% Psfig/TeX 
\def\PsfigVersion{1.9}
% dvips version
%
% All psfig/tex software, documentation, and related files
% in this distribution of psfig/tex are 
% Copyright 1987, 1988, 1991 Trevor J. Darrell
%
% Permission is granted for use and non-profit distribution of psfig/tex 
% providing that this notice is clearly maintained. The right to
% distribute any portion of psfig/tex for profit or as part of any commercial
% product is specifically reserved for the author(s) of that portion.
%
% *** Feel free to make local modifications of psfig as you wish,
% *** but DO NOT post any changed or modified versions of ``psfig''
% *** directly to the net. Send them to me and I'll try to incorporate
% *** them into future versions. If you want to take the psfig code 
% *** and make a new program (subject to the copyright above), distribute it, 
% *** (and maintain it) that's fine, just don't call it psfig.
%
% Bugs and improvements to trevor@media.mit.edu.
%
% Thanks to Greg Hager (GDH) and Ned Batchelder for their contributions
% to the original version of this project.
%
% Modified by J. Daniel Smith on 9 October 1990 to accept the
% %%BoundingBox: comment with or without a space after the colon.  Stole
% file reading code from Tom Rokicki's EPSF.TEX file (see below).
%
% More modifications by J. Daniel Smith on 29 March 1991 to allow the
% the included PostScript figure to be rotated.  The amount of
% rotation is specified by the "angle=" parameter of the \psfig command.
%
% Modified by Robert Russell on June 25, 1991 to allow users to specify
% .ps filenames which don't yet exist, provided they explicitly provide
% boundingbox information via the \psfig command. Note: This will only work
% if the "file=" parameter follows all four "bb???=" parameters in the
% command. This is due to the order in which psfig interprets these params.
%
%  3 Jul 1991	JDS	check if file already read in once
%  4 Sep 1991	JDS	fixed incorrect computation of rotated
%			bounding box
% 25 Sep 1991	GVR	expanded synopsis of \psfig
% 14 Oct 1991	JDS	\fbox code from LaTeX so \psdraft works with TeX
%			changed \typeout to \ps@typeout
% 17 Oct 1991	JDS	added \psscalefirst and \psrotatefirst
%

% From: gvr@cs.brown.edu (George V. Reilly)
%
% \psdraft	draws an outline box, but doesn't include the figure
%		in the DVI file.  Useful for previewing.
%
% \psfull	includes the figure in the DVI file (default).
%
% \psscalefirst width= or height= specifies the size of the figure
% 		before rotation.
% \psrotatefirst (default) width= or height= specifies the size of the
% 		 figure after rotation.  Asymetric figures will
% 		 appear to shrink.
%
% \psfigurepath#1	sets the path to search for the figure
%
% \psfig
% usage: \psfig{file=, figure=, height=, width=,
%			bbllx=, bblly=, bburx=, bbury=,
%			rheight=, rwidth=, clip=, angle=, silent=}
%
%	"file" is the filename.  If no path name is specified and the
%		file is not found in the current directory,
%		it will be looked for in directory \psfigurepath.
%	"figure" is a synonym for "file".
%	By default, the width and height of the figure are taken from
%		the BoundingBox of the figure.
%	If "width" is specified, the figure is scaled so that it has
%		the specified width.  Its height changes proportionately.
%	If "height" is specified, the figure is scaled so that it has
%		the specified height.  Its width changes proportionately.
%	If both "width" and "height" are specified, the figure is scaled
%		anamorphically.
%	"bbllx", "bblly", "bburx", and "bbury" control the PostScript
%		BoundingBox.  If these four values are specified
%               *before* the "file" option, the PSFIG will not try to
%               open the PostScript file.
%	"rheight" and "rwidth" are the reserved height and width
%		of the figure, i.e., how big TeX actually thinks
%		the figure is.  They default to "width" and "height".
%	The "clip" option ensures that no portion of the figure will
%		appear outside its BoundingBox.  "clip=" is a switch and
%		takes no value, but the `=' must be present.
%	The "angle" option specifies the angle of rotation (degrees, ccw).
%	The "silent" option makes \psfig work silently.
%

% check to see if macros already loaded in (maybe some other file says
% "\input psfig") ...
\ifx\undefined\psfig\else\endinput\fi

%
% from a suggestion by eijkhout@csrd.uiuc.edu to allow
% loading as a style file. Changed to avoid problems
% with amstex per suggestion by jbence@math.ucla.edu

\let\LaTeXAtSign=\@
\let\@=\relax
\edef\psfigRestoreAt{\catcode`\@=\number\catcode`@\relax}
%\edef\psfigRestoreAt{\catcode`@=\number\catcode`@\relax}
\catcode`\@=11\relax
\newwrite\@unused
\def\ps@typeout#1{{\let\protect\string\immediate\write\@unused{#1}}}
\ps@typeout{psfig/tex \PsfigVersion}

%% Here's how you define your figure path.  Should be set up with null
%% default and a user useable definition.

\def\figurepath{./}
\def\psfigurepath#1{\edef\figurepath{#1}}

%
% @psdo control structure -- similar to Latex @for.
% I redefined these with different names so that psfig can
% be used with TeX as well as LaTeX, and so that it will not 
% be vunerable to future changes in LaTeX's internal
% control structure,
%
\def\@nnil{\@nil}
\def\@empty{}
\def\@psdonoop#1\@@#2#3{}
\def\@psdo#1:=#2\do#3{\edef\@psdotmp{#2}\ifx\@psdotmp\@empty \else
    \expandafter\@psdoloop#2,\@nil,\@nil\@@#1{#3}\fi}
\def\@psdoloop#1,#2,#3\@@#4#5{\def#4{#1}\ifx #4\@nnil \else
       #5\def#4{#2}\ifx #4\@nnil \else#5\@ipsdoloop #3\@@#4{#5}\fi\fi}
\def\@ipsdoloop#1,#2\@@#3#4{\def#3{#1}\ifx #3\@nnil 
       \let\@nextwhile=\@psdonoop \else
      #4\relax\let\@nextwhile=\@ipsdoloop\fi\@nextwhile#2\@@#3{#4}}
\def\@tpsdo#1:=#2\do#3{\xdef\@psdotmp{#2}\ifx\@psdotmp\@empty \else
    \@tpsdoloop#2\@nil\@nil\@@#1{#3}\fi}
\def\@tpsdoloop#1#2\@@#3#4{\def#3{#1}\ifx #3\@nnil 
       \let\@nextwhile=\@psdonoop \else
      #4\relax\let\@nextwhile=\@tpsdoloop\fi\@nextwhile#2\@@#3{#4}}
% 
% \fbox is defined in latex.tex; so if \fbox is undefined, assume that
% we are not in LaTeX.
% Perhaps this could be done better???
\ifx\undefined\fbox
% \fbox code from modified slightly from LaTeX
\newdimen\fboxrule
\newdimen\fboxsep
\newdimen\ps@tempdima
\newbox\ps@tempboxa
\fboxsep = 3pt
\fboxrule = .4pt
\long\def\fbox#1{\leavevmode\setbox\ps@tempboxa\hbox{#1}\ps@tempdima\fboxrule
    \advance\ps@tempdima \fboxsep \advance\ps@tempdima \dp\ps@tempboxa
   \hbox{\lower \ps@tempdima\hbox
  {\vbox{\hrule height \fboxrule
          \hbox{\vrule width \fboxrule \hskip\fboxsep
          \vbox{\vskip\fboxsep \box\ps@tempboxa\vskip\fboxsep}\hskip 
                 \fboxsep\vrule width \fboxrule}
                 \hrule height \fboxrule}}}}
\fi
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% file reading stuff from epsf.tex
%   EPSF.TEX macro file:
%   Written by Tomas Rokicki of Radical Eye Software, 29 Mar 1989.
%   Revised by Don Knuth, 3 Jan 1990.
%   Revised by Tomas Rokicki to accept bounding boxes with no
%      space after the colon, 18 Jul 1990.
%   Portions modified/removed for use in PSFIG package by
%      J. Daniel Smith, 9 October 1990.
%
\newread\ps@stream
\newif\ifnot@eof       % continue looking for the bounding box?
\newif\if@noisy        % report what you're making?
\newif\if@atend        % %%BoundingBox: has (at end) specification
\newif\if@psfile       % does this look like a PostScript file?
%
% PostScript files should start with `%!'
%
{\catcode`\%=12\global\gdef\epsf@start{%!}}
\def\epsf@PS{PS}
%
\def\epsf@getbb#1{%
%
%   The first thing we need to do is to open the
%   PostScript file, if possible.
%
\openin\ps@stream=#1
\ifeof\ps@stream\ps@typeout{Error, File #1 not found}\else
%
%   Okay, we got it. Now we'll scan lines until we find one that doesn't
%   start with %. We're looking for the bounding box comment.
%
   {\not@eoftrue \chardef\other=12
    \def\do##1{\catcode`##1=\other}\dospecials \catcode`\ =10
    \loop
       \if@psfile
	  \read\ps@stream to \epsf@fileline
       \else{
	  \obeyspaces
          \read\ps@stream to \epsf@tmp\global\let\epsf@fileline\epsf@tmp}
       \fi
       \ifeof\ps@stream\not@eoffalse\else
%
%   Check the first line for `%!'.  Issue a warning message if its not
%   there, since the file might not be a PostScript file.
%
       \if@psfile\else
       \expandafter\epsf@test\epsf@fileline:. \\%
       \fi
%
%   We check to see if the first character is a % sign;
%   if so, we look further and stop only if the line begins with
%   `%%BoundingBox:' and the `(atend)' specification was not found.
%   That is, the only way to stop is when the end of file is reached,
%   or a `%%BoundingBox: llx lly urx ury' line is found.
%
          \expandafter\epsf@aux\epsf@fileline:. \\%
       \fi
   \ifnot@eof\repeat
   }\closein\ps@stream\fi}%
%
% This tests if the file we are reading looks like a PostScript file.
%
\long\def\epsf@test#1#2#3:#4\\{\def\epsf@testit{#1#2}
			\ifx\epsf@testit\epsf@start\else
\ps@typeout{Warning! File does not start with `\epsf@start'.  It may not be a PostScript file.}
			\fi
			\@psfiletrue} % don't test after 1st line
%
%   We still need to define the tricky \epsf@aux macro. This requires
%   a couple of magic constants for comparison purposes.
%
{\catcode`\%=12\global\let\epsf@percent=%\global\def\epsf@bblit{%BoundingBox}}
%
%
%   So we're ready to check for `%BoundingBox:' and to grab the
%   values if they are found.  We continue searching if `(at end)'
%   was found after the `%BoundingBox:'.
%
\long\def\epsf@aux#1#2:#3\\{\ifx#1\epsf@percent
   \def\epsf@testit{#2}\ifx\epsf@testit\epsf@bblit
	\@atendfalse
        \epsf@atend #3 . \\%
	\if@atend	
	   \if@verbose{
		\ps@typeout{psfig: found `(atend)'; continuing search}
	   }\fi
        \else
        \epsf@grab #3 . . . \\%
        \not@eoffalse
        \global\no@bbfalse
        \fi
   \fi\fi}%
%
%   Here we grab the values and stuff them in the appropriate definitions.
%
\def\epsf@grab #1 #2 #3 #4 #5\\{%
   \global\def\epsf@llx{#1}\ifx\epsf@llx\empty
      \epsf@grab #2 #3 #4 #5 .\\\else
   \global\def\epsf@lly{#2}%
   \global\def\epsf@urx{#3}\global\def\epsf@ury{#4}\fi}%
%
% Determine if the stuff following the %%BoundingBox is `(atend)'
% J. Daniel Smith.  Copied from \epsf@grab above.
%
\def\epsf@atendlit{(atend)} 
\def\epsf@atend #1 #2 #3\\{%
   \def\epsf@tmp{#1}\ifx\epsf@tmp\empty
      \epsf@atend #2 #3 .\\\else
   \ifx\epsf@tmp\epsf@atendlit\@atendtrue\fi\fi}


% End of file reading stuff from epsf.tex
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% trigonometry stuff from "trig.tex"
\chardef\psletter = 11 % won't conflict with \begin{letter} now...
\chardef\other = 12

\newif \ifdebug %%% turn me on to see TeX hard at work ...
\newif\ifc@mpute %%% don't need to compute some values
\c@mputetrue % but assume that we do

\let\then = \relax
\def\r@dian{pt }
\let\r@dians = \r@dian
\let\dimensionless@nit = \r@dian
\let\dimensionless@nits = \dimensionless@nit
\def\internal@nit{sp }
\let\internal@nits = \internal@nit
\newif\ifstillc@nverging
\def \Mess@ge #1{\ifdebug \then \message {#1} \fi}

{ %%% Things that need abnormal catcodes %%%
	\catcode `\@ = \psletter
	\gdef \nodimen {\expandafter \n@dimen \the \dimen}
	\gdef \term #1 #2 #3%
	       {\edef \t@ {\the #1}%%% freeze parameter 1 (count, by value)
		\edef \t@@ {\expandafter \n@dimen \the #2\r@dian}%
				   %%% freeze parameter 2 (dimen, by value)
		\t@rm {\t@} {\t@@} {#3}%
	       }
	\gdef \t@rm #1 #2 #3%
	       {{%
		\count 0 = 0
		\dimen 0 = 1 \dimensionless@nit
		\dimen 2 = #2\relax
		\Mess@ge {Calculating term #1 of \nodimen 2}%
		\loop
		\ifnum	\count 0 < #1
		\then	\advance \count 0 by 1
			\Mess@ge {Iteration \the \count 0 \space}%
			\Multiply \dimen 0 by {\dimen 2}%
			\Mess@ge {After multiplication, term = \nodimen 0}%
			\Divide \dimen 0 by {\count 0}%
			\Mess@ge {After division, term = \nodimen 0}%
		\repeat
		\Mess@ge {Final value for term #1 of 
				\nodimen 2 \space is \nodimen 0}%
		\xdef \Term {#3 = \nodimen 0 \r@dians}%
		\aftergroup \Term
	       }}
	\catcode `\p = \other
	\catcode `\t = \other
	\gdef \n@dimen #1pt{#1} %%% throw away the ``pt''
}

\def \Divide #1by #2{\divide #1 by #2} %%% just a synonym

\def \Multiply #1by #2%%% allows division of a dimen by a dimen
       {{%%% should really freeze parameter 2 (dimen, passed by value)
	\count 0 = #1\relax
	\count 2 = #2\relax
	\count 4 = 65536
	\Mess@ge {Before scaling, count 0 = \the \count 0 \space and
			count 2 = \the \count 2}%
	\ifnum	\count 0 > 32767 %%% do our best to avoid overflow
	\then	\divide \count 0 by 4
		\divide \count 4 by 4
	\else	\ifnum	\count 0 < -32767
		\then	\divide \count 0 by 4
			\divide \count 4 by 4
		\else
		\fi
	\fi
	\ifnum	\count 2 > 32767 %%% while retaining reasonable accuracy
	\then	\divide \count 2 by 4
		\divide \count 4 by 4
	\else	\ifnum	\count 2 < -32767
		\then	\divide \count 2 by 4
			\divide \count 4 by 4
		\else
		\fi
	\fi
	\multiply \count 0 by \count 2
	\divide \count 0 by \count 4
	\xdef \product {#1 = \the \count 0 \internal@nits}%
	\aftergroup \product
       }}

\def\r@duce{\ifdim\dimen0 > 90\r@dian \then   % sin(x+90) = sin(180-x)
		\multiply\dimen0 by -1
		\advance\dimen0 by 180\r@dian
		\r@duce
	    \else \ifdim\dimen0 < -90\r@dian \then  % sin(-x) = sin(360+x)
		\advance\dimen0 by 360\r@dian
		\r@duce
		\fi
	    \fi}

\def\Sine#1%
       {{%
	\dimen 0 = #1 \r@dian
	\r@duce
	\ifdim\dimen0 = -90\r@dian \then
	   \dimen4 = -1\r@dian
	   \c@mputefalse
	\fi
	\ifdim\dimen0 = 90\r@dian \then
	   \dimen4 = 1\r@dian
	   \c@mputefalse
	\fi
	\ifdim\dimen0 = 0\r@dian \then
	   \dimen4 = 0\r@dian
	   \c@mputefalse
	\fi
%
	\ifc@mpute \then
        	% convert degrees to radians
		\divide\dimen0 by 180
		\dimen0=3.141592654\dimen0
%
		\dimen 2 = 3.1415926535897963\r@dian %%% a well-known constant
		\divide\dimen 2 by 2 %%% we only deal with -pi/2 : pi/2
		\Mess@ge {Sin: calculating Sin of \nodimen 0}%
		\count 0 = 1 %%% see power-series expansion for sine
		\dimen 2 = 1 \r@dian %%% ditto
		\dimen 4 = 0 \r@dian %%% ditto
		\loop
			\ifnum	\dimen 2 = 0 %%% then we've done
			\then	\stillc@nvergingfalse 
			\else	\stillc@nvergingtrue
			\fi
			\ifstillc@nverging %%% then calculate next term
			\then	\term {\count 0} {\dimen 0} {\dimen 2}%
				\advance \count 0 by 2
				\count 2 = \count 0
				\divide \count 2 by 2
				\ifodd	\count 2 %%% signs alternate
				\then	\advance \dimen 4 by \dimen 2
				\else	\advance \dimen 4 by -\dimen 2
				\fi
		\repeat
	\fi		
			\xdef \sine {\nodimen 4}%
       }}

% Now the Cosine can be calculated easily by calling \Sine
\def\Cosine#1{\ifx\sine\UnDefined\edef\Savesine{\relax}\else
		             \edef\Savesine{\sine}\fi
	{\dimen0=#1\r@dian\advance\dimen0 by 90\r@dian
	 \Sine{\nodimen 0}
	 \xdef\cosine{\sine}
	 \xdef\sine{\Savesine}}}	      
% end of trig stuff
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\psdraft{
	\def\@psdraft{0}
	%\ps@typeout{draft level now is \@psdraft \space . }
}
\def\psfull{
	\def\@psdraft{100}
	%\ps@typeout{draft level now is \@psdraft \space . }
}

\psfull

\newif\if@scalefirst
\def\psscalefirst{\@scalefirsttrue}
\def\psrotatefirst{\@scalefirstfalse}
\psrotatefirst

\newif\if@draftbox
\def\psnodraftbox{
	\@draftboxfalse
}
\def\psdraftbox{
	\@draftboxtrue
}
\@draftboxtrue

\newif\if@prologfile
\newif\if@postlogfile
\def\pssilent{
	\@noisyfalse
}
\def\psnoisy{
	\@noisytrue
}
\psnoisy
%%% These are for the option list.
%%% A specification of the form a = b maps to calling \@p@@sa{b}
\newif\if@bbllx
\newif\if@bblly
\newif\if@bburx
\newif\if@bbury
\newif\if@height
\newif\if@width
\newif\if@rheight
\newif\if@rwidth
\newif\if@angle
\newif\if@clip
\newif\if@verbose
\def\@p@@sclip#1{\@cliptrue}


\newif\if@decmpr

%%% GDH 7/26/87 -- changed so that it first looks in the local directory,
%%% then in a specified global directory for the ps file.
%%% RPR 6/25/91 -- changed so that it defaults to user-supplied name if
%%% boundingbox info is specified, assuming graphic will be created by
%%% print time.
%%% TJD 10/19/91 -- added bbfile vs. file distinction, and @decmpr flag

\def\@p@@sfigure#1{\def\@p@sfile{null}\def\@p@sbbfile{null}
	        \openin1=#1.bb
		\ifeof1\closein1
	        	\openin1=\figurepath#1.bb
			\ifeof1\closein1
			        \openin1=#1
				\ifeof1\closein1%
				       \openin1=\figurepath#1
					\ifeof1
					   \ps@typeout{Error, File #1 not found}
						\if@bbllx\if@bblly
				   		\if@bburx\if@bbury
			      				\def\@p@sfile{#1}%
			      				\def\@p@sbbfile{#1}%
							\@decmprfalse
				  	   	\fi\fi\fi\fi
					\else\closein1
				    		\def\@p@sfile{\figurepath#1}%
				    		\def\@p@sbbfile{\figurepath#1}%
						\@decmprfalse
	                       		\fi%
			 	\else\closein1%
					\def\@p@sfile{#1}
					\def\@p@sbbfile{#1}
					\@decmprfalse
			 	\fi
			\else
				\def\@p@sfile{\figurepath#1}
				\def\@p@sbbfile{\figurepath#1.bb}
				\@decmprtrue
			\fi
		\else
			\def\@p@sfile{#1}
			\def\@p@sbbfile{#1.bb}
			\@decmprtrue
		\fi}

\def\@p@@sfile#1{\@p@@sfigure{#1}}

\def\@p@@sbbllx#1{
		%\ps@typeout{bbllx is #1}
		\@bbllxtrue
		\dimen100=#1
		\edef\@p@sbbllx{\number\dimen100}
}
\def\@p@@sbblly#1{
		%\ps@typeout{bblly is #1}
		\@bbllytrue
		\dimen100=#1
		\edef\@p@sbblly{\number\dimen100}
}
\def\@p@@sbburx#1{
		%\ps@typeout{bburx is #1}
		\@bburxtrue
		\dimen100=#1
		\edef\@p@sbburx{\number\dimen100}
}
\def\@p@@sbbury#1{
		%\ps@typeout{bbury is #1}
		\@bburytrue
		\dimen100=#1
		\edef\@p@sbbury{\number\dimen100}
}
\def\@p@@sheight#1{
		\@heighttrue
		\dimen100=#1
   		\edef\@p@sheight{\number\dimen100}
		%\ps@typeout{Height is \@p@sheight}
}
\def\@p@@swidth#1{
		%\ps@typeout{Width is #1}
		\@widthtrue
		\dimen100=#1
		\edef\@p@swidth{\number\dimen100}
}
\def\@p@@srheight#1{
		%\ps@typeout{Reserved height is #1}
		\@rheighttrue
		\dimen100=#1
		\edef\@p@srheight{\number\dimen100}
}
\def\@p@@srwidth#1{
		%\ps@typeout{Reserved width is #1}
		\@rwidthtrue
		\dimen100=#1
		\edef\@p@srwidth{\number\dimen100}
}
\def\@p@@sangle#1{
		%\ps@typeout{Rotation is #1}
		\@angletrue
%		\dimen100=#1
		\edef\@p@sangle{#1} %\number\dimen100}
}
\def\@p@@ssilent#1{ 
		\@verbosefalse
}
\def\@p@@sprolog#1{\@prologfiletrue\def\@prologfileval{#1}}
\def\@p@@spostlog#1{\@postlogfiletrue\def\@postlogfileval{#1}}
\def\@cs@name#1{\csname #1\endcsname}
\def\@setparms#1=#2,{\@cs@name{@p@@s#1}{#2}}
%
% initialize the defaults (size the size of the figure)
%
\def\ps@init@parms{
		\@bbllxfalse \@bbllyfalse
		\@bburxfalse \@bburyfalse
		\@heightfalse \@widthfalse
		\@rheightfalse \@rwidthfalse
		\def\@p@sbbllx{}\def\@p@sbblly{}
		\def\@p@sbburx{}\def\@p@sbbury{}
		\def\@p@sheight{}\def\@p@swidth{}
		\def\@p@srheight{}\def\@p@srwidth{}
		\def\@p@sangle{0}
		\def\@p@sfile{} \def\@p@sbbfile{}
		\def\@p@scost{10}
		\def\@sc{}
		\@prologfilefalse
		\@postlogfilefalse
		\@clipfalse
		\if@noisy
			\@verbosetrue
		\else
			\@verbosefalse
		\fi
}
%
% Go through the options setting things up.
%
\def\parse@ps@parms#1{
	 	\@psdo\@psfiga:=#1\do
		   {\expandafter\@setparms\@psfiga,}}
%
% Compute bb height and width
%
\newif\ifno@bb
\def\bb@missing{
	\if@verbose{
		\ps@typeout{psfig: searching \@p@sbbfile \space  for bounding box}
	}\fi
	\no@bbtrue
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        \ifno@bb \else \bb@cull\epsf@llx\epsf@lly\epsf@urx\epsf@ury\fi
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\newdimen\p@intvaluey
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%            	calculate x' = x \cos\theta - y \sin\theta
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		     \multiply\count102 by \count241
		     \advance\count240 by -\count102
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		     \count200=#1\count205=0
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			\multiply\count201 by \count100
		 	\advance\count205 by \count201
		     \count201=\count200
			\divide\count201 by 10
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			%
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%		
% Compute any missing values
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% at this point the bounding box has been adjsuted correctly for
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%    \ps@bbw       width=
%    -------  =  ---------- 
%    \@bbw       new width=
% so `new width=' = (width= * \@bbw) / \ps@bbw; where \ps@bbw is the
% width of the original (unrotated) bounding box.
	\if@width
	   \in@hundreds{\@p@swidth}{\@bbw}{\ps@bbw}
	   \edef\@p@swidth{\@result}
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	\if@height
	   \in@hundreds{\@p@sheight}{\@bbh}{\ps@bbh}
	   \edef\@p@sheight{\@result}
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% usage : \psfig{file=, height=, width=, bbllx=, bblly=, bburx=, bbury=,
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%%UNIX --- UPDATED ON 13/8/97  
%====================================================================%
%                  sprocl.tex     27-Feb-1995                        %
% This latex file rewritten from various sources for use in the      %
% preparation of the standard proceedings Volume, latest version     %
% by Susan Hezlet with acknowledgments to Lukas Nellen.              %
% Some changes are due to David Cassel.                              %
%====================================================================%
% contribution to PASCOS98 by D. Wackeroth
% July 20, 1998

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%BEGINNING OF TEXT                           
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\title{LOOP-INDUCED SUSY EFFECTS IN STRONG TOP PAIR PRODUCTION
\footnote{Talk presented at PASCOS-98, Boston, MA, March 1998.}}

\author{D. WACKEROTH}

\address{Theory Group F1, Paul Scherrer Institut,\\
CH-5232 Villigen PSI, Switzerland\\E-mail: wackeroth@psi.ch} 

\maketitle\abstracts{We present some results of 
the calculation of MSSM one-loop contributions to
strong $t \bar t$ production in hadronic collisions.
Here we concentrate on the MSSM
electroweak-like ${\cal O}(\alpha)$ and SQCD ${\cal O}(\alpha_s)$ 
corrections to the main production mechanism at the
Fermilab Tevatron $p \bar p$ collider, $q\bar q\to g \to t \bar t$.
We give results for the total $t \bar t$ production rate and
for parity violating
asymmetries in the production of left and right handed top quark pairs.}
\section{Introduction}
The potential of future hadron colliders 
to perform precision measurements of top quark observables~\cite{topacc} 
is the motivation for an ongoing study on loop effects in strong
$t \bar t$ production within the Electroweak 
Standard Model (SM)~\cite{diplpubsm,smasym} and 
beyond~\cite{topmssm,mssmasym,sqcd,asym}. 
While the SM electroweak (EW) corrections have only marginal effects on
$t \bar t$ observables such as the total $t \bar t$ production rate 
$\sigma(S)$, 
the invariant $t \bar t$ mass distribution $d\sigma/d M_{t \bar t}$
and parity violating asymmetries ${\cal A}_{LR}$ in the production of
left and right handed top quark pairs,
there is the possibility of considerable enhancements
within supersymmetric models. 
The ongoing study of radiative corrections to these observables within the
Minimal Supersymmetric extension of the SM (MSSM) 
reveals promisingly large effects at both the upgraded Tevatron and the LHC. 
At the LHC, for instance, the MSSM EW-like corrections typically
diminish the leading order production cross section $\sigma_B$ by 
$\stackrel{<}{\sim}10\%$, 
they can significantly distort $d\sigma/dM_{t \bar t}$
and they induce asymmetries of ${\cal A}_{LR} \stackrel{<}{\sim} 3\%$. 
In this contribution we will concentrate on 
combining the MSSM EW-like ${\cal O}(\alpha)$ and SQCD ${\cal O}(\alpha_s)$ 
corrections to the main production mechanism at the
Fermilab Tevatron $p \bar p$ collider, $q\bar q\to g \to t \bar t$.
We will present first results 
for $\sigma(\sqrt{S}=2$ TeV) and ${\cal A}_{LR}$. 
\section{Loop-induced SUSY effects in $p \bar p,pp \to t\bar t X$}
At the parton level, the next-to-leading order differential cross sections 
for polarized $t \bar t$ production via 
the $q \bar q$ annihilation and gluon fusion subprocesses are obtained
by contracting the corresponding matrix elements describing the
loop contributions
$\delta {\cal M}_i (i=q\bar q,gg)$~\cite{diplpubsm,topmssm} with the 
Born matrix elements ${\cal M}_B^i$
\be
\frac{d \hat \sigma_i(\hat t,\hat s,\lt,\ltb)}{d \tcd}
= \frac{\beta_t}{32 \pi \hat s} \, \overline{\sum}
\left[ \mid {\cal M}^i_B \mid^2+
2 {\cal R}e \; \overline{\sum}
(\delta {\cal M}_i \times {\cal M}_B^{i*})\right]+\mbox{higher order},
\label{eq:dsig}
\ee
where $\beta_t$ is the top quark velocity and $\lambda_{t,\bar t}$ denote the
$t (\bar t)$ helicity states. $\sd, \hat t$ are Mandelstam variables 
and $\hat \theta$ is the scattering angle of the top quark in the parton CMS.
$\delta {\cal M}_{q \bar q}$ comprises the  
EW(-like) ${\cal O}(\alpha)$ and SQCD ${\cal O}(\alpha_s)$ corrections.
So far we only included EW(-like) corrections to the gluon fusion subprocess
described by $\delta {\cal M}_{gg}$;
the SQCD calculation is work in progress~\cite{sqcd}.
We studied the effects of EW(-like) corrections 
on unpolarized $t \bar t$ observables in~\cite{diplpubsm,smasym,topmssm}.
Parity violating effects in the production of left and right 
handed top quark pairs
which are loop induced through EW(-like) interactions 
are discussed in~\cite{smasym,mssmasym,asym}.
In the following calculation of the SQCD ${\cal O}(\alpha_s)$ corrections 
we will closely follow the notation of~\cite{topmssm,sqcd} 
and refer to it for more details.
\subsection{SQCD ${\cal O}(\alpha_s)$ corrections to 
$q \bar q \to g \to t \bar t$}\label{subsec:sqcd}
In Fig.~\ref{fig:feyn} we display the Feynman diagrams contributing 
to the SQCD ${\cal O}(\alpha_s)$ corrections to $q \bar q \to g \to t \bar t$.
The corresponding contribution to $d \hat \sigma_{q\bar q}$ 
of Eq.~\ref{eq:dsig} reads ($z=\tcd$)~\cite{sqcd}
\bea
\lefteqn{ 2 {\cal R}e \; \overline{\sum}
(\delta {\cal M}_{q\bar q}^{SQCD} \times {\cal M}_B^{q\bar q*}) = }
\nonumber\\
& & \overline{\sum} \mid {\cal M}^{q\bar q}_B \mid^2
\frac{\alpha_s}{2 \pi} {\cal R}e
\left(F_V(\sd,m_q)+F_V(\sd,m_t)+\hat \Pi(\sd)\right)+
\nonumber\\
& & \frac{4 \pi \alpha_s^3}{9} {\cal R}e 
\left(\beta_t^2 (1-z^2) (1+4 \lt \ltb)
 F_M(\sd,m_t) + 2 (\lt-\ltb) \left[2 z G_A(\sd,m_q)+
\right. \right. \nonumber\\
& & \left. \left. \beta_t (1+z^2) G_A(\sd,m_t) \right]\right)
+ \frac{32 \pi \alpha_s^3}{9 \sd} \frac{1}{4} {\cal R}e 
\left( (-1) \frac{7}{3} B_t-\frac{2}{3} B_u \right)(\sd,\hat t) \; ,
\eea
where the vertex corrections 
are parametrized in terms of UV finite form factors $F_V,F_M$ and $G_A$,
$\hat \Pi(\sd)=\Pi(\sd)-\Pi(0)$ denotes the 
subtracted gluon vacuum polarization and
$B_t$ and $B_u$, respectively, parametrize the $t$ and $u$ channel 
box contributions. Their explicit expressions will be provided in~\cite{sqcd}.
In Fig.~\ref{fig:sqcd}(a) we show separately their numerical impact on
the total $t\bar t$ production rate. We observe large cancellations between
the vertex and box contributions
\footnote{Since we differ from~\cite{zack}
in the overall sign of the box diagrams
we feel a brief explanation is in order:~to determine 
the relative sign between the $t$ and $u$ channel box diagram
(as well as between box and Born contribution)
we apply the rules of~\cite{dehk}.
Fixing the reference order as $t \bar t \bar q q$ and choosing the 
fermion flow as indicated in Fig.~\ref{fig:feyn} we only
need to assign an additional minus sign to the $t$ channel box contribution.}.
\begin{figure}[t]
\hfill
\psfig{figure=fd_qq.ps,height=1.5in}
\caption{The Feynman diagrams to the SQCD ${\cal O}(\alpha_s)$ contribution to 
$q \bar q \to g \to t \bar t$.}
\label{fig:feyn}
\end{figure}
\begin{figure}
\hfill
\psfig{figure=sqcd1.ps,height=1.5in} \hfill
\psfig{figure=sqcd2.ps,height=1.5in}
\caption{(a) The different contributions to the
SQCD ${\cal O}(\alpha_s)$ correction in dependence of a common squark mass
with $m_{\tilde g}=200$ GeV. (b) $\Delta$
in dependence of $m_{\tilde g}$ for different choices of $m_{\tilde t_2}$,
$m_{\tilde b_L}$ and $\Phi_{\tilde t}$ (with $m_t=174$ GeV).}
\label{fig:sqcd}
\end{figure}
\begin{figure}
\hfill
\psfig{figure=sqcd3.ps,height=1.5in} \hfill
\psfig{figure=sqcd4.ps,height=1.5in}
\caption{(a) $\Delta$ and (b) ${\cal A}_{LR}$ including the
combined EW-like ${\cal O}(\alpha)$ and 
SQCD ${\cal O}(\alpha_s)$ corrections in dependence of $M_{A^0}$
for different choices of $\tan\beta$, $\Phi_{\tilde t}$ and
$m_{\tilde b_L}$ (with $m_t=174$ GeV and $m_{\tilde t_2}=90$ GeV, 
$\mu=-120$ GeV, $M_2=3 |\mu|$, $m_{\tilde g}=150$ GeV).} 
\label{fig:tot}
\end{figure}
\subsection{Loop-induced EW-like and SQCD effects 
in $p \bar p\rightarrow t \bar t X$ at the Tevatron}\label{subsec:teva}
The hadronic cross section $\sigma_{\lt,\ltb}(S)$
to polarized $t \bar t$ production is obtained by
convoluting $\hat \sigma_i$ of Eq.~\ref{eq:dsig}
with the corresponding parton distribution functions.
We introduce a relative correction $\Delta=\sigma/\sigma_B-1$
(with $\sigma=\sum_{\lt,\ltb} \sigma_{\lt,\ltb}$)
which reveals the numerical impact of the radiative corrections on
the total $t \bar t$ production rate. In Fig.~\ref{fig:sqcd}(b)
we show $\Delta$ in dependence of
the gluino mass $m_{\tilde g}$ for different choices 
of the light stop quark mass $m_{\tilde t_2}$, the sbottom 
quark mass $m_{\tilde b_L}$ and the stop mixing angle $\Phi_{\tilde t}$. 
While the MSSM EW-like corrections are only of the order
of a few $\%$ (apart from the threshold region 
$m_t \approx m_{\tilde t_2}+M_{\tilde\chi^0}$ 
where they can reach $30 \%$)~\cite{topmssm} the
SQCD one-loop corrections can considerably diminish/enhance 
$\sigma_B$ when the gluino is not too heavy.
We now combine the MSSM EW-like corrections of~\cite{topmssm} with 
the SQCD corrections of Sec.~\ref{subsec:sqcd}. 
The resulting relative correction $\Delta$ 
and integrated left-right asymmetry 
${\cal A}_{LR}=[\sigma_{\lt=+1/2,\ltb=-1/2}
-\sigma_{-+}]/[\sigma_{+-}+\sigma_{-+}]$, respectively,
are shown in Fig.~\ref{fig:tot}(a) and (b) for different choices of the
MSSM input parameters. For comparison we also display the SM results
when varying the SM Higgs boson mass.
The SQCD ${\cal O}(\alpha_s)$ contribution induces parity violating asymmetries
when the squarks are non-degenerate in mass (and
$\Phi_{\tilde t} \ne \pi/4$). Then the asymmetries
induced by EW(-like) corrections can be considerably enhanced and are
possibly observable\footnote{
When summing over the $\bar t$ helicities the resulting asymmetry
${\cal A}$ is statistical significant 
($N_s \ge 4$~\cite{smasym,mssmasym}) when  
$|{\cal A}| \stackrel{>}{\sim}1.8\%$ 
which corresponds to $|{\cal A}_{LR}| \stackrel{>}{\sim}2.4\%$.}
at the upgraded Tevatron with ${\cal L}=10\, \mbox{fb}^{-1}$.

\section*{Acknowledgments}
This work is done in collaboration with W. Hollik, C. Kao and W.M. M{\"o}sle.
\section*{References}
\begin{thebibliography}{99}
\bibitem{topacc}Report of the {\em tev2000} Study Group, eds.~D. Amedei 
and R. Brock, FERMILAB-Pub-96/082, April 1996.
\bibitem{diplpubsm}W. Beenakker {\em et al.}, \Journal{\NPB}{411}{343}{1994}
and references therein.
\bibitem{smasym}C. Kao, G.A. Ladinsky and C.-P. Yuan,
{\em Int.~J.~Mod.~Phys.~}A {\bf 12}, 1341 (1997). 
\bibitem{topmssm}W. Hollik, W.M. M\"osle and D. Wackeroth, 
\Journal{\NPB}{516}{29}{1998} and references therein.
\bibitem{mssmasym}C. Kao, \Journal{\PLB}{348}{155}{1995}.
\bibitem{sqcd}W.M. M\"osle and D. Wackeroth, in preparation.
\bibitem{asym}C. Kao and D. Wackeroth, in preparation.
\bibitem{zack}Z. Sullivan, \Journal{\PRD}{56}{451}{1997} 
and references therein.
\bibitem{dehk}A. Denner {\em et al.}, \Journal{\NPB}{387}{467}{1992}.
\end{thebibliography}

\end{document}

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