462 lines
14 KiB
TeX
462 lines
14 KiB
TeX
\documentclass[12pt]{beamer}
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\usetheme{CambridgeUS}
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\usepackage[utf8]{inputenc}
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\usepackage[spanish]{babel}
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\usepackage{amsmath}
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\usepackage{amsfonts}
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\usepackage{amssymb}
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\usepackage{graphicx}
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\usepackage{tikz}
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\usepackage[compat=1.1.0]{tikz-feynman}
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\usepackage{multicol}
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\usepackage{appendixnumberbeamer}
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%\setbeamerfont{page number in head}{size=\large}
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%\setbeamertemplate{footline}{Diapositiva}
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\setbeamertemplate{footline}[frame number]
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\newcommand{\backupbegin}{
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\newcounter{finalframe}
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\setcounter{finalframe}{\value{framenumber}}
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}
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\newcommand{\backupend}{
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\setcounter{framenumber}{\value{finalframe}}
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}
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\author{Física Nuclear y subnuclear }
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\title{Partículas elementales II}
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%\setbeamercovered{transparent}
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%\setbeamertemplate{navigation symbols}{}
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%\logo{}
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%\institute{}
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%\date{}
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%\subject{}
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\begin{document}
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\begin{frame}
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\titlepage
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\end{frame}
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%\begin{frame}{Contenido}
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% \tableofcontents
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%\end{frame}
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\section{Hadrones}
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\begin{frame}{Bariones}
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\textbf{Recordar Lunch Nuclear}
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\begin{table}[ht!]
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\begin{tabular}{|p{0.12\textwidth} | p{0.08\textwidth} | p{0.1\textwidth}|p{0.1\textwidth}|p{0.1\textwidth}|p{0.1\textwidth}|p{0.1\textwidth}|}
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\hline
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Part. & Símb. & Cont. & Espín & Masa $MeV/c^2$ & A-part. & Cont. a-part. \\
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\hline
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protón & $p$ & $uud$ & $1/2$ & 938 & $\bar{p}$ & $\bar{u}\bar{u}\bar{d}$ \\
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\hline
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neutrón & $n$ & $udd$ & $1/2$ & 940 & $\bar{n}$ & $\bar{u}\bar{d}\bar{d}$ \\
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\hline
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Sigma + & $\Sigma^+$ & $uus$ & $1/2$ & 1189 & $\bar{\Sigma^-}$ & $\bar{u}\bar{u}\bar{s}$ \\
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\hline
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Sigma 0 & $\Sigma^0$ & $uds$ & $1/2$ & 1193 & $\bar{\Sigma^0}$ & $\bar{u}\bar{d}\bar{s}$ \\
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\hline
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Sigma - & $\Sigma^0$ & $dds$ & $1/2$ & 1197 & $\bar{\Sigma^-}$ & $\bar{d}\bar{d}\bar{s}$ \\
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\hline
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Lambda & $\Lambda$ & $uds$ & $1/2$ & 1116 & $\bar{\Lambda}$ & $\bar{u}\bar{d}\bar{s}$ \\
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\hline
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Xi 0 & $\Xi^0$ & $uss$ & $1/2$ & 1315 & $\bar{\Xi^0}$ & $\bar{u}\bar{s}\bar{s}$ \\
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\hline
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Xi- & $\Xi^-$ & $dss$ & $1/2$ & 1322 & $\bar{\Xi^-}$ & $\bar{d}\bar{s}\bar{s}$ \\
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\hline
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\end{tabular}
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\end{table}
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\end{frame}
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\begin{frame}{Mesones}
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\begin{table}[ht!]
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\begin{tabular}{|p{0.12\textwidth} | p{0.08\textwidth} | p{0.1\textwidth}|p{0.1\textwidth}|p{0.1\textwidth}|p{0.1\textwidth}|p{0.1\textwidth}|}
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\hline
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Part. & Símb. & Cont. & Espín & Masa $MeV/c^2$ & A-part. & Cont. a-part. \\
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\hline
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pión + & $\pi^+$ & $u\bar{d}$ & $0$ & 140 & $\pi^-$ & $\bar{u}d$ \\
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\hline
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pión 0 & $\pi^0$ & $u\bar{u}$, $d\bar{d}$ & $0$ & 135 & $\pi^0$ & $u\bar{u}$, $d\bar{d}$ \\
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\hline
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eta & $\eta$ & $u\bar{u}$, $d\bar{d}$, $s\bar{s}$ & $0$ & 548 & $\eta$ & $u\bar{u}$, $d\bar{d}$, $s\bar{s}$ \\
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\hline
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kaon + & $K^+$ & $u\bar{s}$ & $0$ & 949 & $K^-$ & $\bar{u}s$ \\
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\hline
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kaon 0 & $K^0$ & $d\bar{s}$ & $0$ & 948 & $\bar{K^0}$ & $\bar{d}s$ \\
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\hline
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rho + & $\rho^+$ & $u\bar{d}$ & $1$ & 770 & $\rho^-$ & $\bar{u}d$ \\
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\hline
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rho 0 & $\rho^0$ & $u\bar{u}$, $d\bar{d}$ & $1$ & 770 & $\rho^0$ & $u\bar{u}$, $d\bar{d}$ \\
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\hline
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omega & $\omega$ & $u\bar{u}$, $d\bar{d}$ & $1$ & 783 & $\omega$ & $u\bar{u}$, $d\bar{d}$ \\
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\hline
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phi & $\phi$ & $s\bar{s}$ & $1$ & 1020 & $\phi$ & $s\bar{s}$ \\
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\hline
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\end{tabular}
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\end{table}
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\end{frame}
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\begin{frame}{Leptones}
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\begin{table}[ht!]
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\begin{tabular}{|p{0.15\textwidth} | p{0.1\textwidth} | p{0.17\textwidth}|p{0.1\textwidth}|p{0.12\textwidth}|}
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\hline
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Part. & Símb. & Masa $MeV/c^2$ & Espín & A-part. \\
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\hline
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electrón & $e^-$ & 0.511 & $1/2$ & $e^+$ \\
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\hline
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neutrino e & $\nu_e$ & $<0.000225$ & $1/2$ & $\bar{\nu_e}$ \\
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\hline
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muón & $\mu^-$ & 106 & $1/2$ & $\bar{\mu}$ \\
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\hline
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neutrino $\mu$ & $\nu_{\mu}$ & $<0.19$ & $1/2$ & $\bar{\nu_{\mu}}$ \\
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\hline
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tau & $\tau^-$ & 1777 & $1/2$ & $\bar{\tau}$ \\
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\hline
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neutrino $\tau$ & $\nu_{\tau}$ & $<18.2$ & $1/2$ & $\bar{\nu_{\tau}}$ \\
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\hline
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\end{tabular}
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\end{table}
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\end{frame}
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\begin{frame}{Tabla de partículas fundamentales}
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\begin{figure}[ht!]
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\begin{center}
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\includegraphics[width=0.5\linewidth]{gen_materia.png}
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\caption{Partículas fundamentales, imagen con licencia \href{https://en.wikipedia.org/wiki/en:GNU_Free_Documentation_License}{GNU Free Documentation License}, File:Generaciones delamateria.png. (2020, March 10). Wikimedia Commons, the free media repository.}
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\label{fig:tab_part}
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\end{center}
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\end{figure}
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\end{frame}
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\begin{frame}{Decaimiento}
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\begin{equation*}
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p \rightarrow e^+ + \pi^0
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\end{equation*}
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\begin{align*}
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\text{Cons. E: }938MeV &\rightarrow 0.511 MeV + 135 MeV \\
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\text{Cons. Q: } +1e &\rightarrow +1e + 0e \\
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\text{Cons. no. bariónico: } 1 &\rightarrow 0 + 0 \\
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\text{Cons. l.e.: } 0 &\rightarrow 1 + 0
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\end{align*}
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\end{frame}
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\begin{frame}{Decaimiento leptónico}
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\begin{equation*}
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\mu^- \rightarrow e^- + \bar{\nu_e} + \nu_{\mu},
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\label{ec:mu}
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\end{equation*}
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\begin{align*}
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\text{Cons. E: }105.6MeV &\rightarrow 0.511 MeV + \sim 0 + \sim 0 \\
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\text{Cons. Q: } -1e &\rightarrow -1e + 0e + 0e \\
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\text{Cons. no. bariónico: } 0 &\rightarrow 0 + 0 +0 \\
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\text{Cons. l.e.: } 0 &\rightarrow 1 + -1 + 0\\
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\text{Cons. l.e.: } 1 &\rightarrow 0 + 0 + 1
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\end{align*}
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\end{frame}
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\begin{frame}{Decaimiento neutrón}
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\begin{equation*}
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n\rightarrow p + e^- + \nu_e
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\end{equation*}
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\end{frame}
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\begin{frame}{Decaimiento neutrón}
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\begin{equation*}
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n\rightarrow p + e^- + \bar{\nu_e}
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\end{equation*}
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\begin{align*}
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\text{Cons. E: }940MeV &\rightarrow 938 MeV + 0.51 MeV + (<225 eV) \\
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\text{Cons. Q: } 0e &\rightarrow 1e + (-1e) + 0e \\
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\text{Cons. no. bariónico: } 1 &\rightarrow 1 + 0 + 0 \\
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\text{Cons. l.e.: } 0 &\rightarrow 0 + 1 + (-1)
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\end{align*}
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\end{frame}
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\begin{frame}{Diagramas de Feynman}
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\begin{itemize}
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\item La teoría cuántica independiente del tiempo no contempla decaimientos
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\item Ecuación de Dirac
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\item Dirección de las flechas: tiempo
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\item En cada vértice se conserva:
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\begin{itemize}
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\item Energía
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\item Carga
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\item Número leptónico de familia
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\item Momento
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\item De cierta forma el número bariónico
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\end{itemize}
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\end{itemize}
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\end{frame}
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\begin{frame}{Ejemplo}
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\feynmandiagram [horizontal=a to b] {
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i1[particle=partícula] -- [fermion] a -- [fermion] i2[particle=anti-partícula],
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a -- [photon, edge label'=\(\gamma\)] b,
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f1[particle= anti-partícula] -- [fermion] b -- [fermion] f2 [particle=partícula],
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};
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\end{frame}
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\begin{frame}{Decaimiento leptónico}
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\begin{equation*}
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\mu^- \rightarrow e^- + \bar{\nu_e} + \nu_{\mu},
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\label{ec:mu}
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\end{equation*}
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\feynmandiagram [horizontal=a to b] {
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a [particle=\(\mu^{-}\)] -- [fermion] b -- [fermion] f1 [particle=\(\nu_{\mu}\)],
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b -- [boson, edge label=\(W^{-}\)] c,
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f2 [particle=\(\overline \nu_{e}\)] -- [fermion] c -- [fermion] f3 [particle=\(e^{-}\)],
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};
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\end{frame}
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\begin{frame}{Decaimineto bariónico}
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\begin{equation*}
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n\rightarrow p + e^- + \bar{\nu_e}
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\end{equation*}
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\end{frame}
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\begin{frame}{Decaimiento bariónico}
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\feynmandiagram [horizontal=a to b] {
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i1[particle=u] -- [] a,
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a -- [fermion] b,
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b -- [] f2 [particle=u],
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};
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\feynmandiagram [horizontal=a to b] {
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i1[particle=d] -- [] a,
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a -- [fermion] b,
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b -- [] f2 [particle=d],
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};
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\feynmandiagram [layered layout, horizontal=a to b] {
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a [particle=d] -- [fermion] b -- [fermion] f1 [particle=u],
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b -- [scalar, edge label'=\(W^{-}\)] c,
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c -- [anti fermion] f2 [particle=\(\overline \nu_{e}\)],
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c -- [fermion] f3 [particle=\(e^{-}\)],
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};
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\end{frame}
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\begin{frame}{Piones}
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\begin{itemize}
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\item Los mesones más ligeros ($140 MeV/c^2$ para $\pi^{\pm}$ y $135MeV/c^2$ para $\pi^0$)
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\item Tiempo de vida media de $2.6\times 10^{-8}s$, propio de interacciones débiles
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\end{itemize}
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\begin{align*}
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\pi^+ &\rightarrow \mu^+ + \nu_{\mu} \\
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\pi^- &\rightarrow \mu^- + \bar{\nu}_{\mu} \\
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\pi^0 &\rightarrow \gamma + \gamma
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\end{align*}
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\end{frame}
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\begin{frame}{Interacción}
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\begin{equation}
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\underset{\bar{u}s}{K^-} + \underset{uud}{p} \rightarrow \underset{d\bar{s}}{K^0} + \underset{u\bar{s}}{K^+} + \underset{sss}{\Omega^-}
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\end{equation}
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\end{frame}
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\begin{frame}{Interacción}
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\begin{equation}
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K^- + p \rightarrow K^0 + K^+ + \Omega^-
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\end{equation}
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\begin{align*}
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\text{Cons. Q: } -1e + 1e &\rightarrow 0e + 1e + (-1e) \\
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\text{Cons. no. bariónico: } 0 + 1 &\rightarrow 0 + 0 + 1 \\
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\text{Cons. extrañeza: } 1 + 0 &\rightarrow (-1) + (-1) + (3)
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\end{align*}
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\end{frame}
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\begin{frame}
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\begin{multicols}{2}
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\begin{equation}
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K^- + p \rightarrow K^0 + K^+ + \Omega^-
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\end{equation}
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\feynmandiagram [horizontal=a to b] {
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i1[particle=u] -- [] a,
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a -- [fermion] b,
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b -- [] f2 [particle=u],
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};
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\feynmandiagram [horizontal=a to b] {
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i1[particle=s] -- [] a,
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a -- [fermion] b,
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b -- [] f2 [particle=s],
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};
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\feynmandiagram [layered layout, horizontal=a to b] {
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a [particle=d] -- [fermion] b -- [fermion] f1 [particle=d],
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b -- [gluon, edge label'=gluón] c,
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c -- [anti fermion] f2 [particle=\(\overline s\)],
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c -- [fermion] f3 [particle=s],
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};
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\feynmandiagram [horizontal=a to b] {
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i1[particle=u] -- [fermion] a -- [fermion] i2[particle=\(\bar{u}\)],
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a -- [gluon, edge label'=gluón] b,
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f1[particle=\(\bar{s}\)] -- [fermion] b -- [fermion] f2 [particle=s],
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};
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\end{multicols}
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\end{frame}
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\begin{frame}{Extrañeza}
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\begin{itemize}
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\item Se descubre en experimentos de rayos cósmicos
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\item Se crean en proceso fuertes
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\item Decaen por procesos débiles
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\item Al crearse aparecen con una pareja específica
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\end{itemize}
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\end{frame}
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\begin{frame}{Producción}
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\begin{equation}
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\pi^- + p \rightarrow K^0 + \Lambda^0
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\end{equation}
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\begin{align*}
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\text{Cons. Q: } -1e + 1e &\rightarrow 0e + 0e \\
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\text{Cons. no. bariónico: } 0 + 1 &\rightarrow 0 + 1 \\
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\text{Cons. extrañeza: } 0 + 0 &\rightarrow 0 + (-1) + 1
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\end{align*}
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\end{frame}
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\begin{frame}{Producción}
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\begin{equation}
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\pi^- + p \rightarrow K^0 + \Lambda^0
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\end{equation}
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\end{frame}
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\begin{frame}{Decaimientos}
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\begin{align*}
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\Lambda^0 \rightarrow \pi^- + p \notag \\
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K^0 \rightarrow \pi^+ + \pi^-
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\end{align*}
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\end{frame}
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\begin{frame}{Decaimientos}
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\begin{equation*}
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K^0 \rightarrow \pi^+ + \pi^-
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\end{equation*}
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\begin{align*}
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\text{Cons. E: } 948MeV &\rightarrow 140 MeV + 140 MeV\\
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\text{Cons. Q: } 0e &\rightarrow 1e + (-1e) \\
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\text{Cons. extrañeza: } -1 &\rightarrow 0 + 0
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\end{align*}
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\end{frame}
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\begin{frame}{Decaimiento}
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\begin{equation*}
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K^0 \rightarrow \pi^+ + \pi^-
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\end{equation*}
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\end{frame}
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\begin{frame}{Decaimiento}
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\feynmandiagram [horizontal=a to b] {
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i1[particle=d] -- [] a,
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a -- [fermion] b,
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b -- [] f2 [particle=d],
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};
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\feynmandiagram [layered layout, horizontal=a to b] {
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a [particle=\( \bar{s} \)] -- [fermion] b -- [fermion] f1 [particle=\( \bar{u} \)],
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b -- [scalar, edge label'=\(W^{+}\)] c,
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c -- [anti fermion] f2 [particle=\(\overline d\)],
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c -- [fermion] f3 [particle=u],
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};
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\end{frame}
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\begin{frame}{Otro decaimiento}
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\feynmandiagram [horizontal=a to b] {
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i1[particle=d] -- [] a,
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a -- [fermion] b,
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b -- [] f2 [particle=d],
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};
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\feynmandiagram [layered layout, horizontal=a to b] {
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a [particle=\( \bar{s} \)] -- [fermion] b -- [fermion] f1 [particle=\( \bar{u} \)],
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b -- [scalar, edge label'=\(W^{+}\)] c,
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c -- [anti fermion] f2 [particle=\(\overline d\)],
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c -- [fermion] f3 [particle=u],
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};
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\end{frame}
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\begin{frame}{Procesos que no suceden}
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\begin{align*}
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\pi^- + p &\not\rightarrow \pi^- + \pi^+ + \Lambda^0 \notag \\
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\pi^- + p &\not\rightarrow K^- + \pi^+ + \Lambda^0 \notag \\
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\pi^- + p &\not\rightarrow \Sigma^+ + K^- \notag \\
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\pi^- + p &\not\rightarrow \Sigma^- + \pi^+
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\end{align*}
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La extrañeza se conserva en procesos fuertes pero se viola en procesos débiles.
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\end{frame}
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\begin{frame}{Conservaciones}
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Sistema cuántico descrito por $\hat{H}$
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\begin{equation}
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-i\hbar \frac{d\Psi}{dt} = \hat{H}\Psi
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\end{equation}
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Relaciones de permutación para un operador $\hat{A}$ con observable $A$:
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|
\begin{equation*}
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[\mathbf{H},\mathbf{A}] = 0 \rightarrow \frac{d}{dt}\langle A \rangle = 0
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\end{equation*}
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La carga
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\begin{equation*}
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\mathbf{Q}\Psi = q\Psi.
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\end{equation*}
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Invariancia de norma
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\begin{equation*}
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\Psi' = e^{i\epsilon Q}\Psi,
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\end{equation*}
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\end{frame}
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\begin{frame}{Isospín}
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\begin{table}[ht!]
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\begin{tabular}{|p{0.2\textwidth}|p{0.2\textwidth}|p{0.2\textwidth}|}
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\hline
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Partícula & $I$ & $I_3$ \\
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\hline
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$p$ & $1/2$ & $1/2$ \\
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$n$ & $1/2$ & $-1/2$ \\
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\hline
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$\pi^+$ & $1$ & $1$ \\
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$\pi^0$ & $1$ & $0$\\
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$\pi^-$ & $1$ & $-1$ \\
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\hline
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$K^+$ & $1/2$ & $1/2$ \\
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$K^0$ & $1/2$ & $-1/2$ \\
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\hline
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$\Sigma^+$ & $1$ & $1$ \\
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$\Sigma^0$ & $1$ & $0$ \\
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$\Sigma^-$ & $1$ & $-1$ \\
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\hline
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\end{tabular}
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\label{tab:lep}
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\caption{Valores del número leptónico por familia para los leptones}
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\end{table}
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\end{frame}
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\begin{frame}{Relación Gell-Mann-Nishima}
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\begin{equation}
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Q = I_3 + \frac{Y}{2} = I_3 + \frac{B-S}{2},
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\end{equation}
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\end{frame}
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\backupbegin
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\section*{Apéndices}
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\begin{frame}[noframenumbering]{}
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\end{frame}
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\backupend
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\end{document}
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