414 lines
10 KiB
TeX
Executable File
414 lines
10 KiB
TeX
Executable File
\documentclass[12pt]{beamer}
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\usetheme{Berlin}
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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{braket}
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\usepackage{tikz}
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\usepackage{tikz-feynman}[compat=1.1.0]
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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{Interacciones y conservaciones}
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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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\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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\section{Resonancias en hadrones}
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\begin{frame}{Resonancia $\Delta(1234)$}
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\begin{figure}[ht!]
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\begin{center}
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\includegraphics[width=0.6\linewidth]{gaussianas.jpg}
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\caption{Esquema de la sección eficaz de las colisiones $\pi-N$ a bajas energías. Imagen adaptada de: \href{http://www.flickr.com/photos/77004318@N00/91432761}{"case3b"} por \href{http://www.flickr.com/photos/77004318@N00}{Samuel Foucher} con licencia \href{https://creativecommons.org/licenses/by-sa/2.0/?ref=ccsearch&atype=rich}{CC BY-SA 2.0}}
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\label{fig:gauss}
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\end{center}
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\end{figure}
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\end{frame}
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\begin{frame}{Vida media}
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\begin{equation*}
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\tau_{\Delta} \approx \frac{\hbar}{\Gamma_{\Delta}c^2}\approx \frac{6.6\times 10^{-22}MeV-sec}{100MeV} \approx 10^{-23} segundos
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\end{equation*}
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\end{frame}
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\begin{frame}{Resonancia $\rho^0$}
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\begin{equation}
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\pi^- + p \rightarrow \pi^+ + \pi^- + n
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\label{ec:rho}
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\end{equation}
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\begin{align*}
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\pi^- + p &\rightarrow \rho^0 + n \\
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\text{después } \rho^0 &\rightarrow \pi^+ + \pi^-
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\end{align*}
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\end{frame}
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\begin{frame}{Tiempo de vida media}
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\begin{equation*}
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\psi \propto e^{\frac{ic^2}{\hbar} (M_0-i\frac{\Gamma}{2})t}, t>0.
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\end{equation*}
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\begin{equation*}
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\tau=\frac{\hbar}{\Gamma c^2}.
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\end{equation*}
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\end{frame}
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\section*{Interacciones}
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\begin{frame}{Interacciones electromagnéticas}
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\begin{itemize}
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\item Una interacción muy estudiada
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\item Aproximaciones clásicas
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\item Teoría de perturbaciones
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\item ?`Si las energías son relativistas y el tratamiento cuántico?
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\begin{itemize}
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\item Electrodinámica cuántica
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\item Radiación multipolar
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\end{itemize}
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\end{itemize}
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\end{frame}
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\begin{frame}{Dispersión electromagnética de leptones}
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\begin{itemize}
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\item Dispersión de M\o{}ller
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\begin{equation*}
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e^- + e^- \rightarrow e^- + e^-
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\end{equation*}
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\item Dispersión de Bhabha
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\begin{equation*}
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e^- + e^+ \rightarrow e^- + e^+
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\end{equation*}
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\end{itemize}
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\end{frame}
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\begin{frame}{Dispersión electromagnética de leptones}
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\feynmandiagram [large, vertical=b to c] {
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a -- [fermion, edge label'=\( e^- \)] b -- [fermion, edge label'=\( e^- \)] j,
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b -- [photon,edge label'=\(\gamma\)] c,
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h -- [anti fermion, edge label'=\( e^+ \)] c -- [anti fermion, edge label'=\( e^+ \)] i;
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};
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\feynmandiagram [horizontal=a to b] {
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i1[particle=\( e^- \)] -- [fermion] a -- [fermion] i2[particle=\( e^+ \)],
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a -- [photon, edge label'=\(\gamma\)] b,
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f1[particle= \( e^- \)] -- [fermion] b -- [fermion] f2 [particle=\( e^+ \)],
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};
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\end{frame}
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\begin{frame}{Interacción fotón-hadrón y mesones mediadores}
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?`Un fotón puede decaer en un par hadrón anti-hadrón?
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\begin{itemize}
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\item $\rho^0$, $\omega^0$ y $\phi^0$
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\end{itemize}
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\end{frame}
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\begin{frame}{Conservaciones y violaciones}
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Conserva
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\begin{itemize}
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\item Extrañeza
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\item Paridad
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\end{itemize}
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\end{frame}
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\begin{frame}{Interacción débil}
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\begin{itemize}
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\item Electrodébil
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\item Radiación nuclear: $\alpha$, $\beta$ y $\gamma$
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\item $\beta = e^{\pm}$
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\begin{itemize}
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\item ?`Elctrones en el núcleo?
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\item Espectro de energías continuo
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\end{itemize}
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\end{itemize}
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\end{frame}
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\begin{frame}{Neutrinos}
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\begin{itemize}
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\item Interacción débil
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\item Partículas neutras
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\item Recuerden
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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{itemize}
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\end{frame}
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\begin{frame}{Corrientes neutras}
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\begin{equation*}
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\nu_{\mu} + e^- \rightarrow \nu_{\mu} + e^-
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\end{equation*}
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\feynmandiagram [large, vertical=b to c] {
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a -- [fermion, edge label'=\( \nu_{\mu} \)] b -- [fermion, edge label'=\( \nu_{\mu} \)] j,
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b -- [scalar,edge label'=\(Z^0\)] c,
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h -- [fermion, edge label'=\( e^- \)] c -- [fermion, edge label'=\( e^- \)] i;
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};
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\end{frame}
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\begin{frame}{Procesos leptónicos}
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\begin{equation*}
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\mu^+ \rightarrow \bar{\nu_{\mu}} + e^+ + \nu_{e}
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\end{equation*}
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\begin{equation*}
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\nu_{\tau} + e^- \rightarrow \nu_{\tau} + e^-
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\end{equation*}
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\end{frame}
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\begin{frame}{Procesos semileptónicos}
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\begin{equation*}
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\underset{\bar{u}d}{\pi^-} \rightarrow \underset{u\bar{u}}{\pi^0} + e^- + \bar{\nu_e}
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\end{equation*}
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\feynmandiagram [horizontal=a to b] {
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i1[particle=\( \bar{u} \)] -- [] a,
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a -- [fermion] b,
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b -- [] f2 [particle=\( \bar{u} \)],
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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}{Procesos semileptónicos}
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\begin{equation*}
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\nu_{\mu} + \underset{udd}{n} \rightarrow \mu^- + \underset{uud}{p}
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\end{equation*}
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\begin{equation*}
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\nu_{\mu} + p \rightarrow \nu_{\mu} + p
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\end{equation*}
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\end{frame}
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\begin{frame}{Procesos hadrónicos}
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\begin{align*}
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K^+ &\rightarrow \pi^+ + \pi^0 \\
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&\rightarrow \pi^+ + \pi^+ + \pi^- \\
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&\rightarrow \pi^+ + \pi^0 + \pi^0.
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\end{align*}
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En ninguno cambia la extrañeza.
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\end{frame}
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\begin{frame}{Violaciones}
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Fuente de muones
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\begin{equation}
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\pi^+ \rightarrow \mu^+ + \nu_{\mu}
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\end{equation}
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\begin{itemize}
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\item Conservación momento
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\item conservación momento angular
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\end{itemize}
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\end{frame}
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\begin{frame}{Paridad}
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\begin{align*}
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\mathbf{P}(\overrightarrow{r}) &= -\overrightarrow{r} \\
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\mathbf{P}(\overrightarrow{p}) &= -\overrightarrow{p}
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\end{align*}
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\end{frame}
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\begin{frame}
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\begin{equation*}
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\mathbf{P}(\overrightarrow{L}) = \mathbf{P}(\overrightarrow{r}) \times \mathbf{P}(\overrightarrow{p}) = (-\overrightarrow{r}) \times (-\overrightarrow{p})= (\overrightarrow{r}) \times (\overrightarrow{p}) = \overrightarrow{L}
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\end{equation*}
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\begin{align*}
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\mathbf{P}\square &= +\square \text{ paridad positiva o par} \\
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\mathbf{P}\square &= -\square \text{ paridad negativa o impar}
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\end{align*}
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\end{frame}
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\begin{frame}
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\begin{equation*}
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\mathbf{P}\ket{\text{estado inicial}} = \mathbf{P}(\ket{a})\mathbf{P}(\ket{b})\mathbf{P}(\ket{\text{movimiento relativo}})
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\label{ec:paridad}
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\end{equation*}
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\begin{align*}
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\eta_p(\text{estado incial}) &= \eta_p(a) \eta_p(b)\eta_p(\text{movimieno relativo}) \notag \\
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\text{función de onda } \eta_p(\text{estado incial}) &= \eta_p(a) \eta_p(b)(-1)^{\ell}
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\label{ec:parorb}
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\end{align*}
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\end{frame}
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\begin{frame}
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\begin{equation}
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\begin{pmatrix}
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u \\
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d
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\end{pmatrix} \begin{pmatrix}
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c \\
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s
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\end{pmatrix} \begin{pmatrix}
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t \\
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b
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\end{pmatrix}.
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\end{equation}
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\begin{align}
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\eta_p(\text{estado incial}) &= \eta_p(a) \eta_p(b)\eta_p(\text{movimieno relativo}) \notag \\
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\text{función de onda } \eta_p(\text{estado incial}) &= \eta_p(a) \eta_p(b)(-1)^{\ell}
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\label{ec:parorb}
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\end{align}
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\end{frame}
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\begin{frame}
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\begin{equation}
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\begin{pmatrix}
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d' \\
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s' \\
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b'
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\end{pmatrix} =
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\begin{pmatrix}
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V_{ud} & V_{us} & V_{ub} \\
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V_{cd} & V_{cs} & V_{cb} \\
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V_{td} & V_{ts} & V_{tb}
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\end{pmatrix}
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\begin{pmatrix}
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d \\
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s \\
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b
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\end{pmatrix}
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\end{equation}
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\end{frame}
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\begin{frame}
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\begin{equation}
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\begin{pmatrix}
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\nu_e \\
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\nu_{\mu} \\
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\nu_{\tau}
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\end{pmatrix} =
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\begin{pmatrix}
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V_{e1} & V_{e2} & V_{e3} \\
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V_{\mu 1} & V_{\mu 2} & V_{\mu 3} \\
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V_{\tau 1} & V_{\tau 2} & V_{\tau 3}
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\end{pmatrix}
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\begin{pmatrix}
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\nu_1 \\
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\nu_2 \\
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\nu_3
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\end{pmatrix}
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\end{equation}
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\end{frame}
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\begin{frame}
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\begin{align*}
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\nu_e =& cos\theta_{12} \nu_1 + sen\theta_{12} \nu_2 \\
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\nu_{\mu} =& -sen\theta_{12} \nu_1 + cos\theta_{12} \nu_2
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\end{align*}
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\begin{equation}
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\ket{\nu_e(t)} = e^{-iE_1t/\hbar}cos\theta_{12} \nu_1 + e^{-iE_2t/\hbar}sen\theta_{12} \nu_2
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\end{equation}
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\begin{equation}
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\mathbb{P}_{\nu_{\mu}}(t) = |\bra{\nu_{\mu}}\ket{\nu_e}(t)|^2 = sen^2 \theta_{12} sen^2\left[ \frac{1}{2} \frac{(E_1 - E_2)t}{\hbar}\right]
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\end{equation}
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\end{frame}
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\begin{frame}{Neutrinos de Majorana}
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\end{frame}
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\end{document}
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