Hadronically decayed tau jet

Isolation (blue) and signal cone (red) of a hadronically decayed tau, and for comparison an electron, muon and quark/gluon-initiated jet. For more related figures, please see the “jet” tag or the Particle Physics category. The cones are constructed with the tangent methods presented here. The neutrinos from τ decay are invisible to the detector and are not shown in the illustrations below.

A hadronic jet initiated by a τ lepton that decays to a charge pion (“one prong”), and two neutral pions (π⁰s):

A hadronic jet initiated by a τ lepton that decays to three charge pions (“three prong”) plus one π⁰:

An isolated electron, that may be misidentified as a τ lepton decay to a single charged pion (“one prong”):

An isolated muon, that may be misidentified as a τ lepton decay to a single charged pion (“one prong”):

A hadronic jet initiated by a quark or gluon, which tend to have more activity in the isolation cone:

Edit and compile if you like:

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% Author: Izaak Neutelings (November 2021)
% Description: jet cones for taus & others
\documentclass[border=3pt,tikz]{standalone}
\usetikzlibrary{calc}
\usetikzlibrary{math} % for \tikzmath
\tikzset{>=latex} % for LaTeX arrow head
\colorlet{myblue}{blue!70!black}
%\colorlet{mydarkblue}{blue!50!black}
\colorlet{mygreen}{green!60!black}
\colorlet{myred}{red!75!black}
\colorlet{isocol}{blue!70!black} % color isolation cone
\colorlet{sigcol}{red!90!black} % color isolation cone
\tikzstyle{track}=[->,line width=0.6,myred]
\tikzstyle{dashed track}=[->,mygreen,line width=0.6,line cap=round,
                          dash pattern=on 2.3 off 2.0]
\newcommand\jetcone[6][sigcol]{{
  \pgfmathanglebetweenpoints{\pgfpointanchor{#2}{center}}{\pgfpointanchor{#3}{center}}
  \pgfmathsetmacro\oang{#4/2} % half-opening angle
  \edef\e{#5} % ratio a/b
  \def\tmpL{tmpL-#2-#3} % unique coordinate name
  \edef\vang{\pgfmathresult} % angle of vector OV
  \tikzmath{
    coordinate \C;
    \C = (#2)-(#3);
    \x = veclen(\Cx,\Cy)*\e*sin(\oang)^2; % x coordinate P
    \y = tan(\oang)*(veclen(\Cx,\Cy)-\x); % y coordinate P
    \a = veclen(\Cx,\Cy)*sqrt(\e)*sin(\oang); % vertical radius
    \b = veclen(\Cx,\Cy)*tan(\oang)*sqrt(1-\e*sin(\oang)^2); % horizontal radius
    \angb = acos(sqrt(\e)*sin(\oang)); % angle of P in ellipse
  }
  \coordinate (\tmpL) at ($(#3)-(\vang:\x pt)+(\vang+90:\y pt)$); % tangency
  \draw[thin,#1!50!black,fill=#1!80!black!50,rotate=\vang] % cone back
    (\tmpL) arc(180-\angb:180+\angb:{\a pt} and {\b pt})
    -- ($(#2)+(0.01,0)$) -- cycle;
  \draw[thin,#1!50!black,rotate=\vang, % cone inside
        top color=#1!60!black!60,bottom color=#1!50!black!75,shading angle=\vang]
    (#3) ellipse({\a pt} and {\b pt});
  #6 % extra tracks
  \draw[thin,#1!50!black,rotate=\vang,fill opacity=0.80, % cone front
 
 
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Click to download: jet_tau.tex • jet_tau.pdf
Open in Overleaf: jet_tau.tex