# Inline Feynman diagrams, Feynman diagrams in equations, very small Feynman diagrams

I'd like to typeset equations like:

I have tried this with the tikz-feynman library, but the diagrams generated by it are just way too large, even with the small option (and also just look awkward). Optimally, I want to type simple diagrams even inline with the text, so that I can avoid awkwardly describing the diagram or using a lot of space and breaking the flow of the document to display the diagram.

• Welcome to TeX.SE. I like very much your question. Good LaTeX. – Sebastiano Feb 23 '19 at 10:44
• Using normal inline TikZ by \tikz may be a solution. – user156344 Feb 23 '19 at 10:55

AFAIK you do not get bent arrows with tikz-feynman. And since you seem not to need the graph drawing algorithms (and since they cannot be uploaded to the arXv), you may just work with plain TikZ.

\documentclass[fleqn]{article}
\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{tikz}
\usetikzlibrary{arrows.meta,bending,decorations.markings}
% from https://tex.stackexchange.com/a/430239/121799
\tikzset{% inspired by https://tex.stackexchange.com/a/316050/121799
arc arrow/.style args={%
to pos #1 with length #2}{
decoration={
markings,
mark=at position 0 with {\pgfextra{%
\pgfmathsetmacro{\tmpArrowTime}{#2/(\pgfdecoratedpathlength)}
\xdef\tmpArrowTime{\tmpArrowTime}}},
mark=at position {#1-\tmpArrowTime} with {\coordinate(@1);},
mark=at position {#1-2*\tmpArrowTime/3} with {\coordinate(@2);},
mark=at position {#1-\tmpArrowTime/3} with {\coordinate(@3);},
mark=at position {#1} with {\coordinate(@4);
\draw[-{Triangle[length=#2,bend]}]
(@1) .. controls (@2) and (@3) .. (@4);},
},
postaction=decorate,
},
fermion arc arrow/.style={arc arrow=to pos #1 with length 2.5mm},
Vertex/.style={fill,circle,inner sep=1.5pt},
insert vertex/.style={decoration={
markings,
mark=at position #1 with {\node[Vertex]{};},
},
postaction=decorate}
}
\DeclareMathOperator{\tr}{tr}
\begin{document}

$\mathscr{P}(\varphi)=-\sum\limits_{n=1}^\infty\tr\left(\Delta L_{12}\right)^n =\vcenter{\hbox{\begin{tikzpicture} \draw[thick,insert vertex=0,fermion arc arrow={0.55}] (0,0) arc(270:-90:0.6); \end{tikzpicture}}}+\frac{1}{2} \vcenter{\hbox{\begin{tikzpicture} \draw[thick,insert vertex/.list={0,0.5}](0,0) arc(270:-90:0.6); \draw[fermion arc arrow/.list={0.3,0.8}] (0,0) arc(270:-90:0.6); \end{tikzpicture}}} +\frac{1}{3} \vcenter{\hbox{\begin{tikzpicture} \draw[thick,insert vertex/.list={0,1/3,2/3}](0,0) arc(270:-90:0.6); \draw[fermion arc arrow/.list={0.21,0.55,0.88}] (0,0) arc(270:-90:0.6); \end{tikzpicture}}}+\dots\;.$

$G(x_1,\dots x_n)=\sum\limits_{m=0}^\infty\frac{1}{m!} \begin{tikzpicture}[baseline={(X.base)}] \node[circle,draw,thick,inner sep=2pt] (X) at (0,0) {n+m}; \foreach \X in {60,90,120} {\draw[thick] (\X:0.6) -- (\X:0.9) node[Vertex]{};} \foreach \X in {-60,-80,-100,-120} {\draw[thick] (\X:0.6) -- (\X:0.9);} \node[rotate=-30,overlay] at (-120:1.1){x_1}; \node[rotate=30,overlay] at (-60:1.1){x_n}; \node at (-90:1.1){\cdots}; \end{tikzpicture}$
\end{document}


• Excuse me very much for this opinion. Can you reduce the size of the three circles? Perhaps the image of the series is more beautiful to see. PS: But what is the matter of this argument in Physics? – Sebastiano Feb 23 '19 at 22:11
• @Sebastiano You can control the appearance by adjusting Vertex/.style={fill,circle,inner sep=1.5pt},. I do not know what this diagram is. If the propagators were dashed, there would be a resemblance to the loops that one has to compute for the Coleman-Weinberg potential. – user121799 Feb 23 '19 at 22:15
• I like very much your work. I not known the Coleman-Weinberg potential. Thank you very much. It will be between my favorities. – Sebastiano Feb 23 '19 at 22:17
• @Sebastiano To the best of my knowledge the term "dimensional transmutation" has been coined in the paper by Coleman and (Eric) Weinberg. It is a rather important observation. Whether or not the above has anything to do with it I do not know. – user121799 Feb 23 '19 at 22:22
• @Sebastiano The second equation is just expressing the greenfunctions of a QFT in terms of a sum over all diagrams. The first is an intermediate step in a proof about the signs of feynman diagrams with fermionic fields. Both are taken from the excellent book "Functional methods in quantum field theory and statistical physics" by Vasiliev. – Leonard Feb 24 '19 at 13:03