\documentclass[11pt]{article}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{physics}

\usepackage{feynmp}
\DeclareGraphicsRule{*}{mps}{*}{}

\begin{document}

\begin{center}
	{\Huge \sc Feynman diagrams}
\end{center}

\noindent
Definition of RPA self-energy
\begin{align*}
\frac{\Sigma_s^\mathrm{RPA}(\vb{k}, i \omega_n^F)}{-\hbar}
&\equiv \:\:
\begin{fmffile}{sigma1order}
	\begin{fmfgraph*}(30,20)
		\fmfipair{i,o}
		\fmfiequ{i}{(0,0h)}
		\fmfiequ{o}{(w,0h)}
		\fmfi{boson}{i -- o}
		\fmfi{fermion}{i{up} .. tension 2 .. {down}o}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
	\end{fmfgraph*}
\end{fmffile}
\:\: + \:\:
\begin{fmffile}{sigma2order}
	\begin{fmfgraph*}(55,20)
		\fmfipair{i,a,b,o}
		\fmfiequ{i}{(0,0h)}
		\fmfiequ{a}{(0.33w,0h)}
		\fmfiequ{b}{(0.67w,0h)}
		\fmfiequ{o}{(w,0h)}
		\fmfi{boson}{i -- a}
		\fmfi{fermion}{a{up} .. tension 1.5 .. {down}b}
		\fmfi{fermion}{b{down} .. tension 1.5 .. {up}a}
		\fmfi{boson}{b -- o}
		\fmfi{fermion}{i{up} .. tension 2 .. {down}o}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{a}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{b}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
	\end{fmfgraph*}
\end{fmffile}
\:\: + \:\:
\begin{fmffile}{sigma3order}
	\begin{fmfgraph*}(90,20)
		\fmfipair{i,a,b,c,d,o}
		\fmfiequ{i}{(0,0h)}
		\fmfiequ{a}{(0.2w,0h)}
		\fmfiequ{b}{(0.4w,0h)}
		\fmfiequ{c}{(0.6w,0h)}
		\fmfiequ{d}{(0.8w,0h)}
		\fmfiequ{o}{(w,0h)}
		\fmfi{boson}{i -- a}
		\fmfi{fermion}{a{up} .. tension 1.5 .. {down}b}
		\fmfi{fermion}{b{down} .. tension 1.5 .. {up}a}
		\fmfi{boson}{b -- c}
		\fmfi{fermion}{c{up} .. tension 1.5 .. {down}d}
		\fmfi{fermion}{d{down} .. tension 1.5 .. {up}c}
		\fmfi{boson}{d -- o}
		\fmfi{fermion}{i{up} .. tension 3 .. {down}o}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{a}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{b}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{c}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{d}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
	\end{fmfgraph*}
\end{fmffile}
\:\: + \: \cdots
\\
&= \:\:
\begin{fmffile}{sigmashort}
	\begin{fmfgraph*}(40,20)
		\fmfipair{i,o}
		\fmfiequ{i}{(0,0h)}
		\fmfiequ{o}{(w,0h)}
		\fmfi{dbl_wiggly}{i -- o}
		\fmfi{fermion}{i{up} .. tension 2 .. {down}o}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
	\end{fmfgraph*}
\end{fmffile}
\end{align*}
Definition of screened interaction
\begin{align*}
\frac{W^\mathrm{RPA}(\vb{k}, i \omega_n^B)}{-\hbar}
= \:\:
\begin{fmffile}{intshort}
	\begin{fmfgraph*}(25,20)
		\fmfipair{i,o}
		\fmfiequ{i}{(0,0h)}
		\fmfiequ{o}{(w,0h)}
		\fmfi{dbl_wiggly}{i -- o}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
	\end{fmfgraph*}
\end{fmffile}
\:\: &\equiv \:\:
\begin{fmffile}{int1order}
	\begin{fmfgraph*}(20,20)
		\fmfipair{i,o}
		\fmfiequ{i}{(0,0h)}
		\fmfiequ{o}{(w,0h)}
		\fmfi{boson}{i -- o}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
	\end{fmfgraph*}
\end{fmffile}
\:\: + \:\:
\begin{fmffile}{int2order}
	\begin{fmfgraph*}(48,20)
		\fmfipair{i,a,b,o}
		\fmfiequ{i}{(0,0h)}
		\fmfiequ{a}{(0.33w,0h)}
		\fmfiequ{b}{(0.67w,0h)}
		\fmfiequ{o}{(w,0h)}
		\fmfi{boson}{i -- a}
		\fmfi{fermion}{a{up} .. tension 1.5 .. {down}b}
		\fmfi{fermion}{b{down} .. tension 1.5 .. {up}a}
		\fmfi{boson}{b -- o}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{a}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{b}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
	\end{fmfgraph*}
\end{fmffile}
\:\: + \:\:
\begin{fmffile}{int3order}
	\begin{fmfgraph*}(80,20)
		\fmfipair{i,a,b,c,d,o}
		\fmfiequ{i}{(0,0h)}
		\fmfiequ{a}{(0.2w,0h)}
		\fmfiequ{b}{(0.4w,0h)}
		\fmfiequ{c}{(0.6w,0h)}
		\fmfiequ{d}{(0.8w,0h)}
		\fmfiequ{o}{(w,0h)}
		\fmfi{boson}{i -- a}
		\fmfi{fermion}{a{up} .. tension 1.5 .. {down}b}
		\fmfi{fermion}{b{down} .. tension 1.5 .. {up}a}
		\fmfi{boson}{b -- c}
		\fmfi{fermion}{c{up} .. tension 1.5 .. {down}d}
		\fmfi{fermion}{d{down} .. tension 1.5 .. {up}c}
		\fmfi{boson}{d -- o}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{a}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{b}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{c}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{d}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
	\end{fmfgraph*}
\end{fmffile}
\:\: + \: \cdots
\end{align*}
Dyson equation for screened interaction:
\begin{align*}
\begin{fmffile}{dysonlhs}
	\begin{fmfgraph*}(30,20)
		\fmfipair{i,o}
		\fmfiequ{i}{(0,0h)}
		\fmfiequ{o}{(w,0h)}
		\fmfi{dbl_wiggly}{i -- o}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
	\end{fmfgraph*}
\end{fmffile}
\:\: &= \:\:
\begin{fmffile}{dyson1rhs}
	\begin{fmfgraph*}(30,20)
		\fmfipair{i,o}
		\fmfiequ{i}{(0,0h)}
		\fmfiequ{o}{(w,0h)}
		\fmfi{boson}{i -- o}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
	\end{fmfgraph*}
\end{fmffile}
\:\: + \:\:
\begin{fmffile}{dyson2rhs}
	\begin{fmfgraph*}(80,20)
		\fmfipair{i,a,b,o}
		\fmfiequ{i}{(0,0h)}
		\fmfiequ{a}{(0.33w,0h)}
		\fmfiequ{b}{(0.67w,0h)}
		\fmfiequ{o}{(w,0h)}
		\fmfi{boson}{i -- a}
		\fmfi{fermion}{a{up} .. tension 1.5 .. {down}b}
		\fmfi{fermion}{b{down} .. tension 1.5 .. {up}a}
		\fmfi{dbl_wiggly}{b -- o}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{a}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{b}
		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
	\end{fmfgraph*}
\end{fmffile}
\end{align*}
Pair bubble definition:
\begin{align*}
-\hbar \Pi_0(\vb{k}, i \omega_n^B)
&= \qquad\:\:
\begin{fmffile}{pairbubble}
	\begin{fmfgraph*}(30,20)
		\fmfipair{i,o}
		\fmfiequ{i}{(0,0h)}
		\fmfiequ{o}{(w,0h)}
		\fmfi{fermion,label=$\tilde{\vb{k}}\!+\!\tilde{\vb{q}}$,label.dist=3}{i{up} .. tension 1.5 .. {down}o}
		\fmfi{fermion,label=$\tilde{\vb{q}}$,label.dist=3}{o{down} .. tension 1.5 .. {up}i}
		\fmfiv{decor.shape=circle,decor.size=4,label=$\tilde{\mathbf{k}}\!\leadsto$,label.angle=160,label.dist=5}{i}
		\fmfiv{decor.shape=circle,decor.size=4,label=$\leadsto\!\tilde{\mathbf{k}}$,label.angle=20,label.dist=5}{o}
	\end{fmfgraph*}
\end{fmffile}
\end{align*}

\end{document}
