41.42 t o} def

 ql 90 r 0 15 t ql 90 r 0 15 t ql 90 r 0 15 t ql 90 r

((c)) 180 15 cbb
stroke
showpage




%%EOF




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%fig2.ps%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%!PS-Adobe-2.0


%   DRAWING COMMANDS

/arrowdict 14 dict def
arrowdict begin
 /mtrx matrix def
end

/arrow
 { arrowdict begin
    /headlength exch def
    /halfheadthickness exch 2 div def
    /halfthickness exch 2 div def
    /tipy exch def /tipx exch def
    /taily exch def /tailx exch def
    /dx tipx tailx sub def
    /dy tipy taily sub def
    /arrowlength dx dx mul dy dy mul add
      sqrt def
    /angle dy dx atan def
    /base arrowlength headlength sub def
    /savematrix mtrx currentmatrix def
    tailx taily translate
    angle rotate
    0 halfthickness neg moveto
    base halfthickness neg lineto
    base halfheadthickness neg lineto
    arrowlength 0 lineto
    base halfheadthickness lineto
    base halfthickness lineto
    0 halfthickness lineto
    closepath
    savematrix setmatrix
    end
  }def

/cir{ stroke  0 0 3 -1 roll 0 360 arc stroke} def
/cirf{ stroke 0 0 3 -1 roll 0 360 arc  fill stroke} def
/cp {currentpoint} def
/l {lineto} def
/m{moveto}def
/r {rotate} def
/rl{rlineto} def
/rm {rmoveto} def
/so {currentpoint translate stroke} def
/o {0 0 moveto} def
/t {translate} def
% ---------------------------------------------
%
%     FONT COMMANDS

%  The number `s' must be filled in by hand and /fh replaced by e.g., /fh1

%/fh {/Helvetica findfont s scalefont setfont moveto show} def
%/fhb {/Helvetica-Bold findfont s scalefont setfont moveto show} def
%/fhbo {/Helvetica-BoldOblique findfont s scalefont setfont moveto show} def
%/fho {/Helvetica-Oblique findfont s scalefont setfont moveto show} def
%/ftb {/Times-Bold  findfont s scalefont setfont moveto show} def
%/ftbi {/Times-BoldItalic findfont s scalefont setfont moveto show} def
%/fti {/Times-Italic findfont s scalefont setfont moveto show} def
%/ftr {/Times-Roman findfont s scalefont  setfont moveto show} def
%/fsym {/Symbol findfont s scalefont setfont moveto show} def

%-------------------------------------------------


/cb{/Courier-BoldOblique
 findfont 20 scalefont setfont moveto show} def
/cbs{/Courier-BoldOblique
 findfont 15 scalefont setfont moveto show} def
/cbb{/Courier-BoldOblique
 findfont 30 scalefont setfont moveto show} def

/fb {/Times-Roman findfont 15 scalefont setfont moveto    show
 } def
/fbb {/Times-Roman findfont 18 scalefont setfont moveto    show
 } def
/fs {/Times-Roman findfont 10 scalefont setfont moveto    show
 } def
/gkb{/Symbol findfont 30 scalefont setfont moveto  show} def
/itb{/Times-Italic findfont 20 scalefont setfont moveto show} def
/itm{/Times-Italic findfont 15 scalefont setfont moveto show} def
/its{/Times-Italic findfont 10 scalefont setfont moveto show} def

%----------------------------------------------------
% ADDITIONAL DEFINITIONS

%thick fermion line of length 50 from cp; cp moved to right end of line
/fer{3 setlinewidth  50 0 rlineto cp t stroke 1 setlinewidth
0 0 m} def

/rtl{3 setlinewidth rlineto cp t stroke 1 setlinewidth 0 0 m} def

%arrowhead with base at 0 0 pointing to the left; cp moved to base
/aro{0 0 -10 0 2 10 10 arrow 0 0 t fill stroke 0 0  m} def

%thin arrowhead pointing left at origin
/aros{0 0 -5 0 2 5 10 arrow  0 0 t fill stroke o} def

%dashed photon line from cp to rel coord (x,y); cp moved to end of line.
%usage: x y pho
/pho{[5 5]0 setdash rlineto cp t stroke o []0 setdash} def

%dot at current point
/dot{cp t  4 cirf o} def

%dotted photon lines up and down
/phoup{90 r [5 5]0 setdash 50 0 rlineto
 -50 0 rm cp t stroke o []0 setdash     -90 r} def

/phodown{-90 r [5 5]0 setdash 50 0 rlineto
 -50 0 rm cp t stroke o []0 setdash     90 r} def
%---------------
%definition for Fig. 6 only
/dot1{0 0 3 0 360 arc closepath gsave fill grestore} def
/ca{ newpath 50 0 10 0 360 arc -90 r stroke (A) -5 45 fb 90 r
 0 0 m} def
/cross{-40 0 m 40 0 lineto 0 40 m 0 5 lineto  0 -5 m 0 -40 lineto stroke
0 0 5 90 -90 arcn stroke 0 0 m} def
/fer1{3 setlinewidth -40 0 m 40 0 lineto stroke 1 setlinewidth
stroke
-20 0 -30 0 2 10 10 arrow fill 30 0 20 0 2 10 10 arrow fill
0 0 m} def
/gh{-40 0 m 40 0 lineto [7] 0 setdash stroke []0 setdash
-20 0 -30 0 .5 4 5 arrow fill 30 0 20 0 .5 4 5 arrow fill
0 0 m} def
/quad{round 180 r round 180 r} def
/round{40 0 m 10 0 lineto 0 0 0 10 5 arct  0 40 lineto stroke
0 0 m} def
/triple{
 0 0 m 40 0 lineto stroke
120 r
0 0 m 40 0 lineto stroke
120 r
10 0 t stroke dot1 0 0 m 30 0 lineto
stroke -10 0 t 0 0 m 120 r} def
%--------------------------------------



.35 .35 scale
300 1000 t
o
fer fer 45 r fer  fer -100 0 rm -135 r fer fer
45 r fer  fer -100 0 rm -135 r fer fer 225 r -100 0 rm fer  fer
45 r fer fer -135 r -100 0 rm fer  fer 45 r  0 -50 rm  fer fer


((a)) 150 -10 cbb



/fers{3 setlinewidth  40 0 rlineto cp t stroke 1 setlinewidth
0 0 m} def
-90 -300 t
o
-45 r -100 0 rm fer aro fer 45 r
fer aro fer 45 r fer aro fer 0 -10 rm 180 r fer aro fer 45 r fer
aro fer 45 r fer aro fer
0 -10 rm 180 r fer aro fer 45 r fer
aro fer 45 r fer aro fer
0 -10 rm 180 r fer aro fer 45 r fer
aro fer 45 r fer aro fer
-95 0 rm 45 r -100 5 rm fers aro fers
90 r fers 90 r fers aro fers 90 r fers
-95 0 rm fers 90 r fers aro fers 90 r fers
90 r fers aro fers

((b)) 150 30 cbb
stroke
showpage




%%EOF




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%fig3.ps
%!PS-Adobe-2.0
%%%%%%BoundingBox: 0 0 468 360

%   DRAWING COMMANDS

/arrowdict 14 dict def
arrowdict begin
 /mtrx matrix def
end

/arrow
 { arrowdict begin
    /headlength exch def
    /halfheadthickness exch 2 div def
    /halfthickness exch 2 div def
    /tipy exch def /tipx exch def
    /taily exch def /tailx exch def
    /dx tipx tailx sub def
    /dy tipy taily sub def
    /arrowlength dx dx mul dy dy mul add
      sqrt def
    /angle dy dx atan def
    /base arrowlength headlength sub def
    /savematrix mtrx currentmatrix def
    tailx taily translate
    angle rotate
    0 halfthickness neg moveto
    base halfthickness neg lineto
    base halfheadthickness neg lineto
    arrowlength 0 lineto
    base halfheadthickness lineto
    base halfthickness lineto
    0 halfthickness lineto
    closepath
    savematrix setmatrix
    end
  }def

/cir{ stroke  0 0 3 -1 roll 0 360 arc stroke} def
/cirf{ stroke 0 0 3 -1 roll 0 360 arc  fill stroke} def
/cp {currentpoint} def
/l {lineto} def
/m{moveto}def
/r {rotate} def
/rl{rlineto} def
/rm {rmoveto} def
/so {currentpoint translate stroke} def
/o {0 0 moveto} def
/t {translate} def
% ---------------------------------------------
%
%     FONT COMMANDS

%  The number `s' must be filled in by hand and /fh replaced by e.g., /fh1

%/fh {/Helvetica findfont s scalefont setfont moveto show} def
%/fhb {/Helvetica-Bold findfont s scalefont setfont moveto show} def
%/fhbo {/Helvetica-BoldOblique findfont s scalefont setfont moveto show} def
%/fho {/Helvetica-Oblique findfont s scalefont setfont moveto show} def
%/ftb {/Times-Bold  findfont s scalefont setfont moveto show} def
%/ftbi {/Times-BoldItalic findfont s scalefont setfont moveto show} def
%/fti {/Times-Italic findfont s scalefont setfont moveto show} def
%/ftr {/Times-Roman findfont s scalefont  setfont moveto show} def
%/fsym {/Symbol findfont s scalefont setfont moveto show} def

%-------------------------------------------------


/cb{/Courier-BoldOblique
 findfont 20 scalefont setfont moveto show} def
/cbs{/Courier-BoldOblique
 findfont 15 scalefont setfont moveto show} def
/cbb{/Courier-BoldOblique
 findfont 30 scalefont setfont moveto show} def

/fb {/Times-Roman findfont 15 scalefont setfont moveto    show
 } def
/fbb {/Times-Roman findfont 18 scalefont setfont moveto    show
 } def
/fs {/Times-Roman findfont 10 scalefont setfont moveto    show
 } def
/gkb{/Symbol findfont 20 scalefont setfont moveto  show} def
/itb{/Times-Italic findfont 20 scalefont setfont moveto show} def
/itm{/Times-Italic findfont 15 scalefont setfont moveto show} def
/its{/Times-Italic findfont 10 scalefont setfont moveto show} def

%----------------------------------------------------
% ADDITIONAL DEFINITIONS

%thick fermion line of length 50 from cp; cp moved to right end of line
/fer{3 setlinewidth  50 0 rlineto cp t stroke 1 setlinewidth
0 0 m} def

/rtl{3 setlinewidth rlineto cp t stroke 1 setlinewidth 0 0 m} def

%arrowhead with base at 0 0 pointing to the left; cp moved to base
/aro{0 0 -10 0 2 10 10 arrow 0 0 t fill stroke 0 0  m} def

%thin arrowhead pointing left at origin
/aros{0 0 -5 0 2 5 10 arrow  0 0 t fill stroke o} def

%dashed photon line from cp to rel coord (x,y); cp moved to end of line.
%usage: x y pho
/pho{[5 5]0 setdash rlineto cp t stroke o []0 setdash} def

%dot at current point
/dot{cp t  4 cirf o} def

%---------------
%definition for Fig. 6 only
/dot1{0 0 3 0 360 arc closepath gsave fill grestore} def
/ca{ newpath 50 0 10 0 360 arc -90 r stroke (A) -5 45 fb 90 r
 0 0 m} def
/cross{-40 0 m 40 0 lineto 0 40 m 0 5 lineto  0 -5 m 0 -40 lineto stroke
0 0 5 90 -90 arcn stroke 0 0 m} def
/fer1{3 setlinewidth -40 0 m 40 0 lineto stroke 1 setlinewidth
stroke
-20 0 -30 0 2 10 10 arrow fill 30 0 20 0 2 10 10 arrow fill
0 0 m} def
/gh{-40 0 m 40 0 lineto [7] 0 setdash stroke []0 setdash
-20 0 -30 0 .5 4 5 arrow fill 30 0 20 0 .5 4 5 arrow fill
0 0 m} def
/quad{round 180 r round 180 r} def
/round{40 0 m 10 0 lineto 0 0 0 10 5 arct  0 40 lineto stroke
0 0 m} def
/triple{
 0 0 m 40 0 lineto stroke
120 r
0 0 m 40 0 lineto stroke
120 r
10 0 t stroke dot1 0 0 m 30 0 lineto
stroke -10 0 t 0 0 m 120 r} def
%--------------------------------------



.45 .45 scale
300 1000 t
o
400 0 rtl cp stroke -300 0 t o
[5 5]0 setdash
0 0 100 100 200 0 curveto
-100 0 rm 0 -75 rl -150 75 rm 135 r 75 0 rl -135 r
150 -20 rm 135 r 75 0 rl -135 r 105 -40 rm 45 r 75 0 rl
-75 0 rm 90 r 75 0 rl -75 0 rm -135 r
45 -10 rm 45 r 75 0 rl -75 0 rm -45 r 40 -20 rm 45 r 75 0 rl -45 r
0 -70 rm 45 r 75 0 rl -75 0 rm -45 r cp stroke
[]0 setdash
(a) -120 50 fbb
(f) -20 80 fbb
(b) 30 90 fbb
(e) 130 90 fbb
(c) 180 85 fbb
(h) 220 65 fbb
(d) 270 45 fbb
(g) 90 -90 fbb
((a)) 70 -150 cbb
stroke

-100 -300 t
2 setlinewidth
o
50 0 rl 135 r 50 0 rl -135 r 5 0 rm 135 r -50 0 rl -135 r 50 0 rl
100 50 rl 135 r 50 0 rl -135 r 5 0 rm 135 r -50 0 rl -135 r 45 -13 rl
45 r 50 0 rl -45 r 5 0 rm 45 r -50 0 rl -45 r 300 0 l
50 0 rl 45 r 50 0 rl -45 r 5 0 rm 45 r -50 0 rl -45 r 50 0 rl
295 0 m 280 14 l 45 r -15 0 rl -45 r -5 0 rm 45 r 18 0 rl -45 r
260 32 l 210 46 l -45 r 40 0 rl 45 r -5 0 rm -45 r -40 0 rl 45 r
163 25 l -45 r  30 0 rl 45 r -5 0 rm -45 r -28 0 rl 45 r 113 0 l
195 0 l 0 20 rl 4 0 rm 0 -20 rl 294 0 l

(a) 0 40 fbb
(b) 170 60 fbb
(c) 270 70 fbb
(d) 370 40 fbb
(e) 220 10 fbb
(f) 160 3 fbb
(g) 203 10 fbb
(h) 260 10 fbb

((b)) 180 -70 cbb


stroke
showpage




%%EOF



%EOF



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%fig4.ps
%!PS-Adobe-2.0
%%BoundingBox: 0 0 468 360

%   DRAWING COMMANDS

/arrowdict 14 dict def
arrowdict begin
 /mtrx matrix def
end

/arrow
 { arrowdict begin
    /headlength exch def
    /halfheadthickness exch 2 div def
    /halfthickness exch 2 div def
    /tipy exch def /tipx exch def
    /taily exch def /tailx exch def
    /dx tipx tailx sub def
    /dy tipy taily sub def
    /arrowlength dx dx mul dy dy mul add
      sqrt def
    /angle dy dx atan def
    /base arrowlength headlength sub def
    /savematrix mtrx currentmatrix def
    tailx taily translate
    angle rotate
    0 halfthickness neg moveto
    base halfthickness neg lineto
    base halfheadthickness neg lineto
    arrowlength 0 lineto
    base halfheadthickness lineto
    base halfthickness lineto
    0 halfthickness lineto
    closepath
    savematrix setmatrix
    end
  }def

/cir{ stroke  0 0 3 -1 roll 0 360 arc stroke} def
/cirf{ stroke 0 0 3 -1 roll 0 360 arc  fill stroke} def
/cp {currentpoint} def
/l {lineto} def
/m{moveto}def
/r {rotate} def
/rl{rlineto} def
/rm {rmoveto} def
/so {currentpoint translate stroke} def
/o {0 0 moveto} def
/t {translate} def
% ---------------------------------------------
%
%     FONT COMMANDS

%  The number `s' must be filled in by hand and /fh replaced by e.g., /fh1

%/fh {/Helvetica findfont s scalefont setfont moveto show} def
%/fhb {/Helvetica-Bold findfont s scalefont setfont moveto show} def
%/fhbo {/Helvetica-BoldOblique findfont s scalefont setfont moveto show} def
%/fho {/Helvetica-Oblique findfont s scalefont setfont moveto show} def
%/ftb {/Times-Bold  findfont s scalefont setfont moveto show} def
%/ftbi {/Times-BoldItalic findfont s scalefont setfont moveto show} def
%/fti {/Times-Italic findfont s scalefont setfont moveto show} def
%/ftr {/Times-Roman findfont s scalefont  setfont moveto show} def
%/fsym {/Symbol findfont s scalefont setfont moveto show} def

%-------------------------------------------------


/cb{/Courier-BoldOblique
 findfont 20 scalefont setfont moveto show} def
/cbs{/Courier-BoldOblique
 findfont 15 scalefont setfont moveto show} def
/cbb{/Courier-BoldOblique
 findfont 30 scalefont setfont moveto show} def

/fb {/Times-Roman findfont 15 scalefont setfont moveto    show
 } def
/fbb {/Times-Roman findfont 18 scalefont setfont moveto    show
 } def
/fs {/Times-Roman findfont 10 scalefont setfont moveto    show
 } def
/gkb{/Symbol findfont 20 scalefont setfont moveto  show} def
/itb{/Times-Italic findfont 20 scalefont setfont moveto show} def
/itm{/Times-Italic findfont 15 scalefont setfont moveto show} def
/its{/Times-Italic findfont 10 scalefont setfont moveto show} def

%----------------------------------------------------
% ADDITIONAL DEFINITIONS

%thick fermion line of length 50 from cp; cp moved to right end of line
/fer{3 setlinewidth  50 0 rlineto cp t stroke 1 setlinewidth
0 0 m} def

/rtl{3 setlinewidth rlineto cp t stroke 1 setlinewidth 0 0 m} def

%arrowhead with base at 0 0 pointing to the left; cp moved to base
/aro{0 0 -10 0 2 10 10 arrow 0 0 t fill stroke 0 0  m} def

%thin arrowhead pointing left at origin
/aros{0 0 -5 0 2 5 10 arrow  0 0 t fill stroke o} def

%dashed photon line from cp to rel coord (x,y); cp moved to end of line.
%usage: x y pho
/pho{[5 5]0 setdash rlineto cp t stroke o []0 setdash} def

%dot at current point
/dot{cp t  4 cirf o} def

%---------------
%definition for Fig. 6 only
/dot1{0 0 3 0 360 arc closepath gsave fill grestore} def
/ca{ newpath 50 0 10 0 360 arc -90 r stroke (A) -5 45 fb 90 r
 0 0 m} def
/cross{-40 0 m 40 0 lineto 0 40 m 0 5 lineto  0 -5 m 0 -40 lineto stroke
0 0 5 90 -90 arcn stroke 0 0 m} def
/fer1{3 setlinewidth -40 0 m 40 0 lineto stroke 1 setlinewidth
stroke
-20 0 -30 0 2 10 10 arrow fill 30 0 20 0 2 10 10 arrow fill
0 0 m} def
/gh{-40 0 m 40 0 lineto [7] 0 setdash stroke []0 setdash
-20 0 -30 0 .5 4 5 arrow fill 30 0 20 0 .5 4 5 arrow fill
0 0 m} def
/quad{round 180 r round 180 r} def
/round{40 0 m 10 0 lineto 0 0 0 10 5 arct  0 40 lineto stroke
0 0 m} def
/triple{
 0 0 m 40 0 lineto stroke
120 r
0 0 m 40 0 lineto stroke
120 r
10 0 t stroke dot1 0 0 m 30 0 lineto
stroke -10 0 t 0 0 m 120 r} def
%--------------------------------------



.4 .4 scale

170 700 t
o
fer fer 135 r fer fer  -135 r
15 0 rm -45 r fer fer 45 r  fer -90 r fer fer 5 0
     rlineto 20 0 rm 25 0 rl -90 r fer 180 r
     cp t 3 setlinewidth 0 0 100 100 200 0 curveto cp t stroke o 80 0 rtl
     -400 0 t o fer fer 10 0 rtl
     3 setlinewidth 0 0 100 110 210 10 curveto cp t stroke o -35 -10 rm
     -50 0 rtl
-55 0 rtl 0 30 rtl 0 20 rm 0 40 rtl
     0 60 rtl
25 0 rtl  25 0 rtl 135 r fer  fer    -135 r
15 0 rm -45 r fer  fer 45 r 40 0 rtl 0 -40 rtl -90 r 90 r 0 -70 rtl
0 -20 rm 0 -20 rtl
25 0 rm -15 0 rtl 0 15 rtl 0 20 rm 0 115 rtl 105 0 rtl

-350 -150 t
(p) -75 0 fbb  (2)-65 -10 fb
(p) 360 0 fbb  (1)370 -10 fb
(p) 360 150  fbb    (3)370 140 fb
(p) -75 150 fbb    (4)-65 140 fb
(p) -20 190  fbb    (5)-10 180 fb
(p) 110 190 fbb    (6)120 180 fb
(k) 40  190  fbb    (5) 50 180 fb
(k) 170 190 fbb    (6)180 180 fb


(q) 90 -15 fbb (3) 100 -25 fb
(q) 180 -15 fbb (5) 190 -25 fb
(q) 245 -15 fbb (1) 255 -25 fb
(q) 150 135 fbb (8) 160 125 fb
(q) 210 135 fbb (6) 220 125 fb
(q) 90 135 fbb (9) 100 125 fb

((b)) 160 -100 cbb
stroke

-30 450 t
o



fer fer fer  fer  fer fer fer



-350 -100 t
o
fer  fer fer fer aro fer fer fer
-100 0 m 0 100 pho -100 0 m 0 -100 pho
50 0 m cp t stroke
[5 5]0 setdash 0 -100 144 45 135 arc
stroke []0 setdash
-100 100 m -80 80 pho 180 -80 m -80 80 pho 80 -180 m
%original at middle of lower thick line
0 44 rm cp t aros -50 10 t -90 r aros 90 r 100 0 t
 -90 r aros 90 r -210 106 t -45 r aros 45 r 100 0 t
-45 r aros 45 r
-90 -160 t  aro 0 100 t aro 125 0 t aro 125 0 t aro
 25 -100 t aro
-275 0 t

%origin at lower thick line, half way between left end of line
%and left end of circular arc

(p   +) -95 0 fbb  (2)-85 -10 fb
(p   +) 310 0 fbb  (1)320 -10 fb
(p   +) 310 100  fbb    (3)320 90 fb
(p   +) -95 100 fbb    (4)-85 90 fb
(p   +) -60 190  fbb    (5)-50 180 fb
(p   ) 40 190 fbb    (6)50 180 fb ( \055) 60 190 gkb

(q) 65 15 fbb (3) 75 5 fb
(q) 140 15 fbb (5) 150 5 fb
(q) 140 60 fbb (4) 150 50 fb
(q) 210 15 fbb (1) 220 5 fb
(q) 110 115 fbb (8) 120 105 fb
(q) 75 60 fbb (2) 85 50 fb
(q) 210 60 fbb (7) 220 50 fb
(q) 160 115 fbb (6) 170 105 fb
(q) 70 115 fbb (9) 80 105 fb


((a)) 130 -80 cbb

50 0 m dot 50 0 m dot 100 0 m dot 50 0 m dot -50 100 m dot
-50 0 m dot -50 0 m dot -50 0 m dot


stroke
showpage




%%EOF


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%fig5.ps
%!PS-Adobe-2.0
%%%%%%BoundingBox: 0 0 468 360

%   DRAWING COMMANDS

/arrowdict 14 dict def
arrowdict begin
 /mtrx matrix def
end

/arrow
 { arrowdict begin
    /headlength exch def
    /halfheadthickness exch 2 div def
    /halfthickness exch 2 div def
    /tipy exch def /tipx exch def
    /taily exch def /tailx exch def
    /dx tipx tailx sub def
    /dy tipy taily sub def
    /arrowlength dx dx mul dy dy mul add
      sqrt def
    /angle dy dx atan def
    /base arrowlength headlength sub def
    /savematrix mtrx currentmatrix def
    tailx taily translate
    angle rotate
    0 halfthickness neg moveto
    base halfthickness neg lineto
    base halfheadthickness neg lineto
    arrowlength 0 lineto
    base halfheadthickness lineto
    base halfthickness lineto
    0 halfthickness lineto
    closepath
    savematrix setmatrix
    end
  }def

/cir{ stroke  0 0 3 -1 roll 0 360 arc stroke} def
/cirf{ stroke 0 0 3 -1 roll 0 360 arc  fill stroke} def
/cp {currentpoint} def
/l {lineto} def
/m{moveto}def
/r {rotate} def
/rl{rlineto} def
/rm {rmoveto} def
/so {currentpoint translate stroke} def
/o {0 0 moveto} def
/t {translate} def
% ---------------------------------------------
%
%     FONT COMMANDS

%  The number `s' must be filled in by hand and /fh replaced by e.g., /fh1

%/fh {/Helvetica findfont s scalefont setfont moveto show} def
%/fhb {/Helvetica-Bold findfont s scalefont setfont moveto show} def
%/fhbo {/Helvetica-BoldOblique findfont s scalefont setfont moveto show} def
%/fho {/Helvetica-Oblique findfont s scalefont setfont moveto show} def
%/ftb {/Times-Bold  findfont s scalefont setfont moveto show} def
%/ftbi {/Times-BoldItalic findfont s scalefont setfont moveto show} def
%/fti {/Times-Italic findfont s scalefont setfont moveto show} def
%/ftr {/Times-Roman findfont s scalefont  setfont moveto show} def
%/fsym {/Symbol findfont s scalefont setfont moveto show} def

%-------------------------------------------------


/cb{/Courier-BoldOblique
 findfont 20 scalefont setfont moveto show} def
/cbs{/Courier-BoldOblique
 findfont 15 scalefont setfont moveto show} def
/cbb{/Courier-BoldOblique
 findfont 30 scalefont setfont moveto show} def

/fb {/Times-Roman findfont 15 scalefont setfont moveto    show
 } def
/fbb {/Times-Roman findfont 18 scalefont setfont moveto    show
 } def
/fs {/Times-Roman findfont 10 scalefont setfont moveto    show
 } def
/gkb{/Symbol findfont 20 scalefont setfont moveto  show} def
/itb{/Times-Italic findfont 20 scalefont setfont moveto show} def
/itm{/Times-Italic findfont 15 scalefont setfont moveto show} def
/its{/Times-Italic findfont 10 scalefont setfont moveto show} def

%----------------------------------------------------
% ADDITIONAL DEFINITIONS

%thick fermion line of length 50 from cp; cp moved to right end of line
/fer{3 setlinewidth  50 0 rlineto cp t stroke 1 setlinewidth
0 0 m} def

/rtl{3 setlinewidth rlineto cp t stroke 1 setlinewidth 0 0 m} def

%arrowhead with base at 0 0 pointing to the left; cp moved to base
/aro{0 0 -10 0 2 10 10 arrow 0 0 t fill stroke 0 0  m} def

%thin arrowhead pointing left at origin
/aros{0 0 -5 0 2 5 10 arrow  0 0 t fill stroke o} def

%dashed photon line from cp to rel coord (x,y); cp moved to end of line.
%usage: x y pho
/pho{[5 5]0 setdash rlineto cp t stroke o []0 setdash} def

%dot at current point
/dot{cp t  4 cirf o} def

%dotted photon lines up and down
/phoup{90 r [5 5]0 setdash 50 0 rlineto
 -50 0 rm cp t stroke o []0 setdash     -90 r} def

/phodown{-90 r [5 5]0 setdash 50 0 rlineto
 -50 0 rm cp t stroke o []0 setdash     90 r} def
%---------------
%definition for Fig. 6 only
/dot1{0 0 3 0 360 arc closepath gsave fill grestore} def
/ca{ newpath 50 0 10 0 360 arc -90 r stroke (A) -5 45 fb 90 r
 0 0 m} def
/cross{-40 0 m 40 0 lineto 0 40 m 0 5 lineto  0 -5 m 0 -40 lineto stroke
0 0 5 90 -90 arcn stroke 0 0 m} def
/fer1{3 setlinewidth -40 0 m 40 0 lineto stroke 1 setlinewidth
stroke
-20 0 -30 0 2 10 10 arrow fill 30 0 20 0 2 10 10 arrow fill
0 0 m} def
/gh{-40 0 m 40 0 lineto [7] 0 setdash stroke []0 setdash
-20 0 -30 0 .5 4 5 arrow fill 30 0 20 0 .5 4 5 arrow fill
0 0 m} def
/quad{round 180 r round 180 r} def
/round{40 0 m 10 0 lineto 0 0 0 10 5 arct  0 40 lineto stroke
0 0 m} def
/triple{
 0 0 m 40 0 lineto stroke
120 r
0 0 m 40 0 lineto stroke
120 r
10 0 t stroke dot1 0 0 m 30 0 lineto
stroke -10 0 t 0 0 m 120 r} def
%--------------------------------------



.45 .45 scale
300 1000 t
o
fer aro phoup fer phodown fer phoup fer aro phoup fer
phodown fer aro phoup fer
-250 -50 t o  fer aro fer fer -90 r fer -90 r phoup fer
aro fer fer phoup  -90 r fer -90 r
-100 -100 t o fer aro fer fer fer aro fer fer aro phoup fer

stroke
showpage




%%EOF


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%ichep.sty follows
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                                                                      %%%
%%%    INSTITUTE OF PHYSICS PUBLISHING                                   %%%
%%%                                                                      %%%
%%%    ICHEP.STY  LaTeX CRC style file for ICHEP94 conference            %%%
%%%                                                                      %%%
%%%    Copyright 1994 IOP Publishing Ltd                                 %%%
%%%                                                                      %%%
%%%    Permission is hereby given to use this file for                   %%%
%%%    material to be submitted to or published by                       %%%
%%%    Institute of Physics Publishing                                   %%%
%%%                                                                      %%%
%%%    Version 1.2  1 July 1994                                          %%%
%%%                                                                      %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% First we have a character check
%
% ! exclamation mark    " double quote
% # hash                ` opening quote (grave)
% & ampersand           ' closing quote (acute)
% $ dollar              % percent
% ( open parenthesis    ) close paren.
% - hyphen              = equals sign
% | vertical bar        ~ tilde
% @ at sign             _ underscore
% { open curly brace    } close curly
% [ open square         ] close square bracket
% + plus sign           ; semi-colon
% * asterisk            : colon
% < open angle bracket  > close angle
% , comma               . full stop
% ? question mark       / forward slash
% \ backslash           ^ circumflex
%
% ABCDEFGHIJKLMNOPQRSTUVWXYZ
% abcdefghijklmnopqrstuvwxyz
% 1234567890
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\typeout{Document Style `ichep.sty'. IOP camera-ready copy
style file for ICHEP Conference Proceedings}

\let\reset@font\empty

\def\refname{References}
\def\figurename{Figure}
\def\tablename{Table}
\def\abstractname{Abstract}

\def\@ptsize{0}
\@namedef{ds@11pt}{\def\@ptsize{0}}
\@namedef{ds@12pt}{\def\@ptsize{0}}
\def\ds@twoside{\@twosidetrue
           \@mparswitchtrue}

\def\ds@draft{\overfullrule 5\p@}

\newif\if@titlepage \@titlepagefalse
\def\ds@titlepage{\@titlepagetrue}

\def\ds@twocolumn{\@twocolumntrue}

%
\newdimen\mathindent
\newlength{\digitwidth}
\newlength{\indentedwidth}
\newcounter{firstpage}
\newbox{\captionbox}
\newcounter{eqnval}
%
\@twosidetrue
%\@mparswitchtrue
\def\ds@draft{\overfullrule 5\p@}
\@options
%
\def\hexnumber@#1{\ifcase#1 0\or 1\or 2\or 3\or 4\or 5\or 6\or 7\or 8\or
 9\or A\or B\or C\or D\or E\or F\fi}
%
\lineskip 1pt \normallineskip 1pt
\def\baselinestretch{1}
\def\@normalsize{\@setsize\normalsize{12pt}\xpt\@xpt
\abovedisplayskip 10pt plus2pt minus5pt
\belowdisplayskip \abovedisplayskip
\abovedisplayshortskip \z@ plus3pt
\belowdisplayshortskip 6pt plus3pt minus3pt}
\def\small{\@setsize\small{11pt}\ixpt\@ixpt
\abovedisplayskip 8.5pt plus 3pt minus 4pt
\belowdisplayskip \abovedisplayskip
\abovedisplayshortskip \z@ plus2pt
\belowdisplayshortskip 4pt plus2pt minus 2pt
\def\@listi{\topsep 4pt plus 2pt minus 2pt\parsep 0pt plus 1pt
\itemsep \parsep}}
\def\footnotesize{\@setsize\footnotesize{9.5pt}\viiipt\@viiipt
\abovedisplayskip 6pt plus 2pt minus 4pt
\belowdisplayskip \abovedisplayskip
\abovedisplayshortskip \z@ plus 1pt
\belowdisplayshortskip 3pt plus 1pt minus 2pt
\def\@listi{\topsep 3pt plus 1pt minus 1pt\parsep 0pt plus 1pt
\itemsep \parsep}}
\def\scriptsize{\@setsize\scriptsize{8pt}\viipt\@viipt}
\def\tiny{\@setsize\tiny{6pt}\vpt\@vpt}
\def\large{\@setsize\large{14pt}\xiipt\@xiipt}
\def\Large{\@setsize\Large{18pt}\xivpt\@xivpt}
\def\LARGE{\@setsize\LARGE{22pt}\xviipt\@xviipt}
\def\huge{\@setsize\huge{25pt}\xxpt\@xxpt}
\def\Huge{\@setsize\Huge{30pt}\xxvpt\@xxvpt}
\normalsize
\oddsidemargin -3pc
\evensidemargin -3pc
\marginparwidth .75in
\marginparsep 7\p@
\topmargin=-72\p@
\headheight=12\p@
\headsep=12\p@
\footheight=12\p@
\footskip=25\p@

\textheight 55pc
\textwidth 18cm
\indentedwidth 15.9cm
\columnsep 1cm
\columnseprule 0\p@
\mathindent = 2pc

\newcommand{\onecol}{\parfillskip=0pt\par\eject
   \onecolumn\parfillskip=0pt plus1fil\noindent}
\newcommand{\twocol}{\parfillskip=0pt\par\eject
   \twocolumn\parfillskip=0pt plus1fil\noindent}
\footnotesep 6.65\p@
\skip\footins 9\p@ plus 4\p@ minus 2\p@
\floatsep 12\p@ plus 2\p@ minus 2\p@
\textfloatsep 18\p@ plus 2\p@ minus 4\p@
\intextsep 12\p@ plus 2\p@ minus 2\p@
\@maxsep 20\p@
\dblfloatsep 12\p@ plus 2\p@ minus 2\p@
\dbltextfloatsep 18\p@ plus 2\p@ minus 4\p@
\@dblmaxsep 20\p@
\@fptop 0\p@
\@fpsep 8\p@ plus 1fil
\@fpbot 0\p@ plus 1fil
\@dblfptop 0\p@
\@dblfpsep 8\p@ plus 1fil
\@dblfpbot 0\p@
\marginparpush 5\p@

\parskip 0\p@
\parindent 16\p@
\topsep 4\p@ plus 2\p@ minus 2\p@
\partopsep 2\p@ plus 1\p@ minus 1\p@
\itemsep 0\p@ plus 2\p@
\@lowpenalty 51
\@medpenalty 151
\@highpenalty 301
\@beginparpenalty -\@lowpenalty
\@endparpenalty -\@lowpenalty
\@itempenalty -\@lowpenalty

\@noskipsecfalse   % new version

%
\def\section{\@startsection{section}{1}{\z@}{-3.5ex plus -1ex minus
 -.2ex}{2.3ex plus .2ex}{\noindent\reset@font\normalsize\bf\raggedright}}
\def\subsection{\@startsection{subsection}{2}{\z@}{-3.25ex plus -1ex minus
 -.2ex}{1.5ex plus .2ex}{\noindent\reset@font
  \normalsize\it\raggedright\nohyphens}}
\def\subsubsection{\@startsection{subsubsection}{3}{\z@}{-3.25ex plus
-1ex minus -.2ex}{-1em}{\reset@font\normalsize\it\nohyphens}}
\def\paragraph{\@startsection
 {paragraph}{4}{\z@}{3.25ex plus 1ex minus .2ex}{-1em}
                                                {\reset@font\normalsize\it}}
\def\subparagraph{\@startsection
 {subparagraph}{4}{\parindent}{3.25ex plus 1ex minus
 .2ex}{-1em}{\reset@font\normalsize\it}}

\def\@sect#1#2#3#4#5#6[#7]#8{\ifnum #2>\c@secnumdepth
     \let\@svsec\@empty\else
     \refstepcounter{#1}\edef\@svsec{\csname the#1\endcsname.\hskip 1em}\fi
     \@tempskipa #5\relax
      \ifdim \@tempskipa>\z@
        \begingroup #6\relax
          \noindent{\hskip #3\relax\@svsec}{\interlinepenalty \@M #8\par}%
        \endgroup
       \csname #1mark\endcsname{#7}\addcontentsline
         {toc}{#1}{\ifnum #2>\c@secnumdepth \else
                      \protect\numberline{\csname the#1\endcsname}\fi
                    #7}\else
        \def\@svsechd{#6\hskip #3\relax  %% \relax added 2 May 90
                   \@svsec #8\csname #1mark\endcsname
                      {#7}\addcontentsline
                           {toc}{#1}{\ifnum #2>\c@secnumdepth \else
                             \protect\numberline{\csname the#1\endcsname}\fi
                       #7}}\fi
     \@xsect{#5}}
%
\def\@ssect#1#2#3#4#5{\@tempskipa #3\relax
   \ifdim \@tempskipa>\z@
     \begingroup #4\noindent{\hskip #1}{\interlinepenalty \@M #5\par}\endgroup
   \else \def\@svsechd{#4\hskip #1\relax #5}\fi
    \@xsect{#3}}


\setcounter{secnumdepth}{3}

\def\appendix{\@@par
 \setcounter{section}{0}
 \setcounter{subsection}{0}
 \setcounter{subsubsection}{0}
 \setcounter{equation}{0}
 \setcounter{figure}{0}
 \setcounter{table}{0}
 \def\thesection{Appendix \Alph{section}}
 \def\theequation{\ifnumbysec
      \Alph{section}.\arabic{equation}\else
      \Alph{section}\arabic{equation}\fi}
 \def\thetable{\ifnumbysec
      \Alph{section}\arabic{table}\else
      A\arabic{table}\fi}
 \def\thefigure{\ifnumbysec
      \Alph{section}\arabic{figure}\else
      A\arabic{figure}\fi}}



\labelsep 4\p@

\leftmargini 16\p@
\leftmarginii 18\p@
\leftmarginiii 16\p@
\leftmarginiv 14\p@
\leftmarginv 10\p@
\leftmarginvi 10\p@
\leftmargin\leftmargini
\labelwidth\leftmargini\advance\labelwidth-\labelsep
\parsep 0\p@ plus 1\p@
\def\@listI{\leftmargin\leftmargini \parsep 4\p@ plus2\p@ minus\p@
\topsep 8\p@ plus2\p@ minus4\p@
\itemsep 4\p@ plus2\p@ minus\p@}

\let\@listi\@listI
\@listi

\def\@listii{\leftmargin\leftmarginii
 \labelwidth\leftmarginii\advance\labelwidth-\labelsep
 \topsep 3\p@ plus 1\p@ minus 1\p@
 \parsep 0\p@ plus 1\p@
 \itemsep \parsep}
\def\@listiii{\leftmargin\leftmarginiii
 \labelwidth\leftmarginiii\advance\labelwidth-\labelsep
 \topsep 2\p@ plus 1\p@ minus 1\p@
 \parsep \z@ \partopsep 1\p@ plus 0\p@ minus 1\p@
 \itemsep \topsep}
\def\@listiv{\leftmargin\leftmarginiv
 \labelwidth\leftmarginiv\advance\labelwidth-\labelsep}
\def\@listv{\leftmargin\leftmarginv
 \labelwidth\leftmarginv\advance\labelwidth-\labelsep}
\def\@listvi{\leftmargin\leftmarginvi
 \labelwidth\leftmarginvi\advance\labelwidth-\labelsep}

\pretolerance=5000
\tolerance=8000
\hbadness=5000
\vbadness=5000
%
\def\labelenumi{\theenumi}
\def\theenumi{\arabic{enumi}}
\def\labelenumii{\theenumii}
\def\theenumii{\alpha{enumii}}
\def\p@enumii{\theenumi.}
\def\labelenumiii{\theenumiii.}
\def\theenumiii{\arabic{enumiii}}
\def\p@enumiii{\p@enumii.\theenumii}
\def\labelenumiv{\theenumiv.}
\def\theenumiv{\arabic{enumiv}}
\def\p@enumiv{\p@enumiii.\theenumiii}

\def\labelitemi{$\m@th\bullet$}
\def\labelitemii{\bf --}
\def\labelitemiii{$\m@th\ast$}
\def\labelitemiv{$\m@th\cdot$}

\def\verse{\let\\=\@centercr
 \list{}{\itemsep\z@ \itemindent -1.5em\listparindent \itemindent
 \rightmargin\leftmargin\advance\leftmargin 1.5em}\item[]}
\let\endverse\endlist
\def\quotation{\list{}{\listparindent 1.5em
 \itemindent\listparindent
 \rightmargin\leftmargin\parsep 0\p@ plus 1\p@}\item[]}
\let\endquotation=\endlist
\def\quote{\list{}{\rightmargin\leftmargin}\item[]}
\let\endquote=\endlist

\def\descriptionlabel#1{\hspace\labelsep \bf #1}
\def\description{\list{}{\labelwidth\z@ \itemindent-\leftmargin
 \let\makelabel\descriptionlabel}}
\let\enddescription\endlist
%
\def\enumerate{\ifnum \@enumdepth >3 \@toodeep\else
      \advance\@enumdepth \@ne
      \edef\@enumctr{enum\romannumeral\the\@enumdepth}\list
      {\csname label\@enumctr\endcsname}{\usecounter
        {\@enumctr}\def\makelabel##1{##1\hss}}\fi}
%
\def\itemize{\ifnum \@itemdepth >3 \@toodeep\else \advance\@itemdepth \@ne
\edef\@itemitem{labelitem\romannumeral\the\@itemdepth}%
\list{\csname\@itemitem\endcsname}{\def\makelabel##1{##1\hss}\topsep=3pt
  \parsep=0pt\listparindent=0pt\itemsep=0pt\partopsep=0pt\rightmargin=0pt
  }\fi}
%
\newenvironment{leqnarray}{\begin{leqnarray}}{\end{leqnarray}}
\def\leqnarray{\stepcounter{equation}\let\@currentlabel=\theequation
\global\@eqnswtrue
\global\@eqcnt\z@\tabskip\mathindent\let\\=\@eqncr
\abovedisplayskip\topsep\ifvmode\advance\abovedisplayskip\partopsep\fi
\belowdisplayskip\abovedisplayskip
\belowdisplayshortskip\abovedisplayskip
\abovedisplayshortskip\abovedisplayskip
$$\halign to
\columnwidth\bgroup\@eqnsel$\displaystyle\tabskip\z@
 {##{}}$&\global\@eqcnt\@ne
                    $\displaystyle{{}##{}}$\hfil    %\hfil delete before 2nd $
 &\global\@eqcnt\tw@ $\displaystyle{{}##}$\hfil
 \tabskip\@centering&\llap{##}\tabskip\z@\cr}
%
\def\endleqnarray{\@@eqncr\egroup
 \global\advance\c@equation\m@ne$$\global\@ignoretrue }
%
\arraycolsep 5\p@
\tabcolsep=6\p@
\arrayrulewidth .4\p@
\doublerulesep 2\p@
\tabbingsep \labelsep
\skip\@mpfootins = \skip\footins
\fboxsep = 3\p@
\fboxrule = .4\p@
\def\titlepage{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn
     \else \newpage \fi \thispagestyle{myheadings}\c@page\z@}

\def\endtitlepage{\if@restonecol\twocolumn \else \newpage \fi}

\newcounter {section}
\newcounter {subsection}[section]
\newcounter {subsubsection}[subsection]
\newcounter {paragraph}[subsubsection]
\newcounter {subparagraph}[paragraph]



\def\thesection {\arabic{section}}
\def\thesubsection {\thesection.\arabic{subsection}}
\def\thesubsubsection {\thesubsection .\arabic{subsubsection}}
\def\theparagraph {\thesubsubsection.\arabic{paragraph}}
\def\thesubparagraph {\theparagraph.\arabic{subparagraph}}
\def\@chapapp{Section}


\def\@pnumwidth{1.55em}
\def\@tocrmarg {2.55em}
\def\@dotsep{4.5}
\setcounter{tocdepth}{2}


\def\tableofcontents{\@restonecolfalse\if@twocolumn\@restonecoltrue
 \onecolumn\fi\section*{Contents}{}\thispagestyle{empty}
 \@starttoc{toc}\if@restonecol\twocolumn\fi}
%
\def\l@section{\@dottedtocline{1}{1.5em}{2.3em}}
\def\l@subsection{\@dottedtocline{2}{3.8em}{3.2em}}
\def\l@subsubsection{\@dottedtocline{3}{7.0em}{4.1em}}
\def\l@paragraph{\@dottedtocline{4}{10em}{5em}}
\def\l@subparagraph{\@dottedtocline{5}{12em}{6em}}
\def\listoffigures{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn
 \fi\section*{List of Figures\@mkboth
 {LIST OF FIGURES}{LIST OF FIGURES}}\@starttoc{lof}\if@restonecol\twocolumn
 \fi}
\def\l@figure{\@dottedtocline{1}{1.5em}{2.3em}}
\def\listoftables{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn
 \fi\section*{List of Tables\@mkboth
 {LIST OF TABLES}{LIST OF TABLES}}\@starttoc{lot}\if@restonecol\twocolumn
 \fi}
\let\l@table\l@figure
%
% Redefinition to remove dotted lines from \@dottedtocline
%
\def\@dottedtocline#1#2#3#4#5{\ifnum #1>\c@tocdepth \else
  \vskip \z@ plus .2\p@
  {\leftskip #2\relax \rightskip \@tocrmarg \parfillskip -\rightskip
    \parindent #2\relax\@afterindenttrue
   \interlinepenalty\@M
   \leavevmode
   \@tempdima #3\relax \advance\leftskip \@tempdima
   \hbox{}\hskip -\leftskip
    #4\nobreak\hfill \nobreak \hbox to\@pnumwidth{\hfil
   \rm #5}\@@par}\fi}

\def\footnoterule{}%
\setcounter{footnote}{0}
\@addtoreset{footnote}{page}
\long\def\@makefntext#1{\parindent 1em\noindent
 \makebox[1em][l]{\footnotesize\rm$\m@th{\fnsymbol{footnote}}$}%
 \footnotesize\rm #1}
\def\@makefnmark{\hbox{${\fnsymbol{footnote}}\m@th$}}
\def\@thefnmark{\fnsymbol{footnote}}
\def\footnote{\@ifnextchar[{\@xfootnote}{\stepcounter{\@mpfn}%
     \begingroup\let\protect\noexpand
       \xdef\@thefnmark{\thempfn}\endgroup
     \@footnotemark\@footnotetext}}
\def\@fnsymbol#1{\ifcase#1\or \dagger\or \ddagger\or \S\or
   \|\or \P\or ^{+}\or ^{\tsty *}\or \sharp
   \or \dagger\dagger \else\@ctrerr\fi\relax}
\newcommand\ftnote[1]{\setcounter{footnote}{#1}%
   \addtocounter{footnote}{-1}\footnote}
\newcommand{\fnm}[1]{\setcounter{footnote}{#1}\footnotetext}
%
\def\center{\trivlist\topsep=0\p@\partopsep=0\p@
   \parsep=0\p@\itemsep=0\p@\centering\item[]}
%
\newenvironment{indented}{\begin{indented}}{\end{indented}}
\def\indented{\list{}{\itemsep=0\p@\labelsep=0\p@\itemindent=0\p@
   \labelwidth=0\p@\leftmargin=1.5cm\rightmargin=1.5cm
   \topsep=0\p@\partopsep=0\p@
   \parsep=0\p@\listparindent=0\p@}\rm}

\let\endindented=\endlist
%
\def\catchline{\hfill}

\def\cpyrtline{\hfill}
%
\def\maketitle{\thispagestyle{myheadings}%
   \vspace*{1.8cm}
   \begin{center}\@title\end{center}
   \vspace*{1.1cm}
   \normalsize\rm
   \begin{center}\@author\end{center}
   \begin{center}\@address\end{center}
   \@collab
   \@abstract}
%
%  Title
%
\def\title#1{\def\@title{\exhyphenpenalty=10000\hyphenpenalty=10000
    \Large\bf#1\par}}
\def\shortitle#1{\def\@shorttitle{#1}}
\let\paper=\title
%
% Authors
%
\renewcommand{\author}[1]{\def\@author{{\large #1\par}}}
%
% Affiliation
%
\newcommand{\address}[1]{\def\@address{\rm #1\par}}
\let\affil=\address
%
\newcommand{\collab}[1]{\def\@collab{\begin{center}%
   {\large\rm #1}\par
   \end{center}}}
%
% Default
%
\def\@collab{}
%
% Abstract
%
\def\abstract#1{\def\@abstract{\begin{center}
{\bf\abstractname}\end{center}%
\begin{indented}
\item[]#1\par
\end{indented}
\vspace{2cm minus1cm}}}%
%
\def\endabstract{}
%
% Command for second and subsequent paragraphs in abstract
%
\def\cabs{\\\hspace*{16\p@}}
%
\def\nosections{\vspace{30\p@ plus12\p@ minus12\p@}
    \noindent\ignorespaces}
%
\def\ack{\ifletter\bigskip\noindent\ignorespaces\else
    \section*{Acknowledgments}\fi}
\def\ackn{\ifletter\bigskip\noindent\ignorespaces\else
    \section*{Acknowledgment}\fi}
%
\newif\ifnumbysec
\def\theequation{\ifnumbysec
      \arabic{section}.\arabic{equation}\else
      \arabic{equation}\fi}
\def\eqnobysec{\numbysectrue\@addtoreset{equation}{section}}
%
\def\eqalign#1{\null\vcenter{\def\\{\cr}\openup\jot\m@th
  \ialign{\strut\hfil$\displaystyle{##{}}$&$\displaystyle{{}##}$\hfil
      \crcr#1\crcr}}\,}
%
\def\eqalignno#1{\displ@y \tabskip\z@skip
  \halign to\if@twocolumn\columnwidth\else\displaywidth\fi
   {\hfil$\@lign\displaystyle{##}$%
    \tabskip\z@skip
    &$\@lign\displaystyle{{}##}$\hfill\tabskip\@centering
    &\llap{$\@lign\hbox{\rm##}$}\tabskip\z@skip\crcr
    #1\crcr}}
%
\def\numparts{\addtocounter{equation}{1}%
     \setcounter{eqnval}{\value{equation}}%
     \setcounter{equation}{0}%
     \def\theequation{\ifnumbysec
     \arabic{section}.\arabic{eqnval}{\it\alph{equation}}%
     \else\arabic{eqnval}{\it\alph{equation}}\fi}}

\def\endnumparts{\def\theequation{\ifnumbysec
     \arabic{section}.\arabic{equation}\else
     \arabic{equation}\fi}%
     \setcounter{equation}{\value{eqnval}}}
%
\def\cases#1{%
     \left\{\,\vcenter{\def\\{\cr}\normalbaselines\openup1\jot\m@th%
     \ialign{\strut$\displaystyle{##}\hfil$&\tqs
     \rm##\hfil\crcr#1\crcr}}\right.}%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  Floats
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  \c@topnumber            : Number of floats allowed at the top of a column.
%
\setcounter{topnumber}{4}
%
%  \topfraction            : Fraction of column that can be devoted to floats.
%
\def\topfraction{1}
%
%  \c@dbltopnumber, \dbltopfraction : Same as above, but for double-column
%                          floats.
%
\setcounter{dbltopnumber}{4}
\def\dbltopfraction{1}
%
%  \c@bottomnumber, \bottomfraction : Same as above for bottom of page.
%
\setcounter{bottomnumber}{2}
\def\bottomfraction{.8}
%
%  \c@totalnumber          : Number of floats allowed in a single column,
%                          including in-text floats.
%
\setcounter{totalnumber}{5}
%
%  \textfraction         : Minimum fraction of column that must contain text.
%
\def\textfraction{0}
%
%  \floatpagefraction    : Minimum fraction of page that must be taken
%                          up by float page.
%
\def\floatpagefraction{.8}
%
%  \dblfloatpagefraction : Same as above, for double-column floats.
%
\def\dblfloatpagefraction{.8}
%
\newcounter{figure}
\def\thefigure{\@arabic\c@figure}
\def\figure{\let\@makecaption\@makeonecolcaption\@float{figure}}
\let\endfigure\end@float
%
\@namedef{figure*}{\let\@makecaption\@makewidecaption
      \@dblfloat{figure}}
\@namedef{endfigure*}{\end@dblfloat}
%
\def\@makewidecaption#1#2{\vspace{10\p@}%
     \sbox{\captionbox}{\noindent\footnotesize\rm\raggedright{\bf #1.} #2}%
     \ifdim\wd\captionbox > \indentedwidth
     \begin{indented}
     \item[]\footnotesize\rm\raggedright{\bf #1.} #2\par
     \end{indented}%
     \else
     \hbox to \hsize{\hfil\box\captionbox\hfil}\fi}
%
\def\@makeonecolcaption#1#2{\vspace{10pt}%
     \parbox{\columnwidth}{\noindent
     \footnotesize\rm\raggedright{\bf #1.} #2}\par}
%
%
% The document style must define the following.
%
%    \fps@TYPE   : The default placement specifier for floats of type TYPE.
%
\def\fps@figure{tb}
\def\fps@table{tb}
%
%    \ftype@TYPE : The type number for floats of type TYPE.
%
\def\ftype@figure{1}
\def\ftype@table{2}
%
%    \ext@TYPE   : The file extension indicating the file on which the
%                  contents list for float type TYPE is stored.  For example,
%                  \ext@figure = 'lof'.
%
\def\ext@table{aux}
\def\ext@figure{aux}
%
%    \fnum@TYPE  : A macro to generate the figure number for a caption.
%                  For example, \fnum@TYPE == Figure \thefigure.
%
\def\fnum@table{\tablename~\thetable}
\def\fnum@figure{\figurename~\thefigure}
%
%    \@makecaption{NUM}{TEXT} : A macro to make a caption, with NUM the value
%                  produced by \fnum@... and TEXT the text of the caption.
%                  It can assume it's in a \parbox of the appropriate width.
%
\newcommand{\Figure}[2]{\def\figspace{\vspace*{#1}}%
    \def\figcap{\caption{#2}}%
    \futurelet\next\@figplace}
\def\@figplace{\ifx\next[\let\next=\@figpl
                 \else\let\next=\@fignopl\fi\next}
\def\@figpl[#1]{\begin{figure}[#1]
   \figspace
   \figcap
   \end{figure}}
\def\@fignopl{\begin{figure}
   \figspace
   \figcap
   \end{figure}}
%
\newcommand{\widefigure}[2]{\def\figspace{\vspace*{#1}}%
    \def\figcap{\caption{#2}}%
    \futurelet\next\@wfigplace}
\def\@wfigplace{\ifx\next[\let\next=\@wfigpl
                 \else\let\next=\@wfignopl\fi\next}
\def\@wfigpl[#1]{\begin{figure*}[#1]
   \figspace
   \figcap
   \end{figure*}}
\def\@wfignopl{\begin{figure*}
   \figspace
   \figcap
   \end{figure*}}
%
% \@float{TYPE}[PLACEMENT] : This macro begins a float environment for a
%     single-column float of type TYPE with PLACEMENT as the placement
%     specifier.  The default value of PLACEMENT is defined by \fps@TYPE.
%     The environment is ended by \end@float.
%     E.g., \figure == \@float{figure}, \endfigure == \end@float.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  Tables
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\newcounter{table}
%
\def\thetable{\@arabic\c@table}
\def\table{\let\@makecaption\@makeonecolcaption
    \footnotesize\rm\@float{table}}
\let\endtable\end@float
%
\@namedef{table*}{\let\@makecaption\@makewidecaption
   \footnotesize\rm
   \@dblfloat{table}}
\@namedef{endtable*}{\end@dblfloat}
%
\def\tabular{\def\@halignto{}\@tabular}
\def\endtabular{\crcr\egroup\egroup $\egroup}
\expandafter \let \csname endtabular*\endcsname = \endtabular
%
\newsavebox{\tablebox}
%
\newcommand{\Table}[2]{\begin{center}
    \lineup
    \begin{tabular}{#1}%
    \hline
    #2
    \hline
    \end{tabular}
    \end{center}}
%
\newcommand{\tabnote}[1]{\begin{indented}
     \item[]\footnotesize\rm\raggedright #1\par
     \end{indented}}
%
% Definitions for centring headings over several columns
% \centre{4}{Results for helium} will centre
% Results for helium over four columns
% \crule{4} will produce a rule centred over four columns
% to go below a centred heading
%
\newcommand{\centre}[2]{\multicolumn{#1}{c}{#2}}
\newcommand{\crule}[1]{\multispan{#1}{\hrulefill}}
%
\def\lineup{\def\0{\hbox{\phantom{\footnotesize\rm 0}}}%
    \def\m{\hbox{$\phantom{-}$}}%
    \def\-{\llap{$-$}}}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% References
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\newcommand{\Bibliography}[1]{\section*{References}\par\numrefs{#1}}
\newcommand{\References}[1]{\section*{References}\footnotesize\rm}
%
\def\thebibliography#1{\list
 {\hfil[\arabic{enumi}]}{\topsep=0\p@\parsep=0\p@
 \partopsep=0\p@\itemsep=0\p@
 \labelsep=5\p@\itemindent=0\p@                %-10
 \settowidth\labelwidth{\footnotesize[#1]}%
 \leftmargin\labelwidth
 \advance\leftmargin\labelsep
% \advance\leftmargin -\itemindent
 \usecounter{enumi}}%
 \def\newblock{\ }
 \sloppy\clubpenalty4000\widowpenalty4000
 \sfcode`\.=1000\footnotesize\rm\relax}
\let\endthebibliography=\endlist
%
\def\numrefs#1{\begin{thebibliography}{#1}}
\def\endnumrefs{\end{thebibliography}}
\let\endbib=\endnumrefs

\mark{{}{}}

\def\ps@headings{\let\@mkboth\markboth
 \def\@oddfoot{}%
 \def\@evenfoot{}%
 \def\@evenhead{\makebox[\mathindent][l]{\normalsize\rm \thepage}%
  \normalsize\it\rightmark\hfill}%
 \def\@oddhead{\makebox[\mathindent][r]{\hfill}{\normalsize\it\leftmark}\hfill
  \normalsize\rm\thepage}%
}%

\def\ps@myheadings{\let\@mkboth\markboth
 \def\@oddhead{\catchline}%
 \def\@oddfoot{\cpyrtline}%
 \def\@evenhead{}%
 \def\@evenfoot{}%
}


\def\today{\ifcase\month\or
 January\or February\or March\or April\or May\or June\or
 July\or August\or September\or October\or November\or December\fi
 \space\number\day, \number\year}

\def\@begintheorem#1#2{\rm \trivlist \item[\hskip \labelsep{\it #1\ #2.}]}
\def\@opargbegintheorem#1#2#3{\rm \trivlist
      \item[\hskip \labelsep{\it #1\ #2\ (#3).}]}

\let\scap=\sc
\renewcommand{\sc}{\protect\scriptsize}
\newcommand{\itsc}{\protect\scriptsize\it}
\newcommand{\bfsc}{\protect\scriptsize\bf}
\def\p@LaTeX{{L\kern-.3em\lower.1em\hbox{$^{\rm A}$}\kern-.15em%
    T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
%
\newcommand{\nohyphens}{\hyphenpenalty=10000\exhyphenpenalty=10000}
\newcommand{\fl}{\hspace*{-\mathindent}}
\newcommand{\Tr}{\mathop{\rm Tr}\nolimits}
\newcommand{\tr}{\mathop{\rm tr}\nolimits}
\newcommand{\Or}{\mathop{\rm O}\nolimits}
\newcommand{\lshad}{[\![}
\newcommand{\rshad}{]\!]}
\newcommand{\case}[2]{{\textstyle\frac{#1}{#2}}}
\def\pt(#1){({\it #1\/})}
\newcommand{\dsty}{\displaystyle}
\newcommand{\tsty}{\textstyle}
\newcommand{\ssty}{\scriptstyle}
\newcommand{\sssty}{\scriptscriptstyle}
\def\lo#1{\llap{${}#1{}$}}
\def\eql{\llap{${}={}$}}
\def\lsim{\llap{${}\sim{}$}}
\def\lsimeq{\llap{${}\simeq{}$}}
\def\lequiv{\llap{${}\equiv{}$}}
%
\def\;{\protect\psemicolon}
\def\psemicolon{\relax\ifmmode\mskip\thickmuskip\else\kern .3333em\fi}
%
\newcommand{\eref}[1]{(\ref{#1})}
\newcommand{\sref}[1]{section~\ref{#1}}
\newcommand{\fref}[1]{figure~\ref{#1}}
\newcommand{\tref}[1]{table~\ref{#1}}
\newcommand{\Eref}[1]{Equation~(\ref{#1})}
\newcommand{\Sref}[1]{Section~\ref{#1}}
\newcommand{\Fref}[1]{Figure~\ref{#1}}
\newcommand{\Tref}[1]{Table~\ref{#1}}

\newcommand{\opencirc}{\raisebox{2\p@}{\;\circle{5}}}
\newcommand{\opensqr}{\mbox{$\Box$}}
\newcommand{\fullcirc}{\raisebox{-2\p@}{\Large$\bullet$}}
\newcommand{\fullsqr}{\mbox{\vrule height6pt width6pt}}
\newcommand{\dotted}
                 {\mbox{${\mathinner{\cdotp\cdotp\cdotp\cdotp\cdotp\cdotp}}$}}
\newcommand{\dashed}{\mbox{-\; -\; -\; -}}
\newcommand{\broken}{\mbox{-- -- --}}
\newcommand{\longbroken}{\mbox{--- --- ---}}
\newcommand{\chain}{\mbox{--- $\cdot$ ---}}
\newcommand{\dashddot}{\mbox{--- $\cdot$ $\cdot$ ---}}
\newcommand{\full}{\mbox{------}}
%
\newcommand{\etal}{{\it et al\/}\ }
\newcommand{\nonum}{\item[]}
%
% abbreviations for IOPP journals
%
\newcommand{\CQG}{{\em Class. Quantum Grav.} }
\newcommand{\HPP}{{\em High Perform. Polym.} }              % added 4/5/93
\newcommand{\IP}{{\em Inverse Problems\/} }
\newcommand{\JHM}{{\em J. Hard Mater.} }                    % added 4/5/93
\newcommand{\JPA}{{\em J. Phys. A: Math. Gen.} }
\newcommand{\JPB}{{\em J. Phys. B: At. Mol. Phys.} }      %1968-87
\newcommand{\jpb}{{\em J. Phys. B: At. Mol. Opt. Phys.} } %1988 and onwards
\newcommand{\JPC}{{\em J. Phys. C: Solid State Phys.} }   %1968--1988
\newcommand{\JPCM}{{\em J. Phys.: Condens. Matter\/} }    %1989 and onwards
\newcommand{\JPD}{{\em J. Phys. D: Appl. Phys.} }
\newcommand{\JPE}{{\em J. Phys. E: Sci. Instrum.} }
\newcommand{\JPF}{{\em J. Phys. F: Met. Phys.} }
\newcommand{\JPG}{{\em J. Phys. G: Nucl. Phys.} }         %1975--1988
\newcommand{\jpg}{{\em J. Phys. G: Nucl. Part. Phys.} }   %1989 and onwards
\newcommand{\MSMSE}{{\em Modelling Simulation Mater. Sci. Eng.} }
\newcommand{\MST}{{\em Meas. Sci. Technol.} }              %1990 and onwards
\newcommand{\NET}{{\em Network\/} }
\newcommand{\NL}{{\em Nonlinearity\/} }
\newcommand{\NT}{{\em Nanotechnology} }
\newcommand{\PAO}{{\em Pure Appl. Optics\/} }
\newcommand{\PM}{{\em Physiol. Meas.} }                        % added 4/5/93
\newcommand{\PMB}{{\em Phys. Med. Biol.} }
\newcommand{\PPCF}{{\em Plasma Phys. Control. Fusion\/} }      % added 4/5/93
\newcommand{\PSST}{{\em Plasma Sources Sci. Technol.} }
\newcommand{\QO}{{\em Quantum Opt.} }
\newcommand{\RPP}{{\em Rep. Prog. Phys.} }
\newcommand{\SLC}{{\em Sov. Lightwave Commun.} }               % added 4/5/93
\newcommand{\SST}{{\em Semicond. Sci. Technol.} }
\newcommand{\SUST}{{\em Supercond. Sci. Technol.} }
\newcommand{\WRM}{{\em Waves Random Media\/} }
%
% Other commonly quoted journals
%
\newcommand{\AC}{{\em Acta Crystallogr.} }
\newcommand{\AM}{{\em Acta Metall.} }
\newcommand{\AP}{{\em Ann. Phys., Lpz.} }
\newcommand{\APNY}{{\em Ann. Phys., NY\/} }
\newcommand{\APP}{{\em Ann. Phys., Paris\/} }
\newcommand{\CJP}{{\em Can. J. Phys.} }
\newcommand{\JAP}{{\em J. Appl. Phys.} }
\newcommand{\JCP}{{\em J. Chem. Phys.} }
\newcommand{\JJAP}{{\em Japan. J. Appl. Phys.} }
\newcommand{\JP}{{\em J. Physique\/} }
\newcommand{\JPhCh}{{\em J. Phys. Chem.} }
\newcommand{\JMMM}{{\em J. Magn. Magn. Mater.} }
\newcommand{\JMP}{{\em J. Math. Phys.} }
\newcommand{\JOSA}{{\em J. Opt. Soc. Am.} }
\newcommand{\JPSJ}{{\em J. Phys. Soc. Japan\/} }
\newcommand{\JQSRT}{{\em J. Quant. Spectrosc. Radiat. Transfer\/} }
\newcommand{\NC}{{\em Nuovo Cimento\/} }
\newcommand{\NIM}{{\em Nucl. Instrum. Methods\/} }
\newcommand{\NP}{{\em Nucl. Phys.} }
\newcommand{\PL}{{\em Phys. Lett.} }
\newcommand{\PR}{{\em Phys. Rev.} }
\newcommand{\PRL}{{\em Phys. Rev. Lett.} }
\newcommand{\PRS}{{\em Proc. R. Soc.} }
\newcommand{\PS}{{\em Phys. Scr.} }
\newcommand{\PSS}{{\em Phys. Status Solidi\/} }
\newcommand{\PTRS}{{\em Phil. Trans. R. Soc.} }
\newcommand{\RMP}{{\em Rev. Mod. Phys.} }
\newcommand{\RSI}{{\em Rev. Sci. Instrum.} }
\newcommand{\SSC}{{\em Solid State Commun.} }
\newcommand{\ZP}{{\em Z. Phys.} }
%

\def\ap#1#2#3 {Ann. Phys. (NY) {\bf#1} (19#2) #3}
\def\apj#1#2#3 {Astrophys. J. {\bf#1} (19#2) #3}
\def\apjl#1#2#3 {Astrophys. J. Lett. {\bf#1} (19#2) #3}
\def\app#1#2#3 {Acta. Phys. Pol. {\bf#1} (19#2) #3}
\def\ar#1#2#3 {Ann. Rev. Nucl. Part. Sci. {\bf#1} (19#2) #3}
\def\cpc#1#2#3 {Computer Phys. Comm. {\bf#1} (19#2) #3}
\def\err#1#2#3 {{\it Erratum} {\bf#1} (19#2) #3}
\def\ib#1#2#3 {{\it ibid.} {\bf#1} (19#2) #3}
\def\jmp#1#2#3 {J. Math. Phys. {\bf#1} (19#2) #3}
\def\ijmp#1#2#3 {Int. J. Mod. Phys. {\bf#1} (19#2) #3}
\def\jetp#1#2#3 {JETP Lett. {\bf#1} (19#2) #3}
\def\jpg#1#2#3 {J. Phys. G. {\bf#1} (19#2) #3}
\def\mpl#1#2#3 {Mod. Phys. Lett. {\bf#1} (19#2) #3}
\def\nat#1#2#3 {Nature (London) {\bf#1} (19#2) #3}
\def\nc#1#2#3 {Nuovo Cim. {\bf#1} (19#2) #3}
\def\nim#1#2#3 {Nucl. Instr. Meth. {\bf#1} (19#2) #3}
\def\np#1#2#3 {Nucl. Phys. {\bf#1} (19#2) #3}
\def\pcps#1#2#3 {Proc. Cam. Phil. Soc. {\bf#1} (#2) #3}
\def\pl#1#2#3 {Phys. Lett. {\bf#1} (19#2) #3}
\def\prep#1#2#3 {Phys. Rep. {\bf#1} (19#2) #3}
\def\prev#1#2#3 {Phys. Rev. {\bf#1} (19#2) #3}
\def\prl#1#2#3 {Phys. Rev. Lett. {\bf#1} (19#2) #3}
\def\prs#1#2#3 {Proc. Roy. Soc. {\bf#1} (19#2) #3}
\def\ptp#1#2#3 {Prog. Th. Phys. {\bf#1} (19#2) #3}
\def\ps#1#2#3 {Physica Scripta {\bf#1} (19#2) #3}
\def\rmp#1#2#3 {Rev. Mod. Phys. {\bf#1} (19#2) #3}
\def\rpp#1#2#3 {Rep. Prog. Phys. {\bf#1} (19#2) #3}
\def\sjnp#1#2#3 {Sov. J. Nucl. Phys. {\bf#1} (19#2) #3}
\def\spj#1#2#3 {Sov. Phys. JEPT {\bf#1} (19#2) #3}
\def\spu#1#2#3 {Sov. Phys.-Usp. {\bf#1} (19#2) #3}
\def\zp#1#2#3 {Zeit. Phys. {\bf#1} (19#2) #3}
%
\ps@headings \pagenumbering{arabic} \onecolumn


