# Feynman diagrams

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 Ti*k*Z-Feynman has been published which uses Ti*k*Z in order to generate Feynman diagrams.

# Ti*k*Z-Feynman

Ti*k*Z-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 Ti*k*Z package and its graph drawing algorithms in order to automate the placement of many vertices. Ti*k*Z-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 Ti*k*Z-Feynman will always be on the project page and in the package documentation on CTAN.

Currently, Ti*k*Z-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.

## Loading the Package

After installing the package, the Ti*k*Z-Feynman package can be loaded with `\usepackage{tikz-feynman}`

in the preamble. It is recommend that you also specify the version of Ti*k*Z-Feynman to use with the `compat`

package option: `\usepackage[compat=1.0.0]{tikz-feynman}`

. This ensures that any new versions of Ti*k*Z-Feynman do not produce any undesirable changes without warning.

## A First Diagram

Feynman diagrams can be declared with the `\feynmandiagram`

command. It is analogous to the `\tikz`

command from Ti*k*Z 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:

- Line 1
`\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.- Line 2
- The left fermion line is drawn by declaring three vertices (
`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. - Line 3
- This edge connects vertices
`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`

. - Line 4
- This line is analogous to line 2 and introduces two new vertices,
`f1`

and`f2`

. It re-uses the previously labelled`b`

vertex. - Line 5
- Finish the declaration of the Feynman diagram. The final semi-colon (
`;`

) 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, Ti*k*Z-Feynman will default to a more rudimentary algorithm and will warn the user instead.

## Adding Styles

So far, the examples have only used the `photon`

and `fermion`

styles. The Ti*k*Z-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 Ti*k*Z 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 Ti*k*Z provides, have a look at the Ti*k*Z manual; it is extremely thorough and provides many usage examples.

## When the Algorithm Isn't Enough

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:

- Add invisible edges
- While still using the default algorithm, it is possible to force certain vertices to be closer together by adding extra edges and making them invisible through
`draw=none`

. The algorithm will treat these extra edges in the same way, but they are simply not drawn at the end; - Use a different algorithm
- In some circumstances, other algorithms may be better suited. Some of the other graph layout algorithms are listed in the package documentation, and an exhaustive list of all algorithms and their parameters is given in the Ti
*k*Z manual; - Manual placement
- As a last resort, very complicated or unusual diagrams will require each vertex to be manually placed.

### Invisible Edges

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,
};
```

### Alternative Algorithms

The graph drawing library from Ti*k*Z 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).

### Manual Placement

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. Ti*k*Z-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 Ti*k*Z-Feynman and declares additional Ti*k*Z-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 Ti*k*Z, 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 Ti*k*Z 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}
```

# FeynMF

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.

## Introduction

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.

## Vertices

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}
```

## Labels

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.

## Line styles

We've seen the 'photon' and 'fermion' line styles above, but the *feynmf* package support many more.

# Further Reading

For more information see:

## Overleaf guides

- Creating a document in Overleaf
- Uploading a project
- Copying a project
- Creating a project from a template
- Including images in Overleaf
- Exporting your work from Overleaf
- Working offline in Overleaf
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- Debugging Compilation timeout errors
- How-to guides

## LaTeX Basics

- Creating your first LaTeX document
- Choosing a LaTeX Compiler
- Paragraphs and new lines
- Bold, italics and underlining
- Lists
- Errors

## Mathematics

- Mathematical expressions
- Subscripts and superscripts
- Brackets and Parentheses
- Fractions and Binomials
- Aligning Equations
- Operators
- Spacing in math mode
- Integrals, sums and limits
- Display style in math mode
- List of Greek letters and math symbols
- Mathematical fonts

## Figures and tables

- Inserting Images
- Tables
- Positioning Images and Tables
- Lists of Tables and Figures
- Drawing Diagrams Directly in LaTeX
- TikZ package

## References and Citations

- Bibliography management in LaTeX
- Bibliography management with biblatex
- Biblatex bibliography styles
- Biblatex citation styles
- Bibliography management with natbib
- Natbib bibliography styles
- Natbib citation styles
- Bibliography management with bibtex
- Bibtex bibliography styles

## Languages

- Multilingual typesetting on Overleaf using polyglossia and fontspec
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## Document structure

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## Formatting

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- Multiple columns
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- Using colours in LaTeX
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## Fonts

## Presentations

## Commands

## Field specific

- Theorems and proofs
- Chemistry formulae
- Feynman diagrams
- Molecular orbital diagrams
- Chess notation
- Knitting patterns
- CircuiTikz package
- Pgfplots package
- Typing exams in LaTeX
- Knitr
- Attribute Value Matrices

## Class files

- Understanding packages and class files
- List of packages and class files
- Writing your own package
- Writing your own class
- Tips