# Jet vectors & the MT2 variable

Illustration of the construction of the MT2 variable from jet and MET vectors used in SUSY searches at CMS & ATLAS, for example in arXiv:1909.03460. The figure was presented at a talk in Pheno2021.

Edit and compile if you like:

% Author: Izaak Neutelings (May 2021)
% Description:
%   Construct variables with vectors of jet en MET,
%   like the MT2 variable in SUSY searches.
% Inspiration: https://slidetodoc.com/presentation_image/ff8a8e4c690b62727b0cc4ed74967d46/image-7.jpg
\documentclass[border=3pt,tikz]{standalone}
\usepackage{amsmath}
\usepackage{physics}
\usepackage{xcolor}
\usetikzlibrary{calc}
\tikzset{>=latex} % for LaTeX arrow head
\usetikzlibrary{decorations.pathreplacing} % for curly braces

\colorlet{myblue}{blue!70!black}
\colorlet{mydarkblue}{blue!40!black}
\colorlet{mygreen}{green!40!black}
\colorlet{myred}{red!65!black}
\tikzstyle{vector}=[->,very thick,myblue,line cap=round]
\tikzstyle{ptmiss}=[->,dashed,thick,myred,line cap=round]
\tikzstyle{cone}=[thin,blue!50!black,fill opacity=0.8]

\newcommand*{\vv}[1]{\vec{\mkern0mu#1}} % aligned vector arrow
\newcommand{\ptmiss}{\vv{p}_\mathrm{T}^\mathrm{miss}}
\newcommand*{\ptmissX}[1]{\vv{p}_\mathrm{T}^\mathrm{miss,#1}}
\newcommand*{\MTX}[1]{M_\mathrm{T}^{(#1)}}
\newcommand\cone[4]{
\pgfmathanglebetweenpoints{\pgfpointanchor{#1}{center}}{\pgfpointanchor{#2}{center}}
\coordinate (tmpR) at ($(#2)+(\pgfmathresult+90:#3)$);
\coordinate (tmpL) at ($(#2)+(\pgfmathresult-90:#3)$);
\draw[blue!80!black] (tmpL) -- (#1) -- (tmpR);
%\draw[red] (#2)++(\pgfmathresult:#4) circle({sqrt(#3^2+#4^2)});
\draw[blue!80!black,rotate=\pgfmathresult] (#2) ellipse({#4} and {#3});
}

\newcommand\jetcone[4]{
\pgfmathanglebetweenpoints{\pgfpointanchor{#1}{center}}{\pgfpointanchor{#2}{center}}
\edef\tmpang{\pgfmathresult}
\coordinate (tmpO) at ($(#1)+(\tmpang:0.02)$); % center
\coordinate (tmpC) at ($(#2)+(\tmpang-180:{abs(#4)+0.4})$); % center
\coordinate (tmpL) at ($(tmpC)+(\tmpang+90:#3)$); % left corner
\coordinate (tmpR) at ($(tmpC)+(\tmpang-90:#3)$); % right corner
\draw[thin,blue!50!black,fill opacity=0.8, %,fill=blue!50!black!30
%(tmpR) arc(90:-90:{#4} and {#3});
(tmpC) ellipse({#4} and {#3});
\begin{scope}
\clip[rotate=\tmpang] (tmpR) -- (tmpO) -- (tmpL) arc(90:-90:{#4+0.6} and {#3});
\draw[vector] (tmpO) -- (#2);
\end{scope}
\draw[thin,blue!50!black,fill opacity=0.8, %,fill=blue!50!black!30
(tmpL) arc(90:270:{#4} and {#3}) -- (tmpO) -- cycle;
}

\begin{document}

% MT2
\begin{tikzpicture}
\def\R{2.8}
\coordinate (O) at (0,0);
\coordinate (J1) at (170:\R); % jet 1 pT
\coordinate (J2) at (-35:\R); % jet 2 pT
\coordinate (M1) at (130:0.9*\R); % pTmiss component 1
\coordinate (M2) at (-10:0.8*\R); % pTmiss component 2
\coordinate (M) at ($(M1)+(M2)$); % total missing momentum

% PTMISS
\draw[ptmiss,-,very thin] (M1) -- (M) -- (M2);
\draw[ptmiss] (O) -- (M1) node[left=2,above=-3] {$\ptmissX{1}$};
\draw[ptmiss] (O) -- (M2) node[right=0] {$\ptmissX{2}$};
\draw[ptmiss] (O) -- (M) node[above right=-1] {$\ptmiss$};

% JET CONES
%\draw[vector] (O) -- (J1);
%\draw[vector] (O) -- (J2);
%\cone{O}{J1}{0.3}{0.15}
\jetcone{O}{J1}{0.4}{0.08}
\jetcone{O}{J2}{0.4}{0.10}
\node[vector,left] at (J1) {$\vv{p}_{j_1}$};
\node[vector,right] at (J2) {$\vv{p}_{j_2}$};

% CURLY BRACE
\draw[line width=0.5,mygreen,decorate,decoration={brace,amplitude=6}]
($(J1)+(195:0.35*\R)$) -- ($(M1)+(120:0.30*\R)$) node[midway,above left=2] {$\MTX{1}$};
\draw[line width=0.5,mygreen,decorate,decoration={brace,amplitude=6}]
($(M2)+(10:0.48*\R)$) -- ($(J2)+(-45:0.26*\R)$) node[midway,below=4,right=4] {$\MTX{2}$};

\end{tikzpicture}

% MT2 in 3D
\begin{tikzpicture}
\def\R{2.8}
\def\M{0.9}
\coordinate (O) at (0,0);
\coordinate (J1) at (170:\R); % jet 1 pT
\coordinate (J2) at (-30:0.90*\R); % jet 2 pT
\coordinate (M1) at (130:1.05*\M); % pTmiss component 1
\coordinate (M2) at (2:0.75*\M); % pTmiss component 2
\coordinate (M) at ($(M1)+(M2)$); % total missing momentum

% JET CONE BACK
\jetcone{O}{J2}{0.35}{-0.1}
\node[vector,right] at (J2) {$\vv{p}_{j_2}$};

% PTMISS
\fill[red!80!black!15,opacity=0.75] % transverse plane
(O) ellipse ({0.8*\M} and {2.5*\M});
\draw[ptmiss,-,very thin] (M1) -- (M) -- (M2);
\draw[ptmiss] (O) -- (M1) node[left=2,above=-3] {$\ptmissX{1}$};
\draw[ptmiss] (O) -- (M2) node[right=0] {$\ptmissX{2}$};
\draw[ptmiss] (O) -- (M) node[above right=-1] {$\ptmiss$};

% JET CONE FRONT
\jetcone{O}{J1}{0.4}{0.1}
\node[vector,left] at (J1) {$\vv{p}_{j_1}$};

\end{tikzpicture}

\end{document}