%\setcounter{page}{874}
\appendix
\chapter{Solution to Exercises}

It should first be noted that none of these solutions is unique.
There are frequently many ways of drawing a diagram with \FEYNMAN,
the number increasing dramatically with the complexity.   The following
are reasonable (although not necessarily optimal) solutions to
all of the problems posed throughout the manual.


\section{Exercises for section 2.9.2}

The question was posed as to why one cannot draw a scalar in two pieces,
commencing from the same point but in opposite directions.
The answer is that all scalars both {\it begin} and {\it end} with a
line segment.  Hence the central segment of such a scalar would be twice
the length of the surrounding segments.  It is easily to overcome this
difficulty by drawing the central segment separately, either as a fermion
or as a scalar of extent [1], and then drawing the two scalars from the
ends.  For example to draw

\begin{picture}(15000,5000)
\drawline\photon[\N\CURLY](7000,0)[4]
\drawline\fermion[\W\REG](7000,0)[400]
\drawline\fermion[\E\REG](7000,0)[400]
\seglength=800  \gaplength=250        
\drawline\scalar[\W\REG](7400,0)[4]
\seglength=800  \gaplength=250
\drawline\scalar[\E\REG](6600,0)[4]
\end{picture}

You could enter:
\begin{verbatim}

\begin{picture}(15000,5000)
\drawline\photon[\N\CURLY](7000,0)[4]
\drawline\fermion[\W\REG](7000,0)[400]
\drawline\fermion[\E\REG](7000,0)[400]
\seglength=800  \gaplength=250   % It is better to use \global\seglength=800 etc.
\drawline\scalar[\W\REG](7400,0)[4]
\seglength=800  \gaplength=250
\drawline\scalar[\E\REG](6600,0)[4]
\end{picture}

\end{verbatim}

The diagram at the conclusion of 2.9.2 may be produced by the following
eight commands (or permutations thereof):

\begin{verbatim}

\begin{picture}(20000,20000)
\thicklines\drawline\photon[\N\FLIPPEDCURLY](3000,3000)[7]
\drawline\fermion[\NW\REG](\pbackx,\pbacky)[\photonlengthy]
\drawline\fermion[\E\REG](\fermionfrontx,\fermionfronty)[\fermionlength]
\drawline\fermion[\SW\REG](\photonfrontx,\photonfronty)[\photonlengthy]
\drawline\fermion[\E\REG](\photonfrontx,\photonfronty)[\photonlengthy]
\drawline\fermion[\N\REG](\pbackx,\pbacky)[\photonlengthy]
\drawline\photon[\SE\REG](\fermionfrontx,\fermionfronty)[7]
\drawline\photon[\NE\FLIPPED](\fermionbackx,\fermionbacky)[7]
\end{picture}

\end{verbatim}

\section{Exercises for section 2.10.2}

\begin{verbatim}

\begin{picture}(20000,20000)
\THINLINES   % Upper-left corner:
\drawline\gluon[\E\REG](18000,18000)[4]
\drawline\fermion[\S\REG](\pfrontx,\pfronty)[\gluonlengthx]
\THICKLINES  % Upper-right corner:
\drawline\gluon[\E\REG](\gluonbackx,\gluonbacky)[4]
\drawline\fermion[\S\REG](\pbackx,\pbacky)[\gluonlengthx]
\THINLINES   % Lower-right corner:
\drawline\fermion[\S\REG](\pbackx,\pbacky)[\gluonlengthx]
\drawline\gluon[\W\REG](\fermionbackx,\fermionbacky)[4]
\THICKLINES  % Lower-left corner:
\drawline\gluon[\W\REG](\gluonbackx,\gluonbacky)[4]
\drawline\fermion[\N\REG](\pbackx,\pbacky)[\fermionlength]
\end{picture}

\end{verbatim}

To draw this with \bs CENTRAL gluons one could divide each side into 
{\em three} pieces, not two.  Alternatively, starting at one point draw the 
\bs THICKLINES segment and then starting {\it at the same point}
draw the \bs THINLINES piece on top of it.  For example:

\begin{verbatim}

\THICKLINES\drawline\gluon[\E\CENTRAL](0,0)[4]
\THINLINES\drawline\gluon[\E\CENTRAL](0,0)[9]

\end{verbatim}

\section{Exercise for section 2.12}
\begin{verbatim}

\begin{picture}(20000,20000)   % Start at upper right.
\thicklines\drawline\photon[\SE\REG](18000,18000)[5]
\drawline\fermion[\S\REG](\photonfrontx,\photonfronty)[\boxlengthy]
\drawarrow[\N\ATBASE](\pmidx,\pmidy)
\drawline\fermion[\W\REG](\photonbackx,\photonbacky)[\photonlengthx]
\drawarrow[\W\ATTIP](\pmidx,\pmidy)
% Draw 2 small lines to connect the vector meson (parallel lines) at a corner:
\thinlines\drawline\fermion[\S\REG](\fermionbackx,\fermionbacky)[150]
\drawline\fermion[\W\REG](\fermionfrontx,\fermionfronty)[150]
\drawline\fermion[\SW\REG](\fermionbackx,\fermionbacky)[7000]  % Upper || line
\drawline\fermion[\S\REG](\fermionbackx,\fermionbacky)[75]  % Find the centre of
\drawline\fermion[\E\REG](\fermionbackx,\fermionbacky)[75]  % the double fermion
\drawline\gluon[\SW\REG](\fermionbackx,\fermionbacky)[5]
\put(\gluonfrontx,\gluonfronty){\circle*{500}} % A `blob': the vector->gauge trans.
\drawline\fermion[\S\REG](\fermionbackx,\fermionbacky)[75]  % Draw to position the
\drawline\fermion[\E\REG](\fermionbackx,\fermionbacky)[75]  % Second parallel line
\drawline\fermion[\NE\REG](\fermionbackx,\fermionbacky)[7000]  % Lower || line
\put(\gluonbackx,\gluonbacky){\circle*{500}}
\gaplength=300
\drawline\scalar[\NW\REG](\gluonbackx,\gluonbacky)[3]  % Left half of scalar
\thicklines\drawarrow[\NW\ATBASE](\scalarbackx,\scalarbacky)
\gaplength=300   \thinlines
\drawline\scalar[\SE\REG](\gluonbackx,\gluonbacky)[3]  % Right half of scalar
\put(\pfrontx,\scalarbacky){{\bf H}$^0$}
\put(\gluonfrontx,\gluonfronty){\ $\,{}_{\longleftarrow f_6(\omega,p_+\cdot q_-)}$}
\drawline\fermion[\NW\REG](\photonfrontx,\photonfronty)[2000]
\drawarrow[\NW\ATBASE](\pbackx,\pbacky)
\drawline\fermion[\SE\REG](\photonbackx,\photonbacky)[2000]
\drawarrow[\NW\ATTIP](\pmidx,\pmidy)
\put(\pbackx,\pbacky){$\,\,p+q$}   % \, gives extra space in math mode.
\end{picture}

\end{verbatim}

\section{Exercise for section 3.8}
\begin{verbatim}

\begin{picture}(20000,12000)(0,6000)
\drawline\photon[\E\REG](4000,0)[7]   % Left half of long photon.
\drawline\fermion[\NW\REG](\photonfrontx,\photonfronty)[4000]
\drawline\fermion[\SW\REG](\photonfrontx,\photonfronty)[4000]
\drawvertex\photon[\E 3](\photonbackx,\photonbacky)[7]  % Continue long photon.
\drawline\fermion[\S\REG](\vertextwox,\vertextwoy)[\vertextwoy]  % Upper half
\drawline\fermion[\N\REG](\vertexthreex,\vertexthreey)[\vertextwoy] % Lower half
\drawline\fermion[\NE\REG](\vertextwox,\vertextwoy)[\vertextwoy]
\drawline\fermion[\SE\REG](\vertexthreex,\vertexthreey)[\vertextwoy]
\end{picture}

\end{verbatim}
Since we've slyly selected the main axis to have zero ordinate (y=0)
that we can use \bs vertextwoy as a length in order to ensure 
that the diagonal fermion segments on the right have half of the length
of the vertical segment.  The vertical fermion is drawn in two sections.

In order to draw a scalar in place of the fermion line on the right
we must set \bs seglength and \bs gaplength so that the vertical line
will connect properly with the photons.  The section on spacing in
chapter four shows how to do this using the \bs phantom commands.
Here we could just divide the line length by an integer.  For instance
the commands
\begin{verbatim}

\global\divide \vertextwoy by 4
\global\gaplength=\vertextwoy
\global\multiply\vertextwoy by 2
\global\seglength=\vertextwoy
\drawline\scalar[\S\REG](\vertextwox,\vertextwoy)[3]

\end{verbatim}
would draw a vertical connecting scalar with three segments.
%\newpage
%\tableofcontents

\section{Exercises for section 4.1.1}
\begin{verbatim}
% Exercise 1: Delbruck Scattering:

\hskip 0.75in                    % Move the whole picture to the right.
\begin{picture}(10000,10000)(0,0)
\drawline\photon[\SE\FLIPPED](0,5000)[6]   % Start from the upper left.
\double\photonlengthx  \multroothalf\photonlengthx  % mults by root 2.
\drawline\fermion[\E\REG](\photonbackx,\photonbacky)[\photonlengthx]
\drawline\photon[\NE\REG](\pbackx,\pbacky)[6]
\drawline\fermion[\S\REG](\photonfrontx,\photonfronty)[\fermionlength]
\drawline\photon[\SE\FLIPPED](\pbackx,\pbacky)[6]
\drawline\fermion[\W\REG](\photonfrontx,\photonfronty)[\fermionlength]
\drawline\photon[\SW\REG](\pbackx,\pbacky)[6]
\drawline\fermion[\N\REG](\photonfrontx,\photonfronty)[\fermionlength]
\end{picture}


% Exercise 2:  Drell-Yan

\documentstyle [12pt]{report}
\begin{document}
\input FEYNMAN
\begin{center}    %  Everything to be centred.
Drell-Yan W-Production

\begin{picture}(10000,10000)    % Note that the 10000 x 10000 box is centred.
\bigphotons   % needed in 12-pt.
\THICKLINES
\drawline\photon[\E\REG](5000,5000)[11]
\drawarrow[\E\ATBASE](\pmidx,4820)
\put(\pmidx,5800){$W^+$}
\drawline\fermion[\NW\REG](\photonfrontx,\photonfronty)[5500]
\drawarrow[\SE\ATBASE](\pmidx,\pmidy)
\put(3500,7100){q}
\drawline\fermion[\SW\REG](\photonfrontx,\photonfronty)[5500]
\drawarrow[\SW\ATBASE](\pmidx,\pmidy)
\put(3500,2300){$\overline{q}'$}
\drawline\fermion[\NE\REG](\photonbackx,\photonbacky)[5500]
\drawarrow[\SW\ATBASE](\pmidx,\pmidy)
\global\advance \pmidx by -1400
\put(\pmidx,7100){$e^+$}
\drawline\fermion[\SE\REG](\photonbackx,\photonbacky)[5500]
\drawarrow[\SE\ATBASE](\pmidx,\pmidy)
\global\advance \pmidx by -1200
\put(\pmidx,2300){$\nu$}
\end{picture}
\end{center}
\end{document}

\end{verbatim}
Note the necessary blank line prior to the \bs begin\{picture\} command.

\section{Exercises for section 4.1.2}
\begin{verbatim}

% Using Phantom Mode.

\begin{picture}(20000,15000)
\drawline\gluon[\E\FLAT](10000,15000)[7]
% Calc lengths of scalars segments and gaps.
\global\Xone=\gluonlengthx
\global\divide\Xone by 8 %  \Xone will be the gap length
\global\Yone=\Xone  % \Xone, \Yone convenient unused variable names.
\double\Yone  %  \Yone will be the segment length
% Draw fermions:
\drawline\fermion[\W\REG](\gluonfrontx,\gluonfronty)[\gluonlengthx]
\drawline\fermion[\S\REG](\gluonfrontx,\gluonfronty)[\gluonlengthx]
\drawline\fermion[\W\REG](\fermionbackx,\fermionbacky)[\gluonlengthx]
% Draw scalars
\global\gaplength=\Xone  \global\seglength=\Yone
\drawline\scalar[\E\REG](\gluonbackx,\gluonbacky)[3]
\global\gaplength=\Xone  \global\seglength=\Yone
\drawline\scalar[\S\REG](\gluonbackx,\gluonbacky)[3]
\global\gaplength=\Xone  \global\seglength=\Yone
\drawline\scalar[\E\REG](\scalarbackx,\scalarbacky)[3]
% Now the photon.  Need to know it's length so use \phantom
\startphantom
\drawline\photon[\E\FLIPPED](0,0)[8]  % Can draw from anywhere
\stopphantom      % Now we have the photon's length!!
\negate\photonlengthx
\global\advance\gluonlengthx by \photonlengthx  % the difference
\global\divide\gluonlengthx by 2
\drawline\fermion[\E\REG](\fermionfrontx,\fermionfronty)[\gluonlengthx]
\drawline\photon[\E\FLIPPED](\pbackx,\pbacky)[8]  % Exactly same as in phantom mode.
\drawline\fermion[\E\REG](\pbackx,\pbacky)[\gluonlengthx]
% This is even easier using stemmed photons!
\end{picture}
\end{verbatim}
The difficulty in rotating this diagram through 45 degrees is that we would
require very short ``stems'' on the ends of the photon and there
is a minimum length which diagonal lines may be drawn.  One could try to
connect the ends by a semi-circle (which may be slightly smaller), a kinked
pair of vertical and horizontal lines, a large dot at the vertex \etc

Almost any diagram may be drawn without recourse to phantom mode
given enough effort and ingenuity.  In this case one could 
\begin{verbatim}
1)  Draw the Gluon
2)  From the point (\pmidx,\pmidy) go down a distance of 
    \gluonlengthx (or \plengthx)
3)  Draw one half of the photon to the left and one half to the right
4)  Draw fermions of length \photonfrontx minus \gluonfrontx or
    \gluonbackx minus \photonbackx at either end
5)  Draw the fermions and scalars as before.
\end{verbatim}
%  good place to break appendix **** !!!
\section{Exercise for section 4.3.2}
\begin{verbatim}

\hskip 1.25in
\begin{picture}(8000,8000)
% Three-gluon vertex:
\stemvertex1\vertexlink3\drawvertex\gluon[\N 3](0,0)[4]
\drawline\fermion[\SW\REG](\vertexonex,\vertexoney)[2000]
\drawline\fermion[\SE\REG](\vertexonex,\vertexoney)[2000]
\drawline\fermion[\N\REG](\vertextwox,\vertextwoy)[2000]
\drawline\fermion[\W\REG](\vertextwox,\vertextwoy)[2000]
% Four-gluon vertex:
\global\stemlength=1500
\stemvertex3\vertexlink4\drawvertex\gluon[\NE 4](\vertexthreex,\vertexthreey)[3]
\drawline\fermion[\W\REG](\vertextwox,\vertextwoy)[2000]
\drawline\fermion[\N\REG](\vertextwox,\vertextwoy)[2000]
\drawline\fermion[\E\REG](\vertexthreex,\vertexthreey)[2000]
\drawline\fermion[\N\REG](\vertexthreex,\vertexthreey)[2000]
% Add loops to line four of the four-gluon vertex (linked above with \vertexlink4).
\global\stemlength=1500
\backstemmed\drawline\gluon[\SE\REG](\vertexfourx,\vertexfoury)[2]
\drawarrow[\SE\ATTIP](\pbackx,\pbacky)
\end{picture}

\end{verbatim}

\section{Exercises for Section 4.3.3}

The capped part of the diagram in section 2.10.2 was produced by:

\begin{verbatim}

\begin{picture}(25000,10000)
\THICKLINES
\put(24000,7000){\circle{3000}}
% We request a circle of diameter 3000 so LaTeX will use it's maximum:  2800.
% First establish the length of half of the doubly capped gluon (since only the
\startphantom      %                                        back may be capped).
\drawline\gluon[\E\REG](0,0)[2]\gluoncap
\stopphantom
\pbackx=22600 \pbacky=7000   % Establish left edge of circle with diameter=2800.
\global\multiply \plengthx by -1    \global\multiply \plengthy by -1
\global\advance \pbackx by \plengthx  \global\advance \pbacky by \plengthy
\drawline\gluon[\E\REG](\pbackx,\pbacky)[2]\gluoncap  % Draw gluon TO the circle
\drawline\gluon[\W\FLIPPED](\gluonfrontx,\gluonfronty)[2]\gluoncap
\drawline\fermion[\NW\REG](\gluonbackx,\gluonbacky)[2000]
\drawline\fermion[\SW\REG](\gluonbackx,\gluonbacky)[2000]
\gluonbackx=25400 \gluonbacky=7000 % Establish right circle edge: diameter=2800.
\negate\gluonlengthx    \negate\gluonlengthy  % Repeat on the right-hand side.
\global\advance\gluonbackx by \gluonlengthx  
\global\advance\gluonbacky by \gluonlengthy
\drawline\gluon[\W\FLIPPED](\gluonbackx,\gluonbacky)[2]\gluoncap
\drawline\gluon[\E\REG](\gluonfrontx,\gluonfronty)[2]\gluoncap
\drawline\fermion[\NE\REG](\gluonbackx,\gluonbacky)[2000]
\drawline\fermion[\SE\REG](\gluonbackx,\gluonbacky)[2000]
\advance \gluonfrontx by -6800  % Experiment or measure out ``CAPPED'' by ruler.
\put(\gluonfrontx,2000){CAPPED}
\THINLINES
\end{picture}

\end{verbatim}
Note that an effective radius of 1400, instead of 1500, was used since
the largest circle which \LaTeX\ can draw has a diameter of 28 points.

The concluding exercise of 4.3.3 may be drawn with:

\begin{verbatim}

\begin{picture}(20000,21000)
% Work our way from the upper left to lower right.
%
\global\stemlength=1500
\backstemmed\drawline\photon[\SE\FLIPPED](0,20000)[8]
\global\advance\pbackx by 1000        % Move to the centre
\global\advance\pbacky by -1000       % of the circle.
\THICKLINES\put(\pbackx,\pbacky){\circle{2830}}\THINLINES   % The circle (thick)
\global\advance\pbackx by 1000        % Move to the SW side 
\global\advance\pbacky by -1000       % of the circle.
\global\stemlength=1500  
% The extra length on this vertex line is from one loop plus a link.
% Note that the following line is both stemmed and linked.
\frontstemmed\drawline\gluon[\SE\FLIPPED](\pbackx,\pbacky)[1]\gluonlink
\vertexcap2\vertexcap3\drawvertex\gluon[\SE 3](\gluonbackx,\gluonbacky)[3]
% Alternately we could have \vertexlink1\vertexcap2\vertexcap3\drawvertex...
% Now draw the fermions, arrows and labels:
\drawline\fermion[\SW\REG](\vertexthreex,\vertexthreey)[2000] % Southmost gluon.
\drawarrow[\SW\ATBASE](\pmidx,\pmidy)
\global\advance\fermionbackx by -700
\global\advance\fermionbacky by -450
\put(\fermionbackx,\fermionbacky){$q$}
\drawline\fermion[\SE\REG](\vertexthreex,\vertexthreey)[2000] % Eastmost gluon.
\drawarrow[\NW\ATBASE](\pmidx,\pmidy)
\global\advance\fermionbackx by 50
\global\advance\fermionbacky by -450
\put(\fermionbackx,\fermionbacky){$\bar q$}
\drawline\fermion[\NE\REG](\vertextwox,\vertextwoy)[2000]
\drawarrow[\NE\ATBASE](\pmidx,\pmidy)
\global\advance\fermionbackx by 50
\global\advance\fermionbacky by -250
\put(\fermionbackx,\fermionbacky){$q$}
\drawline\fermion[\SE\REG](\vertextwox,\vertextwoy)[2000]
\drawarrow[\NW\ATBASE](\pmidx,\pmidy)
\global\advance\fermionbackx by 50
\global\advance\fermionbacky by -750
\put(\fermionbackx,\fermionbacky){$\bar q$}
\end{picture}

\end{verbatim}

\section{Exercises for section 4.5} 

{\Large\bf A:}
\vskip 0.2in
This calligraphic masterpiece was created by:

\begin{verbatim}

\begin{picture}(28000,8000)
%T:
\drawline\gluon[\N\CENTRAL](0,0)[5]
\global\Yone=\gluonbacky
\drawline\fermion[\W\REG](\gluonbackx,\gluonbacky)[4000]
\drawline\fermion[\E\REG](\gluonbackx,\gluonbacky)[4000]
\global\advance\pbackx by 1000
%H:
\drawline\fermion[\N\REG](\pbackx,\gluonfronty)[\gluonlengthy]
\drawline\photon[\E\REG](\pmidx,\pmidy)[5]
\drawline\fermion[\N\REG](\pbackx,\gluonfronty)[\gluonlengthy]
\global\advance\pbackx by 1000
%E:
\drawline\fermion[\N\REG](\pbackx,\gluonfronty)[\gluonlengthy]
\global\advance\pmidy by -300
\drawline\gluon[\E\FLIPPEDCURLY](\pmidx,\pmidy)[4]
\global\advance\plengthx by 500
\drawline\fermion[\E\REG](\gluonfrontx,0)[\plengthx]
\drawline\fermion[\E\REG](\gluonfrontx,\Yone)[\plengthx]
\global\advance\pbackx by 4000
%O:
\drawloop\gluon[\NW 8](\pbackx,300)
\global\advance\loopbackx by 5500
%R:
\drawline\gluon[\S\CURLY](\loopbackx,\Yone)[3]
\global\Xone=\boxlengthy  \double\Xone  \multroothalf\Xone
\put(\pmidx,\pmidy){\oval(\Xone,\boxlengthy)[r]}
\drawline\fermion[\S\REG](\pbackx,\pbacky)[\pbacky]
\global\advance\pfrontx by 500
\global\advance\pfronty by 50
\drawline\photon[\SE\REG](\pfrontx,\pfronty)[4]
%Y:
\global\advance\pbackx by 2600
\drawline\photon[\N\CURLY](\pbackx,0)[3]
\global\Ytwo=\Yone
\negate\plengthy
\global\advance\Ytwo by \plengthy
\double\Ytwo  \multroothalf\Ytwo
\drawline\fermion[\NE\REG](\photonbackx,\photonbacky)[\Ytwo]
\drawline\fermion[\NW\REG](\photonbackx,\photonbacky)[\Ytwo]
\end{picture}

\end{verbatim}
\newpage
{\noindent{\Large\bf B:}}
\vskip 0.2in
The `balloon in the tree' diagram:
\begin{verbatim}

\begin{picture}(28000,28000)
\THICKLINES
%set up position of bottom of loop
\startphantom
\drawloop\gluon[\NE 0](0,0)
\stopphantom
\global\Xone=\loopfrontx  % diameter of `true' loop
\drawloop\gluon[\S 7](12000,18000)
\global\Xtwo=\gluonbackx
\global\Ytwo=\gluonbacky
% draw gluon vertex
\global\advance\loopmidy by \Xone
\global\stemlength=400   % lengthen the stem since this is drawn in BOLD
\stemvertex1\drawvertex\gluon[\S 3](\loopmidx,\loopmidy)[3]
% Determine diameter and centre of fermion loop
\negate\Xtwo
\global\advance\Xtwo by \loopfrontx
\global\Xthree=\Xtwo %  Store.  Will use shortly
\double\Xtwo  % \Xtwo is now the diameter of the fermion loop
\put(\loopfrontx,\Ytwo){\circle{\Xtwo}}
% Lastly the photon
% This begins located at root 1/2 times radius in x & y direction from centre
\multroothalf\Xthree %  This is why we stored it.
\global\advance\loopfrontx by \Xthree
\global\advance\Ytwo by \Xthree
\drawline\photon[\NE\FLIPPED](\loopfrontx,\Ytwo)[5]
\end{picture}

\end{verbatim}
In phantom mode we draw a `central loop' (extent=`0').  This gives us the
position of the {\em east-most} point of a complete loop and thus the true loop
``radius''.  By symmetry we now use this to find the south-most position on an 
{\em incomplete} loop (extent of seven).
We store the (negative of the)
``radius'' under the name \verb@\Xone@ and then draw the gluon
loop, clockwise, storing the endpoints as \verb@(\Xtwo,\Ytwo)@.  We next draw
the three-gluon vertex beginning at the bottom of the gluon loop.  To find this
point we subtract the radius (\bs Xone) from the midpoint of the loop,
(\bs loopmidx,\bs loopmidy).

Next we draw the fermion loop.
We accomplish this by finding the radius of the circle, which is the difference
between the initial and final $x$ (or $y$) coordinates of the gluon loop.
This is now stored as \verb@\Xthree@.  The centre is simply at the $x$
coordinate of the front of the gluon loop and the $y$ coordinate of the rear.
Finally we use \bs multroothalf in order to find the spot, $45^\circ$ around the
circle, from which to draw the photon.