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\section{Photon Vertices}

\subsection{The Anatomy of a Photon Vertex}

%All photons drawn by using \ddrawvertex\ command are drawn in either
%\bs REG or \bs FLIPPED styles.  % changed with PHOTONSETUP27/FEYNMAN34
All four-photon vertices drawn by the \ddrawvertex\ command 
use photons drawn in either
\bs REG or \bs FLIPPED styles.
For three-photon vertices diagonal lines are \bs CURLY or \bs FLIPPEDCURLY.
The format for drawing a photon vertex with \ddrawvertex\ is:
\begin{verbatim}

\drawvertex\photon[&lt;direction of photon one&gt;&lt;number of photons&gt;]
       &lt;(x,y), the co-ordinates of the beginning of the photon one&gt;
       [&lt;number of half-wiggles in each photon line&gt;]

\end{verbatim}
The reason for specifying the number of {\it half}-wiggles is that it
is sometimes convenient to have a line begin and end in either orientation
(see section 2.6.1 for an example of this).

Photonic vertices may be drawn in any of the eight compass directions.
If emboldened photons are desired then \bs THICKLINES, as opposed to
\bs thicklines, should be used.  When the document is [12pt]
the \bs bigphotons statement must be used (as discussed in section 2.9.2).

The following parameters are returned by \ddrawvertex\bs photon:

\begin{verbatim}

\photonfrontx,\photonfronty:   The (x,y) co-ordinates of the front of the last line.
\photonbackx,\photonbacky:     The (x,y) co-ordinates of the back of the last line.
\photonlengthx,\photonlengthy: The (x,y) extent of the last line.
\photoncount:                  The number of photons printed thus far.
\pfrontx,\pfronty:             The (x,y) co-ords of the front of the last line.
\pmidx,\pmidy:                 The (x,y) co-ords of the middle of the last line.
\pbackx,\pbacky:               The (x,y) co-ordinates of the back of the last line.
\plengthx,\plengthy:           The (x,y) extent of the line.
\vertexonex,\vertexoney:       The (x,y) co-ordinates of the back of line one.
\vertextwox,\vertextwoy:       The (x,y) co-ordinates of the back of line two.
\vertexthreex,\vertexthreey:   The (x,y) co-ordinates of the back of line three.
\vertexfourx,\vertexfoury:     The (x,y) co-ordinates of the back of line four.
\vertexmidx,\vertexmidy:       The (x,y) co-ordinates of the middle of the vertex.
\vertexcount:                  The number of vertices printed thus far.

\end{verbatim}

\subsection{Examples and Details}

The following example illustrates that vertices need not only represent
branchings but also crossed lines.
\begin{verbatim}

\begin{picture}(22000,22000)
\drawline\fermion[\NE\REG](0,0)[6000]
\drawvertex\photon[\NE 4](\pbackx,\pbacky)[7]
\drawline\fermion[\N\REG](\vertexonex,\vertexoney)[\photonlengthx]
\drawline\fermion[\N\REG](\fermionbackx,\fermionbacky)[\plengthy]
\drawline\fermion[\NW\REG](\vertextwox,\vertextwoy)[6000]
\end{picture}

\end{verbatim}
Producing
\vskip 1.5in \hskip 1in
\begin{picture}(22000,22000)
\drawline\fermion[\NE\REG](0,0)[6000]
\drawvertex\photon[\NE 4](\pbackx,\pbacky)[7]
\drawline\fermion[\N\REG](\vertexonex,\vertexoney)[\photonlengthx]
\drawline\fermion[\N\REG](\fermionbackx,\fermionbacky)[\plengthy]
\drawline\fermion[\NW\REG](\vertextwox,\vertextwoy)[6000]
\end{picture}
\vskip 0.25in
A couple of items may be noted.  In order to draw the connecting fermion line
we've drawn it in two stages.  We know that \bs photonlengthx will return the
horizontal length of line four of the four-photon vertex.  Since line one is
the one radiating from the centre in the \bs SW direction (since it was drawn
{\it to} the centre of the vertex in the \bs NE direction) we count 
around clockwise to the fourth line being in the \bs SE direction.
Since the picture is symmetric the `x' extent of line four is equal to the
`y' extent of lines one and two.  Therefore to connect the ends of lines one
and two we need to draw a fermion of that length twice.
In the second \ddrawline\ statement we use \bs plengthy so that it is the
same length as the previous line.\ \ If, instead, we'd wanted
\vskip 1.25in \hskip 1in

\begin{picture}(12000,12000)(-10000,0)
\drawline\fermion[\NE\REG](0,0)[6000]
\THICKLINES\drawvertex\photon[\NE 4](\pbackx,\pbacky)[7]
\THINLINES\drawline\fermion[\N\REG](\vertexonex,\vertexoney)[\photonlengthx]
\drawline\fermion[\N\REG](\fermionbackx,\fermionbacky)[\plengthy]
\drawline\fermion[\NW\REG](\vertextwox,\vertextwoy)[6000]
\THICKLINES\flipvertex\drawvertex\photon[\NE 3](\vertexthreex,\vertexthreey)[4]
\flipvertex\drawvertex\photon[\SE 3](\vertexfourx,\vertexfoury)[4]
\end{picture}
\vskip 0.25in \hskip -1.52em
we'd have added
\begin{verbatim}

\THICKLINES\flipvertex\drawvertex\photon[\NE 3](\vertexthreex,\vertexthreey)[4]
\flipvertex\drawvertex\photon[\SE 3](\vertexfourx,\vertexfoury)[4]

\end{verbatim}
and put a \verb@\THICKLINES@ modifier before the first \ddrawvertex\ and a
\verb)\THINLINES) following it.
Note how we had to use \bs flipvertex in order to make the sets of vertices
connect properly.  In point of fact the above pictures are flawed since
the two fermion-fermion-photon vertices are not symmetric.  In this instance
it would be more appropriate to draw two long photons instead of a vertex.

Photonic vertices may also be {\it stemmed}, as will be discussed in
the next chapter (however see section 2.9 for an example of stemmed
photons).  Finally we point out, in the form of an exercise, that
being able to produce photons with an odd number of half-wiggles again
has its uses.
\vskip 0.5in
Exercise:  Draw the following using \ddrawvertex.  How could you
replace the fermion on the right by a scalar?
\vskip 0.5in \hskip 0.5in
\begin{picture}(20000,12000)(0,-6000)
\drawline\photon[\E\REG](4000,0)[7]
\drawline\fermion[\NW\REG](\photonfrontx,\photonfronty)[4000]
\drawline\fermion[\SW\REG](\photonfrontx,\photonfronty)[4000]
\drawvertex\photon[\E 3](\photonbackx,\photonbacky)[7]
\drawline\fermion[\S\REG](\vertextwox,\vertextwoy)[\vertextwoy]
\drawline\fermion[\N\REG](\vertexthreex,\vertexthreey)[\vertextwoy]
\drawline\fermion[\NE\REG](\vertextwox,\vertextwoy)[\vertextwoy]
\drawline\fermion[\SE\REG](\vertexthreex,\vertexthreey)[\vertextwoy]
\end{picture}
\vskip 2.05in
Note that the diagonal fermion segments on the right have half of the length
of the vertical segment.
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