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Next: Conclusion and Outlook Up: Drawing Feynman Diagrams Previous: Description of the


All examples are presented with full source code, no tuning ``behind the scenes'' has been performed.

Tree Diagrams

 1 \begin{fmfgraph*}(50,30)
 2   \fmfleftn{i}{2} \fmfrightn{o}{4}
 3   \fmflabel{$e_-$}{i1}\fmflabel{$e_+$}{i2}
 4   \fmflabel{$\mu_+$}{o1}
 5   \fmflabel{$\nu_{\mu}$}{o2}
 6   \fmflabel{$s$}{o3}
 7   \fmflabel{$\noexpand\bar c$}{o4}
 8   \fmf{fermion}{i1,v1,i2}
 9   \fmf{boson,label=$\gamma,,Z$}{v1,v2}
10   \fmf{boson}{v3,v2,v4}
11   \fmf{fermion}{o1,v3,o2}
12   \fmf{fermion}{o4,v4,o3}
13   \fmfdot{v1,v3,v4}\fmfblob{.12w}{v2}
14 \end{fmfgraph*}

Figure: Resonant s-channel contribution.

 1 \begin{fmfgraph*}(50,30)
 2   \fmfleftn{i}{2}\fmfrightn{o}{4}
 3   \fmflabel{$e_-$}{i1}\fmflabel{$e_+$}{i2}
 4   \fmflabel{$\noexpand\bar c$}{o1}
 5   \fmflabel{$\nu_{\mu}$}{o2}
 6   \fmflabel{$\mu_+$}{o3}
 7   \fmflabel{$s$}{o4}
 8   \fmf{boson,label=$\gamma,,Z$}{v1,v2}
 9   \fmf{fermion}{i1,v1,i2}
10   \fmf{fermion}{o1,v2,v3,o4}
11   \fmffreeze
12   \fmf{boson}{v3,v4}\fmf{fermion}{o3,v4,o2}
13   \fmfdotn{v}{4}
14 \end{fmfgraph*}

Figure: Singly resonant contribution.

 1 \newenvironment{Zff}
 2   {\begin{fmfgraph*}(35,30)
 3      \fmfleft{Z}\fmfright{fb,f}
 4      \fmflabel{$\noexpand\bar f$}{fb}
 5      \fmflabel{$Z$}{Z}\fmflabel{$f$}{f}}
 6   {\end{fmfgraph*}}
 7 \begin{Zff}
 8   \fmf{boson}{Z,v}\fmf{fermion}{fb,v,f}
 9   \fmfdot{v}
10 \end{Zff}
11 \begin{Zff}
12   \fmf{boson}{Z,v1}\fmf{boson}{v2,v3}
13   \fmf{fermion,left,tension=.5}{v1,v2,v1}
14   \fmf{fermion}{fb,v3,f}\fmfdotn{v}{3}
15 \end{Zff}\\ [3\baselineskip]
16 \begin{Zff}
17   \fmf{boson}{Z,Zv}
18   \fmf{fermion}{fb,fbv,Zv,fv,f}
19   \fmffreeze
20   \fmf{boson}{fbv,fv}\fmfdot{Zv,fbv,fv}
21 \end{Zff}
22 \begin{Zff}
23   \fmf{boson}{Z,Zv}\fmf{boson}{fbv,Zv,fv}
24   \fmf{fermion}{fb,fbv}\fmf{fermion}{fv,f}
25   \fmffreeze
26   \fmf{fermion}{fbv,fv}\fmfdot{Zv,fbv,fv}
27 \end{Zff}

Figure: Z-decay at tree level and at one loop.

Let us start with a view into the immediate future: figure gif shows the s-channel diagram for the production of W's at LEP2 with semileptonic decay. The starred form of the |fmfgraph| environment is used because we want to include labels for vertices and arcs. Its arguments are the width and height of the diagram, measured in |.

In line 2 we declare two incoming particles and four outgoing particles, which will be placed on the left or right side of the diagram, respectively. These vertices are labelled in lines 3--7. On line 7, the | command has been protected from premature expansion by |. This is similar to LaTeX's moving arguments, but necessary for some robust commands too. If in doubt, protect all commands; it will never hurt. Line 8 connects the incoming fermions with the first inner vertex. This vertex is connected in line 9 with the vertex. Here it should be noted that the line style (|boson| in this case, which is an alias for |wiggly|) takes comma-separated options. Here we add a label with the |label| option. Inside the argument of an option, commas have to be doubled to distinguish them from option separators.

 1 In supersymmetrical field theories, the
 2 quadratic divergences cancel:
 3 \begin{equation}
 4   \parbox{20mm}{\begin{fmfgraph}(20,15)
 5     \fmfleft{i} \fmfright{o}
 6     \fmf{dashes}{i,v,v,o}
 7   \end{fmfgraph}}
 8   + \parbox{20mm}{\begin{fmfgraph}(20,15)
 9     \fmfleft{i} \fmfright{o}
10     \fmf{dashes}{i,v1} \fmf{dashes}{v2,o}
11     \fmf{fermion,left,tension=.3}{v1,v2,v1}
12  \end{fmfgraph}}
13  = \ln\Lambda^2
14 \end{equation}

Figure: Graphical equations can be created easily, the '134 parbox is used for centering the diagram vertically.

1 \begin{fmfgraph}(50,20)
2   \fmfleft{w}\fmfright{e}
3   \fmf{boson}{w,vw}\fmf{boson}{ve,e}
4   \fmf{fermion,tension=.5}{vw,vn,ve,vs,vw}
5   \fmf{gluon}{vn,vs}
6   \fmffixed{(0,h)}{vn,vs}
7   \fmfdot{vw,vn,ve,vs}
8 \end{fmfgraph}

Figure: contribution to the Z-selfenergy, both in a graph mode rendition and and in an improved version that needs immediate mode commands.

The rest is now straightforward: lines 10--12 connect the remaining vertices with W's and fermions and line 13 add dots and a blob at the respective vertices. The width of the blob is specified in the first argument of | to be 12% of the total width of the diagram, |6mm| would be as good in this example.

Figure gif is a variation on that theme, this time showing a singly resonant contribution to the same final state. Lines 3--7 are identical to the preceding diagram, but permute the flavours of the final states. In lines 8--10 the outer string of fermion lines is connected to the incoming particles.

Line 11 uses the important | command for the first time. It will fix the layout of the vertices that have been entered so far, producing a skeleton for the vertices entered later. If we would no do this, the W and fermions entered in line 12 would pull the diagram over to the right too much (try this by processing the diagram with and without the | yourself!).

Loop Diagrams

Returning to the physics currently produced at LEP, we show the tree-level and one-loop diagrams contributing to the decay of Z's into two fermions in figure gif. Since we will have to produce four diagrams with identical external particles, it is convenient to define an environment in lines 1--6 that will take care of the repetitive tasks. The rest of the example should be almost self-explanatory by now. Let me just point to the | commands in lines 19 and 25 that are used to obtain a straight continuation of the external arcs into the triangles.

Line 13 uses the |tension| option to reduce the tension of the ares in the fermion loop, effectively blowing up the loop from its default size. Also new is the |left| option that instructs not to draw the arc on a straight line, but to take a left detour instead.


Feynman diagrams can be used everywhere in a LaTeX document, in particular in centered | for creating graphical equations like the one in figure gif. The only new feature in this figure is the arc connecting a vertex with itself in line 6. Such lines do not affect the layout and will draw them into the biggest available gap.

Multi-loop Diagrams

Multi-loop diagrams can pose special challenges: in figure gif two renditions of a contribution to the Z-selfenergy are shown. The first uses graph mode commands exclusively but its visual appearance is certainly inferior to the second version that uses immediate mode commands to generate arcs that are more generally curved than half-circles. The full source code of this version will be discussed in the next issue of the CNL.

In the graph mode version, we note the first use of the | command which is used to specify a linear constraint (cf. (gif)). If this command were not be used, the loop would collapse, therefore we force it open by demanding that the upper vertex |vn| be exactly the full height of the diagram above the lower vertex |vs|: .

Figure: A ``penguin'' diagram in fancy layout and a Higgs production process. These examples will be discussed in the next article of this series.

next up previous
Next: Conclusion and Outlook Up: Drawing Feynman Diagrams Previous: Description of the

Janne Saarela
Fri Jun 23 10:54:20 METDST 1995