Feynman diagrams provide a very compact and intuitive way of representing interactions between particles. These diagrams can be included into LaTeX documents thanks to a few packages. One of the older packages is feynmf
which uses MetaPost in order to generate the diagrams. More recently, a new package called TikZ-Feynman has been published which uses TikZ in order to generate Feynman diagrams.
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TikZ-Feynman is a LaTeX package allowing Feynman diagrams to be easily generated within LaTeX with minimal user instructions and without the need of external programs. It builds upon the TikZ package and its graph drawing algorithms in order to automate the placement of many vertices. TikZ-Feynman still allows fine-tuned placement of vertices so that even complex diagrams can be generated with ease.
The most up-to-date information for TikZ-Feynman will always be on the project page and in the package documentation on CTAN.
Currently, TikZ-Feynman is too new to have made it into ShareLaTeX's installation, but we are working to get it included soon. In the meantime, it is possible to include the package files manually in a ShareLaTeX project as shown in this template.
After installing the package, the TikZ-Feynman package can be loaded with \usepackage{tikz-feynman}
in the preamble. It is recommend that you also specify the version of TikZ-Feynman to use with the compat
package option: \usepackage[compat=1.0.0]{tikz-feynman}
. This ensures that any new versions of TikZ-Feynman do not produce any undesirable changes without warning.
Feynman diagrams can be declared with the \feynmandiagram
command. It is analogous to the \tikz
command from TikZ and requires a final semi-colon (;
) to finish the environment. For example, a simple s-channel diagram is:
\feynmandiagram [horizontal=a to b] { i1 -- [fermion] a -- [fermion] i2, a -- [photon] b, f1 -- [fermion] b -- [fermion] f2, };
Let's go through this example line by line:
\feynmandiagram
introduces the Feynman diagram and allows for optional arguments to be given in the brackets [<options>]
. In this instance, horizontal=a to b
orients the algorithm outputs such that the line through vertices a
and b
is horizontal.
i1
, a
and i2
) and connecting them with edges --
. Just like the \feynmandiagram
command above, each edge also take optional arguments specified in brackets [<options>]
. In this instance, we want these edges to have arrows to indicate that they are fermion lines, so we add the fermion
style to them. As you will see later on, optional arguments can also be given to the vertices in exactly the same way.
a
and b
with an edge styled as a photon. Since there is already a vertex labelled a
, the algorithm will connect it to a new vertex labeled b
.
f1
and f2
. It re-uses the previously labelled b
vertex.
;
) is important.
The name given to each vertex in the graph does not matter. So in this example, i1
, i2
denote the initial particles; f1
, f2
denotes the final particles; and a
, b
are the end points of the propagator. The only important aspect is that what we called a
in line 2 is also a
in line 3 so that the underlying algorithm treats them as the same vertex.
The order in which vertices are declared does not matter as the default algorithm re-arranges everything. For example, one might prefer to draw the fermion lines all at once, as with the following example (note also that the way we named vertices is completely different):
\feynmandiagram [horizontal=f2 to f3] { f1 -- [fermion] f2 -- [fermion] f3 -- [fermion] f4, f2 -- [photon] p1, f3 -- [photon] p2, };
As a final remark, the calculation of where vertices should be placed is usually done through an algorithm written in Lua. As a result, LuaTeX is required in order to make use of these algorithms. If LuaTeX is not used, TikZ-Feynman will default to a more rudimentary algorithm and will warn the user instead.
So far, the examples have only used the photon
and fermion
styles. The TikZ-Feynman package comes with quite a few extra styles for edges and vertices which are all documented over in the package documentation. For example, it is possible to add momentum arrows with momentum=<text>
, and in the case of end vertices, the particle can be labelled with particle=<text>
. To demonstrate how they are used, we take the generic s-channel diagram from earlier and make it a electron-positron pairs annihilating into muons:
\feynmandiagram [horizontal=a to b] { i1 [particle=\(e^{-}\)] -- [fermion] a -- [fermion] i2 [particle=\(e^{+}\)], a -- [photon, edge label=\(\gamma\), momentum'=\(k\)] b, f1 [particle=\(\mu^{+}\)] -- [fermion] b -- [fermion] f2 [particle=\(\mu^{-}\)], };
In addition to the style keys documented below, style keys from TikZ can be used as well:
\feynmandiagram [horizontal=a to b] { i1 [particle=\(e^{-}\)] -- [fermion, very thick] a -- [fermion, opacity=0.2] i2 [particle=\(e^{+}\)], a -- [red, photon, edge label=\(\gamma\), momentum'={[arrow style=red]\(k\)}] b, f1 [particle=\(\mu^{+}\)] -- [fermion, opacity=0.2] b -- [fermion, very thick] f2 [particle=\(\mu^{-}\)], };
For a list of all the various styles that TikZ provides, have a look at the TikZ manual; it is extremely thorough and provides many usage examples.
By default, the \feynmandiagram
and \diagram
commands use the spring layout
algorithm to place all the edges. The spring layout
algorithm attempts to `spread out' the diagram as much as possible which—for most simpler diagrams—gives a satisfactory result; however in some cases, this does not produce the best diagram and this section will look at alternatives. There are three main alternatives:
draw=none
. The algorithm will treat these extra edges in the same way, but they are simply not drawn at the end;
The underlying algorithm treats all edges in exactly the same way when calculating where to place all the vertices, and the actual drawing of the diagram (after the placements have been calculated) is done separately. Consequently, it is possible to add edges to the algorithm, but prevent them from being drawn by adding draw=none
to the edge style.
This is particularly useful if you want to ensure that the initial or final states remain closer together than they would have otherwise as illustrated in the following example (note that opacity=0.2
is used instead of draw=none
to illustrate where exactly the edge is located).
% No invisible to keep the two photons together \feynmandiagram [small, horizontal=a to t1] { a [particle=\(\pi^{0}\)] -- [scalar] t1 -- t2 -- t3 -- t1, t2 -- [photon] p1 [particle=\(\gamma\)], t3 -- [photon] p2 [particle=\(\gamma\)], };
% Invisible edge ensures photons are parallel \feynmandiagram [small, horizontal=a to t1] { a [particle=\(\pi^{0}\)] -- [scalar] t1 -- t2 -- t3 -- t1, t2 -- [photon] p1 [particle=\(\gamma\)], t3 -- [photon] p2 [particle=\(\gamma\)], p1 -- [opacity=0.2] p2, };
The graph drawing library from TikZ has several different algorithms to position the vertices. By default, \diagram
and \feynmandiagram
use the spring layout
algorithm to place the vertices. The spring layout
attempts to spread everything out as much as possible which, in most cases, gives a nice diagram; however, there are certain cases where this does not work. A good example where the spring layout
doesn't work are decays where we have the decaying particle on the left and all the daughter particles on the right.
% Using the default spring layout \feynmandiagram [horizontal=a to b] { a [particle=\(\mu^{-}\)] -- [fermion] b -- [fermion] f1 [particle=\(\nu_{\mu}\)], b -- [boson, edge label=\(W^{-}\)] c, f2 [particle=\(\overline \nu_{e}\)] -- [fermion] c -- [fermion] f3 [particle=\(e^{-}\)], };
% Using the layered layout \feynmandiagram [layered layout, horizontal=a to b] { a [particle=\(\mu^{-}\)] -- [fermion] b -- [fermion] f1 [particle=\(\nu_{\mu}\)], b -- [boson, edge label'=\(W^{-}\)] c, c -- [anti fermion] f2 [particle=\(\overline \nu_{e}\)], c -- [fermion] f3 [particle=\(e^{-}\)], };
You may notice that in addition to adding the layered layout
style to \feynmandiagram
, we also changed the order in which we specify the vertices. This is because the layered layout
algorithm does pay attention to the order in which vertices are declared (unlike the default spring layout
); as a result, c--f2, c--f3
has a different meaning to f2--c--f3
. In the former case, f2
and f3
are both on the layer below c
as desired; whilst the latter case places f2
on the layer above c
(that, the same layer as where the W-boson originates).
In more complicated diagrams, it is quite likely that none of the algorithms work, no matter how many invisible edges are added. In such cases, the vertices have to be placed manually. TikZ-Feynman allows for vertices to be manually placed by using the \vertex
command.
The \vertex
command is available only within the feynman
environment (which itself is only available inside a tikzpicture
). The feynman
environment loads all the relevant styles from TikZ-Feynman and declares additional TikZ-Feynman-specific commands such as \vertex
and \diagram
. This is inspired from PGFPlots and its use of the axis
environment.
The \vertex
command is very much analogous to the \node
command from TikZ, with the notable exception that the vertex contents are optional; that is, you need not have {<text>}
at the end. In the case where {}
is specified, the vertex automatically is given the particle
style, and otherwise it is a usual (zero-sized) vertex.
To specify where the vertices go, it is possible to give explicit coordinates though it is probably easiest to use the positioning
library from TikZ which allows vertices to be placed relative to existing vertices. By using relative placements, it is possible to easily tweak one part of the graph and everything will adjust accordingly—the alternative being to manually adjust the coordinates of every affected vertex.
Finally, once all the vertices have been specified, the \diagram*
command is used to specify all the edges. This works in much the same way as \diagram
(and also \feynmandiagram
), except that it uses an very basic algorithm to place new nodes and allows existing (named) nodes to be included. In order to refer to an existing node, the node must be given in parentheses.
This whole process of specifying the nodes and then drawing the edges between them is shown below for the muon decay:
\begin{tikzpicture} \begin{feynman} \vertex (a) {\(\mu^{-}\)}; \vertex [right=of a] (b); \vertex [above right=of b] (f1) {\(\nu_{\mu}\)}; \vertex [below right=of b] (c); \vertex [above right=of c] (f2) {\(\overline \nu_{e}\)}; \vertex [below right=of c] (f3) {\(e^{-}\)}; \diagram* { (a) -- [fermion] (b) -- [fermion] (f1), (b) -- [boson, edge label'=\(W^{-}\)] (c), (c) -- [anti fermion] (f2), (c) -- [fermion] (f3), }; \end{feynman} \end{tikzpicture}
The feynmf package lets you easily draw Feynman diagrams in your LaTeX documents. All you need to do is specify the vertices, the particles and the labels, and it will automatically layout and draw your diagram for you.
Let's start with a quick example:
\begin{fmffile*}{diagram} \begin{fmfgraph}(40,25) \fmfleft{i1,i2} \fmfright{o1,o2} \fmf{fermion}{i1,v1,o1} \fmf{fermion}{i2,v2,o2} \fmf{photon}{v1,v2} \end{fmfgraph} \end{fmffile*}
Open this example in ShareLaTeX
The fmffile* environment must be put around all of your Feynman diagrams. You can use fmffile environment for multiple diagrams, so you can put one around your whole document and forget about it. The second argument to the fmffile environment tells LaTeX where to write the files that it uses to store the diagram. You can name this whatever you want, but you need to run metafont on your diagram between LaTeX runs in order for your diagram to show up (ShareLaTeX does this automatically):
pdflatex feynmf.tex mf '\mode:=laserjet; input diagram' pdflatex feynmf.tex
The 'fmfgraph' environment starts a Feynman diagram, and the figures in brackets afterwards specify the width and height of the diagram.
The first thing you need to do is specify your external vertices, and where they should be positioned. You can name your vertices anything you like, and say where they should be positioned with the commands \fmfleft, \fmfright, \fmftop, \fmfbottom. For example
% Creates two vertices on the left called i1 and i2 \fmfleft{i1,i2} % Creates two vertices on the right called o1 and o2 \fmfright{o1,o2}
You can connect up vertices with the \fmf, which will create new vertices if you pass in names that haven't been created yet. For example
% Will create a fermion line between i1 and % the newly created v1, and between v1 and o1. \fmf{fermion}{i1,v1,o1} % Will create a photon line between v1 and the newly created v2 \fmf{photon}{v2,v2}
A vertex can be labelled using the \fmflabel command, which takes two arguments: the label to apply to the vertex, and the name of the vertex to apply it to. For example, in the above diagram, if we add in the following labels, we get the updated diagram below:
Open this example in ShareLaTeX
Note that math mode can used inside the vertex labels, as we have done above.
We've seen the 'photon' and 'fermion' line styles above, but the feynmf package support many more.
For more information see: