Vertices & Tension

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External vertices

It is easy to define the external vertices in feynmp by using one or more these five commands:

  • \fmfleft,
  • \fmfright,
  • \fmftop,
  • \fmfbottom,
  • \fmfsurround.

For example:

A simple example of tension
External vertices
External vertices 
\documentclass[11pt,border=4pt]{standalone}
\usepackage{feynmp-auto}
\begin{document}
\begin{fmffile}{feyngraph}
  \begin{fmfgraph*}(100,80) % dimensions (WH)
    \fmfcurved % redundant
    % define external vertices
    \fmfleft{l1,l2,l3,l4}
    \fmfright{r1,r2,r3,r4}
    % draw circles at points with labels
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=l1,l.a=45}{l1}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=l2,l.a=0}{l2}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=l3,l.a=0}{l3}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=l4,l.a=-45}{l4}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=r1,l.a=135}{r1}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=r2,l.a=180}{r2}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=r3,l.a=180}{r3}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=r4,l.a=-135}{r4}
  \end{fmfgraph*}
\end{fmffile}
\end{document}

\documentclass[11pt,border=4pt]{standalone}
\usepackage{feynmp-auto}
\begin{document}
\begin{fmffile}{feyngraph}
  \begin{fmfgraph*}(100,80) % dimensions (WH)
    \fmfcurved % redundant
    % define external vertices
    \fmftop{t1,t2,t3,t4}
    \fmfbottom{b1,b2,b3,b4}
    % draw circles at points with labels
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=t1,l.a=-45}{t1}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=t2,l.a=-90}{t2}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=t3,l.a=-90}{t3}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=t4,l.a=-135}{t4}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=b1,l.a=45}{b1}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=b2,l.a=90}{b2}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=b3,l.a=90}{b3}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=b4,l.a=135}{b4}
  \end{fmfgraph*}
\end{fmffile}
\end{document}

\documentclass[11pt,border=4pt]{standalone}
\usepackage{feynmp-auto}
\begin{document}
\begin{fmffile}{feyngraph}
  \begin{fmfgraph*}(100,80) % dimensions (WH)
    \fmfcurved % redundant
    % define external vertices
    \fmfsurround{v1,v2,v3,v4,v5,v6,v7}
    % draw circles at points with labels
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=v1,l.a=180}{v1}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=v2,l.a=-130}{v2}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=v3,l.a=-50}{v3}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=v4,l.a=-20}{v4}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=v5,l.a=20}{v5}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=v6,l.a=80}{v6}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=v7,l.a=120}{v7}
  \end{fmfgraph*}
\end{fmffile}
\end{document}

The external vertices are by default positioned on a curve. One can set this behavior explicitly with \fmfcurved, In code example above it is redundant, but it can be used if one wants to turn off the behavior of \fmfstraight (see next) in the same diagram.

One can set the external vertices on a straight line along the edges for the rectangular frame with \fmfstraight instead. Below is an illustration of \fmfstraight.

A simple example of tension
External vertices
External vertices 
\documentclass[11pt,border=4pt]{standalone}
\usepackage{feynmp-auto}
\begin{document}
\begin{fmffile}{feyngraph}
  \begin{fmfgraph*}(100,80) % dimensions (WH)
    \fmfstraight
    % define external vertices
    \fmfleft{l1,l2,l3,l4}
    \fmfright{r1,r2,r3,r4}
    % draw circles at points with labels
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=l1,l.a=45}{l1}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=l2,l.a=0}{l2}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=l3,l.a=0}{l3}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=l4,l.a=-45}{l4}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=r1,l.a=135}{r1}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=r2,l.a=180}{r2}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=r3,l.a=180}{r3}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=r4,l.a=-135}{r4}
  \end{fmfgraph*}
\end{fmffile}
\end{document}

\documentclass[11pt,border=4pt]{standalone}
\usepackage{feynmp-auto}
\begin{document}
\begin{fmffile}{feyngraph}
  \begin{fmfgraph*}(100,80) % dimensions (WH)
    \fmfstraight
    % define external vertices
    \fmftop{t1,t2,t3,t4}
    \fmfbottom{b1,b2,b3,b4}
    % draw circles at points with labels
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=t1,l.a=-45}{t1}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=t2,l.a=-90}{t2}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=t3,l.a=-90}{t3}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=t4,l.a=-135}{t4}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=b1,l.a=45}{b1}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=b2,l.a=90}{b2}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=b3,l.a=90}{b3}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=b4,l.a=135}{b4}
  \end{fmfgraph*}
\end{fmffile}
\end{document}

\documentclass[11pt,border=4pt]{standalone}
\usepackage{feynmp-auto}
\begin{document}
\begin{fmffile}{feyngraph}
  \begin{fmfgraph*}(100,80) % dimensions (WH)
    \fmfstraight
    % define external vertices
    \fmfsurround{v1,v2,v3,v4,v5,v6,v7}
    % draw circles at points with labels
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=v1,l.a=180}{v1}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=v2,l.a=-130}{v2}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=v3,l.a=-50}{v3}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=v4,l.a=-20}{v4}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=v5,l.a=20}{v5}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=v6,l.a=80}{v6}
    \fmfv{d.sh=circle,d.f=full,d.si=4,l.d=5,label=v7,l.a=120}{v7}
  \end{fmfgraph*}
\end{fmffile}
\end{document}

Tension

Below is a simple example of how one can use tension to make lines between vertices shorter (tension>1) or longer (tension<1). The default tension is 1. 

\documentclass[11pt,border=4pt]{standalone}
\usepackage{feynmp-auto}
\begin{document}
\begin{fmffile}{feyngraph}
  \begin{fmfgraph*}(120,60) % dimensions (WH)
    % external vertices
    \fmfleft{i2,i1}
    \fmfright{o2,o1}
    % main
    \fmf{fermion,tension=2}{i1,v1} % make shorter
    \fmf{fermion}{i2,v2}
    \fmf{fermion,tension=2}{v2,o2} % make shorter
    \fmf{fermion}{v1,o1}
    \fmf{photon,tension=0.5}{v1,v2} % make longer
  \end{fmfgraph*}
\end{fmffile}
\end{document}
A simple example of tension

(Note that the figure on the left column has some extra labels highlighter in red for illustrative purposes. Press the big compile button to get the actual result of the code example.)

Internal vertices

Using tension to pull internal vertices

The goal is to make the fermion lines from the boson decay equal in length:

A simple example of tension
External vertices

Step 1

\documentclass[11pt,border=4pt]{standalone}
\usepackage{feynmp-auto}
\begin{document}
\begin{fmffile}{feyngraph}
  \begin{fmfgraph*}(120,120) % dimensions (WH)
    % external vertices
    \fmfleft{i1}
    \fmfright{o4,o3,o2,o1}
    % main
    \fmf{fermion}{o1,v1,o2}
    \fmf{fermion}{o3,v2,o4}
    \fmf{dashes,tension=1.5}{i1,v}
    % decay
    \fmf{boson}{v,v1}
    \fmf{boson}{v,v2}
  \end{fmfgraph*}
\end{fmffile}
\end{document}
An example of how to use tension

Step 2

\documentclass[11pt,border=4pt]{standalone}
\usepackage{feynmp-auto}
\begin{document}
\begin{fmffile}{feyngraph}
  \begin{fmfgraph*}(120,120) % dimensions (WH)
    % external vertices
    \fmfstraight
    \fmfleft{i2,i1,i0}
    \fmfright{o4,o3,o2,o1}
    % skeleton
    \fmf{dashes,foreground=(1,,.6,,.6)}{i0,v1}
    \fmf{dashes,foreground=(1,,.6,,.6),tension=0.4}{v1,v2}
    \fmf{dashes,foreground=(1,,.6,,.6)}{i2,v2}
    % main
    \fmf{fermion}{o1,v1,o2}
    \fmf{fermion}{o3,v2,o4}
  \end{fmfgraph*}
\end{fmffile}
\end{document}
An example of how to use tension

Step 3

\documentclass[11pt,border=4pt]{standalone}
\usepackage{feynmp-auto}
\begin{document}
\begin{fmffile}{feyngraph}
  \begin{fmfgraph*}(120,120) % dimensions (WH)
    % external vertices
    \fmfstraight
    \fmfleft{i2,i1,i0}
    \fmfright{o4,o3,o2,o1}
    % skeleton
    \fmf{dashes,foreground=(1,,.7,,.7)}{i0,v1}
    \fmf{dashes,foreground=(1,,.7,,.7),tension=0.4}{v1,v2}
    \fmf{dashes,foreground=(1,,.7,,.7)}{i2,v2}
    % main
    \fmf{fermion}{o1,v1,o2}
    \fmf{fermion}{o3,v2,o4}
    \fmffreeze
    \fmf{dashes,tension=1.5}{i1,v}
    % decay
    \fmf{boson}{v,v1}
    \fmf{boson}{v,v2}
  \end{fmfgraph*}
\end{fmffile}
\end{document}
An example of how to use tension

Using \fmfforce for exact placement

Full code

The example figures above were generated with the following code: …