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I'd like to typeset equations like: Inline Feynman diagrams 1 Inline Feynman diagrams 2

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.

Leonard
  • 233

1 Answers1

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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}

enter image description here

  • 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
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    @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. –  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
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    @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. –  Feb 23 '19 at 22:22
  • @marmot Thank you very much! This is a beautiful solution, though it takes a lot of typing work. I guess one could program some macros to do this. – Leonard Feb 24 '19 at 12:56
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    @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
  • @Leonard Upvote your comment and give you my best thank you. – Sebastiano Feb 24 '19 at 13:04
  • @Leonard I agree. (Some days I feel like writing a very basic TikZ library for that might be worthwhile.) What do the dots/blobs represent? –  Feb 24 '19 at 14:32
  • @marmot In the first equation they are the 2 line vertices, that is the quadratic part of the action, here denoted as L_1,2. Vasiliev uses a special kind of notation that does not differentiate between fermions and bosons (or any type of field), but in this chapter he reverts to the standard notation for fermions to discuss signs. In the second they represent the source contract with part of the external legs of the diagrams, while the others are evaluated at x_1 through x_n – Leonard Feb 24 '19 at 22:55
  • @Leonard Interesting! (I was under the impression that the solid lines should always represent the same dof, but then this does not make too much sense since fermions do not have cubic interactions. But after the explanations it does make sense. Thanks!) –  Feb 24 '19 at 23:00
  • just a comment: It is in fact possible to use the TiKZ graph drawing library also for arXiv uploads. The arXiv does not allow for LuaLaTeX, but it still works using tikzexternalize. – Julian Feb 26 '19 at 08:21
  • @Julian This is effectively the same as compiling the diagram as standalone and uploading and including the resulting pdf. Yes, that works, of course. –  Feb 26 '19 at 14:35