% Read in by FEYNMANDOC: FEYNMANDOC3B. Called by FD3B.COM \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[] <(x,y), the co-ordinates of the beginning of the photon one> [] \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.