# Helicity in pion decay

Study of helicity (handedness) and spin correlations in pion decay, and parity violation.
See the “decay” tag for more similar figures.

Edit and compile if you like:

% Author: Izaak Neutelings (March 2020)
\documentclass[border=3pt,tikz]{standalone}
\usepackage{amsmath} % for \dfrac
\usepackage{mathabx} % for \Earth
\usepackage{bm} % \bm
\usepackage{physics}
\usepackage{tikz}
\usetikzlibrary{arrows.meta}
\usetikzlibrary{calc}
\tikzset{>=latex} % for LaTeX arrow head
\usepackage{xcolor}
\colorlet{vcol}{red!50!black}
\colorlet{veccol}{green!50!black}
\colorlet{Icol}{blue!70!black}
\colorlet{wcol}{orange!90!black}
\colorlet{Scol}{green!60!black}
\tikzstyle{current}=[->,Icol] %thick,
\tikzstyle{force}=[->,very thick,BFcol]
\tikzstyle{velocity}=[->,thick,vcol]
\tikzstyle{spin}=[->,very thick,Scol]
\tikzstyle{charge+}=[very thin,draw=black,top color=red!50,bottom color=red!90!black,shading angle=20,circle,inner sep=0.2]
\tikzstyle{charge-}=[very thin,draw=black,top color=blue!50,bottom color=blue!80,shading angle=20,circle,inner sep=0.2]
\tikzstyle{charge0}=[very thin,draw=black,top color=green!60!black!40,bottom color=green!60!black!80,shading angle=20,circle,inner sep=0.2]
\tikzstyle{vector}=[->,thick,veccol]
\tikzset{
pics/spin/.style={
code={
\def\L{0.38}
\draw[-{Latex[length=3,width=2.5]},pic actions,rotate=#1,line width=0.9,Scol] (-\L/2,0) --++ (\L,0);
\draw[pic actions,rotate=#1,thin,white]
(-0.15*\L,0)++(170:{0.16*\L} and {0.22*\L}) arc (170:190:{0.16*\L} and {0.22*\L});
\draw[-{Latex[length=1.2,width=1]},pic actions,rotate=#1,very thin]
(-0.15*\L,0)++(25:{0.16*\L} and {0.22*\L}) arc (25:305:{0.16*\L} and {0.22*\L}) --++ (50:0.09*\L);
}},
pics/spin/.default=90,
}

\begin{document}

% PION DECAY
\begin{tikzpicture}
\def\L{1.0}
\def\v{0.9*\L}
\coordinate (O) at (0,0);
\coordinate (L) at (-0.85*\L,0);
\coordinate (R) at (\L,0);
\coordinate (LS) at ($(L)+(-0.45*\v,0.15*\L)$);
\coordinate (RS) at ($(R)+( 0.50*\v,0.15*\L)$);

% VECTORS
\draw[->,velocity] (R)++(0.1*\L,0) --++ (\v,0);
\draw[->,velocity] (L) --++ (-\v,0);
\pic at (RS) {spin={180}};
\pic at (LS) {spin={  0}};
%\node[vcol,above=5,scale=0.8] at (O) {$S_\pi = 0$};

% PARTICLES
\draw[charge0] (L) circle (0.07*\L);
\node[charge+,scale=1.00] at (O) {$+$};
\node[charge+,scale=0.78] at (R) {$+$};
\node at ($(L)+(0,-0.45*\L)$) {\strut$\nu_\mu$};
\node at ($(O)+(0,-0.45*\L)$) {\strut$\pi^+$};
\node[right=-8] at ($(R)+(0,-0.45*\L)$) {\strut$\mu^+$};

\end{tikzpicture}

% MU DECAY - maximal positron energy
\begin{tikzpicture}
\def\L{0.8}
\def\v{1.2*\L}
\coordinate (O) at (0,0);
\coordinate (LT) at (-\L, 0.18*\L);
\coordinate (LB) at (-\L,-0.18*\L);
\coordinate (R) at (\L,0);
\coordinate (T) at (0,0.42*\L);
\coordinate (LTS) at ($(LT)+(-0.3*\v, 0.18*\L)$);
\coordinate (LBS) at ($(LB)+(-0.3*\v,-0.18*\L)$);
\coordinate (RS) at ($(R)+( 0.50*\v,0.18*\L)$);

% VECTORS
\draw[->,velocity] (R)++(0.05*\L,0) --++ (\v,0);
\draw[->,velocity] (LT) --++ (-0.7*\v,0) node[black,above=2,left=-2] {\strut$\nu_\mathrm{e}$};
\draw[->,velocity] (LB) --++ (-0.7*\v,0) node[black,below=3,left=-3] {\strut$\overline{\nu}_\mu$};
\pic at (RS)  {spin={  0}};
\pic at (LTS) {spin={  0}};
\pic at (LBS) {spin={180}};
\pic at (T)   {spin={  0}};

% PARTICLES
\draw[charge0] (LT) circle (0.07*\L);
\draw[charge0] (LB) circle (0.07*\L);
\node[charge+,scale=1.00] at (O) {$+$};
\node[charge+,scale=0.60] at (R) {$+$};
\node at ($(O)+(0,-0.56*\L)$) {\strut$\mu^+$};
\node[right=-7] at ($(R)+(0,-0.56*\L)$) {\strut$\mathrm{e}^+$};

\end{tikzpicture}

% MU DECAY - minimal positron energy
\begin{tikzpicture}
\def\L{0.8}
\def\v{1.2*\L}
\coordinate (O) at (0,0);
\coordinate (LT) at (-\L, 0.18*\L);
\coordinate (LB) at (-\L,-0.18*\L);
\coordinate (R) at (\L,0);
\coordinate (T) at (0,0.42*\L);
\coordinate (LTS) at ($(LT)+(-0.4*\v, 0.18*\L)$);
\coordinate (LBS) at ($(LB)+(-0.3*\v,-0.18*\L)$);
\coordinate (RS) at ($(R)+( 0.4*\v,0.18*\L)$);

% VECTORS
\draw[->,velocity] (R)++(0.05*\L,0) --++ (0.9*\v,0) node[black,right=-1] {\strut$\overline{\nu}_\mu$};
\draw[->,velocity] (LT) --++ (-0.9*\v,0) node[black,above=2,left=-2] {\strut$\nu_\mathrm{e}$};
\draw[->,velocity] (LB) --++ (-0.6*\v,0) node[black,below=3,left=-3] {\strut$\mathrm{e}^+$};
\pic at (RS)  {spin={  0}};
\pic at (LTS) {spin={  0}};
\pic at (LBS) {spin={180}};
\pic at (T)   {spin={  0}};

% PARTICLES
\draw[charge0] (LT) circle (0.07*\L);
\draw[charge0] (R) circle (0.07*\L);
\node[charge+,scale=1.00] at (O) {$+$};
\node[charge+,scale=0.60] at (LB) {$+$};
\node at ($(O)+(0,-0.56*\L)$) {\strut$\mu^+$};

\end{tikzpicture}

\end{document}