diff --git a/00_Bachelor_Thesis.tex b/00_Bachelor_Thesis.tex deleted file mode 100644 index 52ec2d6..0000000 --- a/00_Bachelor_Thesis.tex +++ /dev/null @@ -1,71 +0,0 @@ -\documentclass[10pt,a4paper]{article} -\usepackage[utf8]{inputenc} -\usepackage[english]{babel} -\usepackage{amsmath} -\usepackage{amsfonts} -\usepackage{amssymb} -\usepackage{graphicx} -\usepackage{fancyhdr} -\usepackage{paralist} -\pagestyle{fancy} -\usepackage{pdfpages} -\usepackage{subcaption} -\usepackage{float} -\usepackage{hyperref} -\usepackage{physics} -\usepackage{tabstackengine} -\usepackage[bottom]{footmisc} -\setlength{\skip\footins}{1.2pc plus 5pt minus 2pt} -\title{Particle Track reconstruction using a recurrent neural network at the $\mu-3e$ experiment} -\author{\textit{Bachelor thesis of}\\ Sascha Liechti} -\date{06.04.2018} - -\lhead{} -\chead{} -\rhead{} - -\lfoot{} -\cfoot{\thepage} -\rfoot{} - -\renewcommand{\headrulewidth}{0.4pt} -\renewcommand{\footrulewidth}{0pt} - -\setlength{\parindent}{0pt} - -\begin{document} - -\maketitle - -\begin{center} -\textit{Supervised by}\\ -Prof. Nicola Serra\\ -Dr. Patrick Owen\\ -\end{center} - -\textbf{\large{Abstract}} -During the $\mu-3e$ experiment we faced the challenge of reconstructing the paths of certain low momentum particles that curled back into the detector and cause additional hits. To face this, a recurrent neural network was used which found the right track for $87 \%$ of these particles. - -\newpage -\tableofcontents - -\newpage - -\include{01_Standard_Model} - -%\include{02_Setup} -% -%\include{03_Datatable} -% -%\include{04_Results} -% -%\include{05_Graphics} -% -%\include{06_Calculus} -% -%\include{07_Error-Calculus} - -%\include{08_Bibliography} - - -\end{document} diff --git a/01_Standard_Model.tex b/01_Standard_Model.tex deleted file mode 100644 index daf4a15..0000000 --- a/01_Standard_Model.tex +++ /dev/null @@ -1,174 +0,0 @@ -\section{Standard Model} -\subsection{Elementary particles and forces} -\label{intro_elem_part} - -The Standard Model(SM) describes all known elementary particles as well as three of the four known forces\footnote{Strong, weak and electromagnetic forces}.\\ -The elementary particles that make up matter can be split into two categories, namely quarks and leptons. There are 6 types of quarks and six types of leptons. The type of a particle is conventionally called flavour. The six quark flavours and the six lepton flavours are separated over 3 generations (each which two quarks and two leptons in it). -Experimental evidence suggests that there exist exactly three generations of particles. Each particle of the first generation has higher energy versions of itself with the same characteristics (e.g. $e^- \rightarrow \mu^- \rightarrow \tau^-$)as in other generations. Contrary, each following generation has a higher mass than the generation before.\\ - - -\begin{table}[H] -\begin{center} -\caption{Quarks in the Standard Model} \label{Quark_SM_table} -\begin{tabular}{l l l l l} -&&Quarks&&\\\hline - & Particle& & Q$[e]$ & $\frac{mass}{GeV}$\\ \hline -1. Gen. & up & u & $-\frac{1}{3}$ & $0.003$\\ - & down & d & $\frac{2}{3}$ & $0.005$\\ -2. Gen. & strange & s & $-\frac{1}{3}$ & $0.1$\\ - & charm & c & $\frac{2}{3}$ & $1.3$\\ -3. Gen. & bottom & b & $-\frac{1}{3}$ & $4.5$\\ - & top & t & $\frac{2}{3}$ & $174$\\ -\end{tabular} -\end{center} -\end{table} - - -One category consists of quarks($q$)(see Table \ref{Quark_SM_table}). In this, we differentiate between up-type quarks, with charge $-\frac{1}{3}e$, and down-type, quarks with charge $\frac{2}{3}e$. Quarks interact with all fundamental forces.\\ -Each quark carries a property called colour-charge. The possible color charges are red(r), green(gr), blue(bl) in which anti-quarks carry anti-colour. Quarks can only carry one colour, whilst every free particle has to be colorless\footnote{Colour confinement}. In conclusion we cannot observe a single quark.\\ -Free particles can achieve being colourless in two ways. Either by having all three colors present in the same amount (one quark of each color), which creates the characteristic group of baryons($qqq$) and anti-baryons($\bar{q}\bar{q}\bar{q}$) or by having a color and its anticolor present, which creates the group of mesons($q\bar{q}$).\\\newpage - - -\begin{table}[H] -\begin{center} -\caption{Leptons in the standard model} \label{Lepton_SM_table} -\begin{tabular}{l l l l l} - & & Leptons & &\\ \hline - & Particle& & Q$[e]$ & $\frac{mass}{GeV}$\\ \hline -1. Gen. & electron & $e^-$ & $-1$ & $0.005$\\ - & neutrino & $\nu_e$ & $0$ & $<10^{-9}$\\ -2. Gen. & muon & $\mu^-$ & $-1$ & $0.106$\\ - & neutrino & $\nu_{\mu}$ & $0$ & $<10^{-9}$\\ -3. Gen. & tau & $\tau^-$ & $-1$ & $1.78$\\ - & neutrino & $\nu_{\tau}$ & $0$ & $<10^{-9}$ - \label{Lepton_table} -\end{tabular} -\end{center} -\end{table} - -The other group consists of leptons(l)(see Table \ref{Lepton_SM_table}). They only interact through the weak and the electromagnetic force. Each generation consists of a lepton of charge -1 and a corresponding neutrally charged neutrino. The electron has the lowest energy of all charged leptons. This makes the electron stable while the higher generation particles decay to lower energy particles. \\ -\\ - -The leptons of one generation, namely the charged lepton and its corresponding neutrino are called a lepton family. A lepton of a family counts as 1 to its corresponding lepton family number whilst a anti-lepton counts as -1. - -\begin{table}[H] -\begin{center} -\caption{Fundamental forces} \label{fund_forces_table} -\begin{tabular}{l l l l l l l} -Force & Strengh & Boson & & Spin & Charge & $\frac{mass}{GeV}$ \\\hline -Strong & 1 & gluon & $g$ & 1& 0& 0 \\ -Electromagnetism & $10^{-3}$ & photon & $\gamma$ & 1& 0& 0 \\ -Weak & $10^{-8}$ & Z boson & $Z$ & 1& 0& 80.4\\ - & $10^{-8}$ & W boson & $W^{\pm}$& 1& $\pm 1$& 91.2\\ -\end{tabular} -\end{center} -\end{table} - - -The particles of the SM interact through the 3 fundamental forces of the SM. In these interactions, particles called bosons are being exchanged which are the carriers of their respective force (see Table \ref{fund_forces_table}).\\ -As mentioned above, only quarks can interact through the strong force, in which they exchange gluons. Gluons are massless and neutrally charged. The strong force has the biggest coupling strengh of 1 (though it decreases with higher energies as a result of gluon-gluon self interaction loops, which interfere negatively in perturbation theory)\footnote{Mark Thomson - Modern Particle physics - 10.5.2}. A gluon carries colour charge and hence can change the colour of a quark but it conserves its flavour. The strong interaction has an underlying gauge symmetry of SU(3). Therefore, it can be derived that color charge is conserved through the strong interaction\footnote{E.g. through Gell-Mann matrices}.\\ -The electromagnetic(EM) force is propagated through the photon. It carries zero charge and no invariant mass. Exclusively charged particles can interact through the electromagnetic force. The coupling strength is $\alpha \approx \frac{1}{137}$, contrary to the strong force the coupling constant increases with higher energies\footnote{Mark Thomson - Modern Particle physics - 10.5.1}. This difference stems from the fact that photon-photon interaction loops are not allowed whereas gluon-gluon interaction loops are. In perturbation theory this results in only positive terms being added to the coupling strength. The underlying gauge symmetry is of SU(1). The electromagnetic force also conserves flavour.\\ -The weak force has two types of bosons. The bosons of the weak force are the only bosons to have an inertial mass.\\ -First we will discuss the neutrally charged Z boson\footnote{Discovered at Super Proton Synchrotron accelerator - Cern - 1983}. Even though the Z boson belongs to the weak force it, it also has an electromagnetic part additionally to the weak force part\footnote{$Z \rightarrow EM_{part} + W^3$, Modern Particle Physics - Blababla}. It follows directly, that the Z boson couples weaker to uncharged particles.\\ -The other boson of the weak force is the W boson\footnote{Discovered at Super Proton Synchrotron accelerator - Cern - 1983}. In the classical SM, the only way particles can change flavour is through the weak force by emitting or absorbing W boson. It is important to notice that, besides of having an invariant mass, the W boson is the only boson with a non zero charge ($Q_{W^\pm} = \pm 1e$). In the gauge symmetry of the weak force the $W^\pm$ are actually the creation and annihilation operators of said symmetry\footnote{$W^\pm = W_1 \pm i W_2$}.\\ -An important characteristic of the weak force is that it exclusively couples to lefthanded(LH) particles and righthanded(RH) antiparticles (describing chirality states)\footnote{In the ultrarelativistic limit helicity and chirality eigenstates are the same}.\\ -The chirality operators for left- and righthandedness are: \\\\ -LH: $\frac{1}{2}(1-\gamma^5)$, RH: $\frac{1}{2}(1+\gamma^5)$\\\\ -As a consequence RH particles and LH anti-particles cant couple to the W boson at all. This also results in charged RH particles and LH anti-particles to couple to the Z boson only through the electromagnetic part of the itself, while uncharged RH particles and LH anti particles (e.g. RH $\nu$, LH $\bar{\nu}$) don't couple with the EM force nor the weak force. - - -\subsection{Interaction rules} - -Now we will establish the general rules for interactions in the SM.\\\\ -\textbf{Baryon number is conserved}\\ -As we already established before, the only interaction that can change flavour is the weak force through the W boson. We directly see that all other interactions baryon number has to be conserved. So any up-type quark can be changed to a down-type quark and backwards by emitting or absorbing a W boson. In the end however, there are still 3 quarks which form a baryon\footnote{I exclude $q\bar{q}$ pair-antipair production to form a pentaquark($qqqq\bar{q}$) and other exotic states}, even though it changed its type and charge. A well known example is the beta decay, where a down quark in a neutron decays into a an up quark to form now a proton(e.g. see Figure \ref{beta-decay_feynman}). We easily see that the baryon number is conserved.\\\\ - -\begin{figure}[H] -\begin{center} -\begin{subfigure}{0.45\textwidth} -\includegraphics[width=0.8\textwidth]{beta_decay_feynman.png} -\caption{Feynman diagram of the $\beta$-decay} -\label{beta-decay_feynman} -\end{subfigure} -\begin{subfigure}{0.45\textwidth} -\includegraphics[width=0.8\textwidth]{muon-decay-feynman.png} -\caption{Feynman diagram of a $\mu$-decay} -\label{muon-decay_feynman} -\end{subfigure} -\end{center} -\end{figure} - -\textbf{Lepton family number is conserved}\\ -According to the SM lepton family number is conserved. As all interactions beside the W conserve particle flavour, it is easy to see that lepton family number is conserved.\\ -Whenever a lepton interaction with a W boson, it just changes a lepton to its corresponding lepton neutrino and or the other way around (e.g. see Figure \ref{muon-decay_feynman}).\\\\ - -\section{Physics beyond the SM} - -\subsection{Neutrino Oscillation} - -Classically the SM considers neutrinos to be massless. While this assumption works well for a lot of cases, we know nowadays that at least two of the three neutrinos have to have mass\footnote{The mass difference between neutrinos is non zero: $m_i - m_j = \Delta m_{i,j} \neq 0$, $\forall j \neq i$}. Neutrinos are known to oscillate between all three states of flavour, as the eigenstates of flavour are not eigenstates of mass. As a consequence $\nu_e$, $\nu_\mu$ and $\nu_\tau$ are not fundamental particle states but a mixture of the mass eigenstates $\nu_1$, $\nu_2$ and $\nu_3$. They are connected through the PMNS matrix: - -\begin{equation} -\begin{pmatrix}\nu_e \\ \nu_\mu\\ \nu_\tau\end{pmatrix} = -\begin{pmatrix} -U_{e_1} & U_{e_2} & U_{e_3} \\ -U_{\mu_1} & U_{\mu_2} & U_{\mu_1} \\ -U_{\tau_1} & U_{\tau_2} & U_{\tau_3} -\end{pmatrix} -\begin{pmatrix} -\nu_1 \\ \nu_2 \\ \nu_3 -\end{pmatrix} -\label{PMNS_neutrino} -\end{equation} - -As a result neutrinos propagate as a superposition of all mass eigenstates. Additionally we can describe the PMNS matrix through three mixing angles $\theta_{12}$, $\theta_{13}$ and $\theta_{23}$ and a complex phase $\delta$ \footnote{Measurements: $\theta_{12} \approx 35^\circ$, $\theta_{13} \approx 10^\circ$, $\theta_{23} \approx 45^\circ$ - KamLAND, MINOS, ...}. The electron superposition looks then like this:\\\\ - -$\ket{\nu_e} = U_{e_1} \ket{\nu_1} e^{{-i \Phi_1}} + U_{e_2} \ket{\nu_2} e^{{-i \Phi_2}} + U_{e_3} \ket{\nu_3} e^{{-i \Phi_3}}$ with $\Phi_i = E_i \times t$\\\\ - - -As a result lepton family number is not a conserved quantity anymore as neutrino flavour oscillates over time.\\ -We can calculate the probability for a neutrino to transition from flavour $\alpha$ to $\beta$ like: - -\begin{equation} -\begin{split} -P(\nu_\alpha \rightarrow \nu_\beta) = 2 Re \left( U_{\alpha_1} U^\ast_{\beta_1} U^\ast_{\alpha_2} U_{\beta_2} e^{-i(\Phi_1-\Phi_2)}\right)\\ -+ 2 Re \left( U_{\alpha_1} U^\ast_{\beta_1} U^\ast_{\alpha_3} U_{\beta_3} e^{-i(\Phi_1-\Phi_3)}\right)\\ -+ 2 Re \left( U_{\alpha_2} U^\ast_{\beta_2} U^\ast_{\alpha_3} U_{\beta_3} e^{-i(\Phi_2-\Phi_3)}\right) -\end{split} -\label{neutrino_flavour_change_prob} -\end{equation} - -An important thing to note is, that if any elements of the PMNS matrix are complex, this process is not invariant under time reversal ($t \rightarrow -t$)\footnote{\alignLongunderstack{% - \text{The probability does not change if we add a complex phase to the PMNS}\\ \text{ matrix, just if one of the elements has a phase different from the others}}}\\ -$P(\nu_\alpha \rightarrow \nu_\beta) \neq P(\nu_\beta \rightarrow \nu_\alpha)$. - -\begin{figure}[H] -\begin{center} -\includegraphics[width=0.8\textwidth]{neutrino_oscillation.png} -\caption{Process that violates lepton family number conservation through neutrino oscillation} -\label{neutrino_osc_feyn} -\end{center} -\end{figure} - -Nowadays it's a well accepted fact that lepton family number gets violated through neutrino oscillation.\\ -But why should flavour oscillation be exclusive to neutrinos?\\ -Maybe there are ways for the EM charged leptons as well to directly transition to another lepton family\footnote{Maybe also possible for quarks?}? - -\subsection{New physics} - -As a consequence of neutrino oscillation lepton flavour is a broken symmetry. The SM has to be adapted to include lepton flavour violation (LFV) and massive neutrinos. LFV is also expected for charged neutrinos.\\ -Although it has yet to be determined how LFV violation exactly works to which scale it exists.\\\\ - -This may raise the question on why charged LFV has never been observed yet. This is especially surprising as the mixing angles of the neutrinos have been measured to be big.\\ -There are two reasons why charged LFV is strongly surpressed: -The first is that charged leptons are much heavier than neutrinos and the other that the mass differences between neutrino flavour are tiny compared to the W boson mass.\\\\ -In the classical SM, charged LFV is already forbidden at tree level. Though it can be induced indirectly through higher order loop diagrams (using neutrino oscillation). By adding new particles beyond the SM, we generate new ways for LFV in the charged sector to happen. As LFV is naturally generated in many models beyond the SM, finding charged LFV is a strong hint for new physics. - - -\begin{figure}[H] -\begin{center} -\includegraphics[width=0.8\textwidth]{charged_lfv_neutrino_osc.png} -\caption{Process that violates lepton family number conservation through neutrino oscillation} -\label{neutrino_osc_feyn} -\end{center} -\end{figure} \ No newline at end of file diff --git a/375px-Standard_Model_of_Elementary_Particles.svg.png b/375px-Standard_Model_of_Elementary_Particles.svg.png deleted file mode 100644 index 64c9730..0000000 --- a/375px-Standard_Model_of_Elementary_Particles.svg.png +++ /dev/null Binary files differ diff --git a/Report/00_Bachelor_Thesis.aux b/Report/00_Bachelor_Thesis.aux new file mode 100644 index 0000000..5356cf7 --- /dev/null +++ b/Report/00_Bachelor_Thesis.aux @@ -0,0 +1,20 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} 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Defining command \comm with sig. '' on line 202. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \anticommutator with sig. ' l m m ' on line 203. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \acommutator with sig. '' on line 204. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \acomm with sig. '' on line 205. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \vectorbold with sig. ' s m ' on line 208. +................................................. +................................................. +. 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LaTeX info: "xparse/define-command" +. +. Defining command \dotproduct with sig. '' on line 217. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \vdot with sig. '' on line 218. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \crossproduct with sig. '' on line 220. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \cross with sig. '' on line 221. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \cp with sig. '' on line 222. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \gradient with sig. ' g o d() ' on line 232. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \grad with sig. '' on line 233. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \divergence with sig. ' g o d() ' on line 243. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \div with sig. '' on line 246. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \curl with sig. ' g o d() ' on line 256. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \laplacian with sig. ' g o d() ' on line 266. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \principalvalue with sig. 'g' on line 274. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \pv with sig. '' on line 275. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \PV with sig. 'g' on line 276. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \Re with sig. 'g' on line 277. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \Im with sig. 'g' on line 278. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \opbraces with sig. ' m g o d() ' on line 299. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \trigbraces with sig. ' m o d() ' on line 305. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \sin with sig. '' on line 309. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \cos with sig. '' on line 310. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \tan with sig. '' on line 311. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \csc with sig. '' on line 312. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \sec with sig. '' on line 313. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \cot with sig. '' on line 314. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \arcsin with sig. '' on line 316. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \arccos with sig. '' on line 317. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \arctan with sig. '' on line 318. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \arccsc with sig. '' on line 320. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \arcsec with sig. 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'' on line 333. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \asec with sig. '' on line 335. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \acot with sig. '' on line 337. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \sinh with sig. '' on line 339. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \cosh with sig. '' on line 340. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \tanh with sig. '' on line 341. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \csch with sig. '' on line 343. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \sech with sig. '' on line 345. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \coth with sig. '' on line 346. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \exp with sig. '' on line 348. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \log with sig. '' on line 349. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \ln with sig. '' on line 350. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \det with sig. '' on line 351. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \Pr with sig. '' on line 352. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \tr with sig. '' on line 353. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \Tr with sig. '' on line 354. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \Res with sig. '' on line 355. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qqtext with sig. ' s m ' on line 377. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qq with sig. '' on line 378. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qcomma with sig. '' on line 380. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qc with sig. '' on line 381. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qif with sig. 's' on line 383. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qthen with sig. 's' on line 384. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qelse with sig. 's' on line 385. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qotherwise with sig. 's' on line 386. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qunless with sig. 's' on line 387. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qgiven with sig. 's' on line 388. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qusing with sig. 's' on line 389. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qassume with sig. 's' on line 390. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qsince with sig. 's' on line 391. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qlet with sig. 's' on line 392. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qfor with sig. 's' on line 393. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qall with sig. 's' on line 394. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qeven with sig. 's' on line 395. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qodd with sig. 's' on line 396. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qinteger with sig. 's' on line 397. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qand with sig. 's' on line 398. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qor with sig. 's' on line 399. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qas with sig. 's' on line 400. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qin with sig. 's' on line 401. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qcc with sig. 's' on line 402. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \differential with sig. ' o g d() ' on line 415. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \dd with sig. '' on line 416. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \derivative with sig. ' s o m g d() ' on line 435. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \dv with sig. '' on line 436. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \partialderivative with sig. ' s o m g g d() ' on line 460. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \pderivative with sig. '' on line 461. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \pdv with sig. '' on line 462. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \variation with sig. ' o g d() ' on line 474. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \var with sig. '' on line 475. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \functionalderivative with sig. ' s o m g d() ' on line +. 494. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \fderivative with sig. '' on line 495. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \fdv with sig. '' on line 496. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \bra with sig. ' s m t\ket s g ' on line 526. +................................................. +************************************************* +* LaTeX warning: "xparse/not-single-char" +* +* Argument delimiter '\ket ' for the command '\bra' should be a single +* character. +************************************************* +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \ket with sig. ' s m ' on line 533. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \innerproduct with sig. ' s m g ' on line 548. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \braket with sig. '' on line 549. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \ip with sig. '' on line 550. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \outerproduct with sig. ' s m g ' on line 565. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \dyad with sig. '' on line 566. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \op with sig. '' on line 567. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \ketbra with sig. '' on line 568. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \expectationvalue with sig. ' s s m g ' on line 587. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \expval with sig. '' on line 588. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \ev with sig. '' on line 589. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \vev with sig. ' m ' on line 590. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \matrixelement with sig. ' s s m m m ' on line 601. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \matrixel with sig. '' on line 602. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \mel with sig. '' on line 603. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \identitymatrix with sig. 'm' on line 633. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \imat with sig. '' on line 634. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \xmatrix with sig. ' s m m m ' on line 659. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \xmat with sig. '' on line 660. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \zeromatrix with sig. ' m g ' on line 662. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \zmat with sig. '' on line 663. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \paulixmatrix with sig. '' on line 665. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \pauliymatrix with sig. '' on line 666. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \paulizmatrix with sig. '' on line 667. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \paulimatrix with sig. 'm' on line 678. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \pmat with sig. '' on line 679. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \diagonalmatrix with sig. 'O{} >{\SplitList {,}}m ' on line +. 681. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \@dmat with sig. 'mmggggggg' on line 714. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \dmat with sig. '' on line 715. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \antidiagonalmatrix with sig. 'O{} >{\SplitList {,}}m ' on +. line 717. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \@admat with sig. 'mmggggggg' on line 750. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \admat with sig. '' on line 751. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \flatfrac with sig. ' m m ' on line 754. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \homework with sig. 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+\BOOKMARK [2][-]{subsection.2.2}{New physics}{section.2}% 6 +\BOOKMARK [1][-]{section.3}{Experimental setup of the eee-experiment}{}% 7 +\BOOKMARK [1][-]{section.4}{Machine learning}{}% 8 diff --git a/Report/00_Bachelor_Thesis.pdf b/Report/00_Bachelor_Thesis.pdf new file mode 100644 index 0000000..1843e60 --- /dev/null +++ b/Report/00_Bachelor_Thesis.pdf Binary files differ diff --git a/Report/00_Bachelor_Thesis.synctex.gz b/Report/00_Bachelor_Thesis.synctex.gz new file mode 100644 index 0000000..0f20fae --- /dev/null +++ b/Report/00_Bachelor_Thesis.synctex.gz Binary files differ diff --git a/Report/00_Bachelor_Thesis.tex b/Report/00_Bachelor_Thesis.tex new file mode 100644 index 0000000..52ec2d6 --- /dev/null +++ b/Report/00_Bachelor_Thesis.tex @@ -0,0 +1,71 @@ +\documentclass[10pt,a4paper]{article} +\usepackage[utf8]{inputenc} +\usepackage[english]{babel} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{graphicx} +\usepackage{fancyhdr} +\usepackage{paralist} +\pagestyle{fancy} +\usepackage{pdfpages} +\usepackage{subcaption} +\usepackage{float} +\usepackage{hyperref} +\usepackage{physics} +\usepackage{tabstackengine} +\usepackage[bottom]{footmisc} +\setlength{\skip\footins}{1.2pc plus 5pt minus 2pt} +\title{Particle Track reconstruction using a recurrent neural network at the $\mu-3e$ experiment} +\author{\textit{Bachelor thesis of}\\ Sascha Liechti} +\date{06.04.2018} + +\lhead{} +\chead{} +\rhead{} + +\lfoot{} +\cfoot{\thepage} +\rfoot{} + +\renewcommand{\headrulewidth}{0.4pt} +\renewcommand{\footrulewidth}{0pt} + +\setlength{\parindent}{0pt} + +\begin{document} + +\maketitle + +\begin{center} +\textit{Supervised by}\\ +Prof. Nicola Serra\\ +Dr. Patrick Owen\\ +\end{center} + +\textbf{\large{Abstract}} +During the $\mu-3e$ experiment we faced the challenge of reconstructing the paths of certain low momentum particles that curled back into the detector and cause additional hits. To face this, a recurrent neural network was used which found the right track for $87 \%$ of these particles. + +\newpage +\tableofcontents + +\newpage + +\include{01_Standard_Model} + +%\include{02_Setup} +% +%\include{03_Datatable} +% +%\include{04_Results} +% +%\include{05_Graphics} +% +%\include{06_Calculus} +% +%\include{07_Error-Calculus} + +%\include{08_Bibliography} + + +\end{document} diff --git a/Report/00_Bachelor_Thesis.toc b/Report/00_Bachelor_Thesis.toc new file mode 100644 index 0000000..aec68db --- /dev/null +++ b/Report/00_Bachelor_Thesis.toc @@ -0,0 +1,9 @@ +\babel@toc {english}{} +\contentsline {section}{\numberline {1}Standard Model}{3}{section.1} +\contentsline {subsection}{\numberline {1.1}Elementary particles and forces}{3}{subsection.1.1} +\contentsline {subsection}{\numberline {1.2}Interaction rules}{5}{subsection.1.2} +\contentsline {section}{\numberline {2}Physics beyond the SM}{6}{section.2} +\contentsline {subsection}{\numberline {2.1}Neutrino Oscillation}{6}{subsection.2.1} +\contentsline {subsection}{\numberline {2.2}New physics}{7}{subsection.2.2} +\contentsline {section}{\numberline {3}Experimental setup of the $\mu \rightarrow eee$-experiment}{8}{section.3} +\contentsline {section}{\numberline {4}Machine learning}{8}{section.4} diff --git a/Report/01_Standard_Model.aux b/Report/01_Standard_Model.aux new file mode 100644 index 0000000..b38ae8c --- /dev/null +++ b/Report/01_Standard_Model.aux @@ -0,0 +1,61 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\@writefile{toc}{\contentsline {section}{\numberline {1}Standard Model}{3}{section.1}} +\@writefile{toc}{\contentsline {subsection}{\numberline {1.1}Elementary particles and forces}{3}{subsection.1.1}} +\newlabel{intro_elem_part}{{1.1}{3}{Elementary particles and forces}{subsection.1.1}{}} +\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Quarks in the Standard Model\relax }}{3}{table.caption.2}} 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There are 6 types of quarks and six types of leptons. The type of a particle is conventionally called flavour. The six quark flavours and the six lepton flavours are separated over 3 generations (each which two quarks and two leptons in it). +Experimental evidence suggests that there exist exactly three generations of particles. Each particle of the first generation has higher energy versions of itself with the same characteristics (e.g. $e^- \rightarrow \mu^- \rightarrow \tau^-$)as in other generations. Contrary, each following generation has a higher mass than the generation before.\\ + + +\begin{table}[H] +\begin{center} +\caption{Quarks in the Standard Model} \label{Quark_SM_table} +\begin{tabular}{l l l l l} +&&Quarks&&\\\hline + & Particle& & Q$[e]$ & $\frac{mass}{GeV}$\\ \hline +1. Gen. & up & u & $-\frac{1}{3}$ & $0.003$\\ + & down & d & $\frac{2}{3}$ & $0.005$\\ +2. Gen. & strange & s & $-\frac{1}{3}$ & $0.1$\\ + & charm & c & $\frac{2}{3}$ & $1.3$\\ +3. Gen. & bottom & b & $-\frac{1}{3}$ & $4.5$\\ + & top & t & $\frac{2}{3}$ & $174$\\ +\end{tabular} +\end{center} +\end{table} + + +One category consists of quarks($q$)(see Table \ref{Quark_SM_table}). In this, we differentiate between up-type quarks, with charge $-\frac{1}{3}e$, and down-type, quarks with charge $\frac{2}{3}e$. Quarks interact with all fundamental forces.\\ +Each quark carries a property called colour-charge. The possible color charges are red(r), green(gr), blue(bl) in which anti-quarks carry anti-colour. Quarks can only carry one colour, whilst every free particle has to be colorless\footnote{Colour confinement}. In conclusion we cannot observe a single quark.\\ +Free particles can achieve being colourless in two ways. Either by having all three colors present in the same amount (one quark of each color), which creates the characteristic group of baryons($qqq$) and anti-baryons($\bar{q}\bar{q}\bar{q}$) or by having a color and its anticolor present, which creates the group of mesons($q\bar{q}$).\\\newpage + + +\begin{table}[H] +\begin{center} +\caption{Leptons in the standard model} \label{Lepton_SM_table} +\begin{tabular}{l l l l l} + & & Leptons & &\\ \hline + & Particle& & Q$[e]$ & $\frac{mass}{GeV}$\\ \hline +1. Gen. & electron & $e^-$ & $-1$ & $0.005$\\ + & neutrino & $\nu_e$ & $0$ & $<10^{-9}$\\ +2. Gen. & muon & $\mu^-$ & $-1$ & $0.106$\\ + & neutrino & $\nu_{\mu}$ & $0$ & $<10^{-9}$\\ +3. Gen. & tau & $\tau^-$ & $-1$ & $1.78$\\ + & neutrino & $\nu_{\tau}$ & $0$ & $<10^{-9}$ + \label{Lepton_table} +\end{tabular} +\end{center} +\end{table} + +The other group consists of leptons(l)(see Table \ref{Lepton_SM_table}). They only interact through the weak and the electromagnetic force. Each generation consists of a lepton of charge -1 and a corresponding EM neutrally charged neutrino. The electron has the lowest energy of all charged leptons. This makes the electron stable while the higher generation particles decay to lower energy particles. \\ +\\ + +The leptons of one generation, namely the charged lepton and its corresponding neutrino are called a lepton family. A lepton of a family counts as 1 to its corresponding lepton family number whilst a anti-lepton counts as -1. + +\begin{table}[H] +\begin{center} +\caption{Fundamental forces} \label{fund_forces_table} +\begin{tabular}{l l l l l l l} +Force & Strengh & Boson & & Spin & Charge & $\frac{mass}{GeV}$ \\\hline +Strong & 1 & gluon & $g$ & 1& 0& 0 \\ +Electromagnetism & $10^{-3}$ & photon & $\gamma$ & 1& 0& 0 \\ +Weak & $10^{-8}$ & Z boson & $Z$ & 1& 0& 80.4\\ + & $10^{-8}$ & W boson & $W^{\pm}$& 1& $\pm 1$& 91.2\\ +\end{tabular} +\end{center} +\end{table} + + +The particles of the SM interact through the 3 fundamental forces of the SM. In these interactions, particles called bosons are being exchanged which are the carriers of their respective force (see Table \ref{fund_forces_table}).\\ +As mentioned above, only quarks can interact through the strong force, in which they exchange gluons. Gluons are massless and EM neutrally charged. The strong force has the biggest coupling strengh of 1 (though it decreases with higher energies as a result of gluon-gluon self interaction loops, which interfere negatively in perturbation theory)\footnote{Mark Thomson - Modern Particle physics - 10.5.2}. A gluon carries colour charge and hence can change the colour of a quark but it conserves its flavour. The strong interaction has an underlying gauge symmetry of SU(3). Therefore, it can be derived that color charge is conserved through the strong interaction\footnote{E.g. through Gell-Mann matrices}.\\ +The electromagnetic(EM) force is propagated through the photon. It carries zero charge and no invariant mass. Exclusively charged particles can interact through the electromagnetic force. The coupling strength is $\alpha \approx \frac{1}{137}$, contrary to the strong force the coupling constant increases with higher energies\footnote{Mark Thomson - Modern Particle physics - 10.5.1}. This difference stems from the fact that photon-photon interaction loops are not allowed whereas gluon-gluon interaction loops are. In perturbation theory this results in only positive terms being added to the coupling strength. The underlying gauge symmetry is of SU(1). The electromagnetic force also conserves flavour.\\ +The weak force has two types of bosons. The bosons of the weak force are the only bosons to have an inertial mass.\\ +First we will discuss the EM neutrally charged Z boson\footnote{Discovered at Super Proton Synchrotron accelerator - Cern - 1983}. Even though the Z boson belongs to the weak force it, it also has an electromagnetic part additionally to the weak force part\footnote{$Z \rightarrow EM_{part} + W^3$, Modern Particle Physics - Blababla}. It follows directly, that the Z boson couples weaker to uncharged particles.\\ +The other boson of the weak force is the W boson\footnote{Discovered at Super Proton Synchrotron accelerator - Cern - 1983}. In the classical SM, the only way particles can change flavour is through the weak force by emitting or absorbing W boson. It is important to notice that, besides of having an invariant mass, the W boson is the only boson with a non zero charge ($Q_{W^\pm} = \pm 1e$). In the gauge symmetry of the weak force the $W^\pm$ are actually the creation and annihilation operators of said symmetry\footnote{$W^\pm = W_1 \pm i W_2$}.\\ +An important characteristic of the weak force is that it exclusively couples to lefthanded(LH) particles and righthanded(RH) antiparticles (describing chirality states)\footnote{In the ultrarelativistic limit helicity and chirality eigenstates are the same}.\\ +The chirality operators for left- and righthandedness are: \\\\ +LH: $\frac{1}{2}(1-\gamma^5)$, RH: $\frac{1}{2}(1+\gamma^5)$\\\\ +As a consequence RH particles and LH anti-particles cant couple to the W boson at all. This also results in charged RH particles and LH anti-particles to couple to the Z boson only through the electromagnetic part of the itself, while uncharged RH particles and LH anti particles (e.g. RH $\nu$, LH $\bar{\nu}$) don't couple with the EM force nor the weak force. + + +\subsection{Interaction rules} + +Now we will establish the general rules for interactions in the SM.\\\\ +\textbf{Baryon number is conserved}\\ +As we already established before, the only interaction that can change flavour is the weak force through the W boson. We directly see that all other interactions baryon number has to be conserved. So any up-type quark can be changed to a down-type quark and backwards by emitting or absorbing a W boson. In the end however, there are still 3 quarks which form a baryon\footnote{I exclude $q\bar{q}$ pair-antipair production to form a pentaquark($qqqq\bar{q}$) and other exotic states}, even though it changed its type and charge. A well known example is the beta decay, where a down quark in a neutron decays into a an up quark to form now a proton(e.g. see Figure \ref{beta-decay_feynman}). We easily see that the baryon number is conserved.\\\\ + +\begin{figure}[H] +\begin{center} +\begin{subfigure}{0.45\textwidth} +\includegraphics[width=0.8\textwidth]{beta_decay_feynman.png} +\caption{Feynman diagram of the $\beta$-decay} +\label{beta-decay_feynman} +\end{subfigure} +\begin{subfigure}{0.45\textwidth} +\includegraphics[width=0.8\textwidth]{muon-decay-feynman.png} +\caption{Feynman diagram of a $\mu$-decay} +\label{muon-decay_feynman} +\end{subfigure} +\end{center} +\end{figure} + +\textbf{Lepton family number is conserved}\\ +According to the SM lepton family number is conserved. As all interactions beside the W conserve particle flavour, it is easy to see that lepton family number is conserved.\\ +Whenever a lepton interaction with a W boson, it just changes a lepton to its corresponding lepton neutrino and or the other way around (e.g. see Figure \ref{muon-decay_feynman}).\\\\ + +\section{Physics beyond the SM} + +\subsection{Neutrino Oscillation} + +Classically the SM considers neutrinos to be massless. While this assumption works well for a lot of cases, we know nowadays that at least two of the three neutrinos have to have mass\footnote{The mass difference between neutrinos is non zero: $m_i - m_j = \Delta m_{i,j} \neq 0$, $\forall j \neq i$}. Neutrinos are known to oscillate between all three states of flavour, as the eigenstates of flavour are not eigenstates of mass. As a consequence $\nu_e$, $\nu_\mu$ and $\nu_\tau$ are not fundamental particle states but a mixture of the mass eigenstates $\nu_1$, $\nu_2$ and $\nu_3$. They are connected through the PMNS matrix: + +\begin{equation} +\begin{pmatrix}\nu_e \\ \nu_\mu\\ \nu_\tau\end{pmatrix} = +\begin{pmatrix} +U_{e_1} & U_{e_2} & U_{e_3} \\ +U_{\mu_1} & U_{\mu_2} & U_{\mu_1} \\ +U_{\tau_1} & U_{\tau_2} & U_{\tau_3} +\end{pmatrix} +\begin{pmatrix} +\nu_1 \\ \nu_2 \\ \nu_3 +\end{pmatrix} +\label{PMNS_neutrino} +\end{equation} + +As a result neutrinos propagate as a superposition of all mass eigenstates. Additionally we can describe the PMNS matrix through three mixing angles $\theta_{12}$, $\theta_{13}$ and $\theta_{23}$ and a complex phase $\delta$ \footnote{Measurements: $\theta_{12} \approx 35^\circ$, $\theta_{13} \approx 10^\circ$, $\theta_{23} \approx 45^\circ$ - KamLAND, MINOS, ...}. The electron superposition looks then like this:\\\\ + +$\ket{\nu_e} = U_{e_1} \ket{\nu_1} e^{{-i \Phi_1}} + U_{e_2} \ket{\nu_2} e^{{-i \Phi_2}} + U_{e_3} \ket{\nu_3} e^{{-i \Phi_3}}$ with $\Phi_i = E_i \times t$\\\\ + + +As a result lepton family number is not a conserved quantity anymore as neutrino flavour oscillates over time.\\ +We can calculate the probability for a neutrino to transition from flavour $\alpha$ to $\beta$ like: + +\begin{equation} +\begin{split} +P(\nu_\alpha \rightarrow \nu_\beta) = 2 Re \left( U_{\alpha_1} U^\ast_{\beta_1} U^\ast_{\alpha_2} U_{\beta_2} e^{-i(\Phi_1-\Phi_2)}\right)\\ ++ 2 Re \left( U_{\alpha_1} U^\ast_{\beta_1} U^\ast_{\alpha_3} U_{\beta_3} e^{-i(\Phi_1-\Phi_3)}\right)\\ ++ 2 Re \left( U_{\alpha_2} U^\ast_{\beta_2} U^\ast_{\alpha_3} U_{\beta_3} e^{-i(\Phi_2-\Phi_3)}\right) +\end{split} +\label{neutrino_flavour_change_prob} +\end{equation} + +An important thing to note is, that if any elements of the PMNS matrix are complex, this process is not invariant under time reversal ($t \rightarrow -t$)\footnote{\alignLongunderstack{% + \text{The probability does not change if we add a complex phase to the PMNS}\\ \text{ matrix, just if one of the elements has a phase different from the others}}}\\ +$P(\nu_\alpha \rightarrow \nu_\beta) \neq P(\nu_\beta \rightarrow \nu_\alpha)$. + +\begin{figure}[H] +\begin{center} +\includegraphics[width=0.8\textwidth]{neutrino_oscillation.png} +\caption{Process that violates lepton family number conservation through neutrino oscillation} +\label{neutrino_osc_feyn} +\end{center} +\end{figure} + +Nowadays it's a well accepted fact that lepton family number gets violated through neutrino oscillation.\\ +But why should flavour oscillation be exclusive to neutrinos?\\ +Maybe there are ways for the EM charged leptons as well to directly transition to another lepton family\footnote{Maybe also possible for quarks?}? + +\subsection{New physics} + +As a consequence of neutrino oscillation lepton flavour is a broken symmetry. The SM has to be adapted to include lepton flavour violation (LFV) and massive neutrinos. LFV is also expected for charged neutrinos.\\ +Although it has yet to be determined how LFV violation exactly works to which scale it exists.\\\\ + +This may raise the question on why charged LFV has never been observed yet. This is especially surprising as the mixing angles of the neutrinos have been measured to be big.\\ +There are two reasons why charged LFV is strongly surpressed: +The first is that charged leptons are much heavier than neutrinos and the other that the mass differences between neutrino flavour are tiny compared to the W boson mass.\\\\ +In the classical SM, charged LFV is already forbidden at tree level. Though it can be induced indirectly through higher order loop diagrams (using neutrino oscillation). By adding new particles beyond the SM, we generate new ways for LFV in the charged sector to happen. As LFV is naturally generated in many models beyond the SM, finding charged LFV is a strong hint for new physics.\\ + +\textbf{Image LFV via neutrino, at tree level and directly involving supersymmetric particles}\\ + +One way charged LFV can occur is through super symmetric particles (see Figure Bratenene). By observing charged LFV supersymmetry would gain new importance.\\ +Together with supersymmetric models, other extensions of the SM such as left-right symmetric models, grand unified models, models with an extended Higgs sector and models where electroweak symmetry is broken dynamically are all good candidates to explain charged LFV and most importantly experimentally accessible in a large region of the parameter space. + +\section{Experimental setup of the $\mu \rightarrow eee$-experiment} + +\section{Machine learning} + +Machine learning has already proven itself to be very successful in resolving many problems in numerous other areas of science and also in the private sector. Based on these promising results, scientists are eager to study the potential of machine learning in physics. \ No newline at end of file diff --git a/Report/02_Setup.tex b/Report/02_Setup.tex new file mode 100644 index 0000000..e69de29 --- /dev/null +++ b/Report/02_Setup.tex diff --git a/Report/04_Results.tex b/Report/04_Results.tex new file mode 100644 index 0000000..fdfdc03 --- /dev/null +++ b/Report/04_Results.tex @@ -0,0 +1,19 @@ +\section{Results} +In the positronium experiment we measured the time spectrum of a positronium decay. \\ + +For the positron in aluminium we find a small life time. The fit gives us: + +\begin{center} +$\tau = 0.134(10) ns$ +\end{center} +According to the manual of this experiment we should see a lifetime of $100 - 200 ps$. So our result appears to be correct.\\ +\\ + +In POM positronium forms and we measured for the lifetimes of ortho- and para-positronium: + +\begin{center} +\begin{align*} +\tau_{para} = 0.246(5) ns\\ +\tau_{ortho} = 1.800(34) ns +\end{align*} +\end{center} diff --git a/Report/05_Graphics.tex b/Report/05_Graphics.tex new file mode 100644 index 0000000..3d43789 --- /dev/null +++ b/Report/05_Graphics.tex @@ -0,0 +1,74 @@ +\section{Data and Measurement} +\subsection{Energies} +The data that we measured in this experiment was stored in form of a histogram. There is no need to show the raw data, it's not very useful.\\ +We started with a raw measurement of the energies of the gamma photons, which came out of our experiment for both photomultipliers. We expected two peaks at $511keV$ and $1275keV$ for the photons. + + +\begin{figure}[H] +\begin{center} +\includegraphics[width=0.8\textwidth]{picture/FullSpectrum.png} +\caption{Full spectrum by both photomultiplier} +\end{center} +\end{figure} + + +We recognise the two peaks at bin number $\sim 800$ and $\sim 1900$.\\ +As next we chose the window of measurement for both photomultiplier. This will then give us the gate for the measurement on the digital converter. + + +\begin{figure}[H] +\begin{center} +\begin{subfigure}{0.45\textwidth} +\includegraphics[width=0.8\textwidth]{picture/511keV.png} +\caption{$511keV$ peak by PM 1} +\end{subfigure} +\begin{subfigure}{0.45\textwidth} +\includegraphics[width=0.8\textwidth]{picture/1275keV.png} +\caption{$1275keV$ peak by PM 2} +\end{subfigure} +\caption{Measured energy windows by both photomultiplier} +\end{center} +\end{figure} +\newpage + + +\subsection{Lifetime measurement} +Next we measured the lifetime of the positronium in aluminium and polymer. + + +\begin{figure}[H] +\begin{center} +\includegraphics[width=0.8\textwidth]{picture/LifetimeData.png} +\caption{Lifetime in polymer} +\end{center} +\end{figure} + + +\begin{figure}[H] +\begin{center} +\includegraphics[width=0.8\textwidth]{picture/LifetimeAluminium.png} +\caption{Lifetime in aluminium} +\end{center} +\end{figure} + + +At this point we quickly realize an asymmetric peak on the right side, about $100$ bins after the main peak. After some tries we changed a constant fraction discriminator and started a new set of measurement. + + +\begin{figure}[H] +\begin{center} +\includegraphics[width=0.8\textwidth]{picture/LifeTimeData2.png} +\caption{Lifetime in polymer} +\end{center} +\end{figure} + + +\begin{figure}[H] +\begin{center} +\includegraphics[width=0.8\textwidth]{picture/LifetimeAluminium2.png} +\caption{Lifetime in aluminium} +\end{center} +\end{figure} + + +Which looks much cleaner. We used this last measurements for future evaluation.\\ \ No newline at end of file diff --git a/Report/06_Calculus.tex b/Report/06_Calculus.tex new file mode 100644 index 0000000..9afa7b4 --- /dev/null +++ b/Report/06_Calculus.tex @@ -0,0 +1,195 @@ +\section{Evaluation} + +\subsection{Time Calibration} +All our measurements have been done through a time to amplitude converter, which through a digital converter, produces a histogram of all measured times. This needs to be calibrated to be anyhow useful. To calibrate, two set of measurements have been done with a clock. + + +\begin{figure}[H] +\begin{center} +\includegraphics[width=0.75\textwidth]{picture/TimeCal_Clock.png} +\caption{Calibration with counter} +\label{clock} +\end{center} +\end{figure} + +The first measurement (figure \ref{clock}) was done with a counter and gave us $10ns$ intervals but the measured peaks are not very stable. + + +\begin{figure}[H] +\begin{center} +\includegraphics[width=0.75\textwidth]{picture/TimeCal_Cable.png} +\caption{Calibration with cable} +\label{cable-clock} +\end{center} +\end{figure} + +The second measurement was done with cables as a delay (figure \ref{cable-clock}). For that we used $8ns$ cables. This time the peaks appear to be much more sharp and we decided to use this to calibrate the bins of the histogram.\\ +One interval is bigger due to the use of a $10ns$ cable by accident.\\ +Using a quick fit of 7 gaussians, we could determine precisely the position of the peak and we put it against the real time delay. + + +\begin{figure}[H] +\begin{center} +\includegraphics[width=0.8\textwidth]{picture/Calibration_Cable.png} +\caption{Slope of calibration} +\label{fig.calib} +\end{center} +\end{figure} + + +A linear fit (figure \ref{fig.calib}) gave us the slope of the line and at the same moment this will be the factor to transform the bin number in relative time, since the $0s$ is not defined. + +\begin{center} +$t_{rel} = n^{\circ}_{bin} \cdot a_{slope} + b$ +\end{center} + +We then get the following value: +\begin{center} + +$a_{slope} = 0.02443 ns$ +\end{center} + +We applied this factor to all of our future calculations. At this point we assumed that we do not need to consider errors. +\newpage + +\subsection{Positron lifetime in Aluminium} +\subsubsection{Simple fit} + +A set of measurements have been done with the aluminium frame in order to find the starting point of the process. For that we used a gaussian to fit the measured lifetime distribution with the least square method. + +\begin{equation} +f_{gauss}(t) = A \cdot \exp^{-\frac{(t-\mu)^2}{2\sigma^2}} + const +\end{equation} + +This fit (figure \ref{fig.gau_alu}) gives us the following results: + +\begin{center} +$A = 1990(10)$\\ +$\mu = 17.122(1) ns$ \\ +$\sigma = 0.301(1) ns$ +\end{center} + +We then quickly see that the function does not quite fit the data. We see a clear structure in the residual (figure \ref{fig.res_alu}) and a little tail on the right side of the gaussian. The last is an evidence for a small life time of the positron diffusing through aluminium frame.\\ +We note that all the errors shown here have been given by the least square method we used on python. + +\begin{figure}[H] +\begin{center} +\includegraphics[width=0.8\textwidth]{picture/FittedAlu.png} +\caption{Fitted gaussian} +\label{fig.gau_alu} +\end{center} +\end{figure} + +\begin{figure}[H] +\begin{center} +\includegraphics[width=0.8\textwidth]{picture/ResidualsAlu.png} +\caption{Residual} +\label{fig.res_alu} +\end{center} +\end{figure} + +\subsubsection{Advanced fit} +Since a simple gaussian was not enough to fit the the data correctly, we decided to use a combination of distributions (see formula \ref{eq.spec_distrib_A}). \\ + +\begin{equation} +F(t) = g_{\text{gauss}}(t) + a_{\text{convol}}(t) + \text{const.} +\label{eq.spec_distrib_A} +\end{equation} + +where $g_{\text{gauss}}(t)$ is a simple gaussian distribution (formula \ref{eq.gauss}) + +\begin{equation} +g_{\text{gauss}}(t) = G e^{-\frac{(t-\mu_{g})^2}{2\sigma_{g}^2}} +\label{eq.gauss} +\end{equation} + +and the distribution $a_{\text{convul}}(t)$ is a convulsion between a gaussian and exponential distribution (formula \ref{eq.convul}). We used this to equate for the fact that the positron diffuses through the metal before it annihilates. + +\begin{equation} +a_{convol}(t) = A e^{-\frac{1}{\tau}\left((t-\mu) - \frac{\sigma^2}{2\tau}\right)} \cdot \left(1 + erf\left(\frac{(t-\mu)-{\sigma^2}{\frac{1}{\tau}}}{\sqrt{2}\sigma}\right)\right) +\label{eq.convul} +\end{equation} + +This distribution gave us a much cleaner fit on the data (figure \ref{fig.gau_alu_s}). + +\begin{figure}[H] +\begin{center} +\includegraphics[width=0.8\textwidth]{picture/FittedAlu_s.png} +\caption{Fitted distribution} +\label{fig.gau_alu_s} +\end{center} +\end{figure} + +\begin{figure}[H] +\begin{center} +\includegraphics[width=0.8\textwidth]{picture/ResidualsAlu_s.png} +\caption{Residual} +\label{fig.res_alu_s} +\end{center} +\end{figure} +\newpage +The residuals looks almost perfect. We get diverse values from this fit. +\begin{center} +\begin{tabular}{r l} +&Convolution 1 \\ +$A = $&$4501(283)$\\ +$\tau = $&$0.134(10) ns $\\ +$\mu = $&$16.994(7) ns $\\ +$\sigma = $&$0.228(3) ns$\\ +\\ +&Gaussian \\ +$A = $&$267(29)$\\ +$\mu = $&$17.098(81) ns$\\ +$\sigma = $&$0.692(62) ns$ +\end{tabular} +\end{center} + +Also here, the errors have been calculated by the least square method on python. A good indication if our fit now works, is to take a look at the $\chi^2$-value. In our case its $\chi^2 = 1.081$, which is very good. This confirmed our fit.\\ +Even though no positronium forms in aluminium, we see a small lifetime, due to the positron diffusing through the aluminium before it annihilates with an electron of the electron gas. + +\subsection{Lifetime in POM} +For the lifetime distribution in a polymer basically reused the same function as developed above (formula \ref{eq.spec_distrib_POM}) and just added another convoluted gaussian (2 particles decaying: para- and ortho-positronium) and fit the data. + +\begin{equation} +F(t) = g_{\text{gauss}}(t) + a_{\text{convol}}(t) + b_{\text{convol}}(t) + \text{const.} +\label{eq.spec_distrib_POM} +\end{equation} + +\begin{figure}[H] +\begin{center} +\includegraphics[width=0.8\textwidth]{picture/FittedPOM.png} +\caption{Fitted distribution} +\label{fig.gau_pom_s} +\end{center} +\end{figure} + +Also here we can describe precisely our data with this distribution. From the fit we then get the following result: + +\begin{center} +\begin{tabular}{r l r l} +&Convolution 1 & &Convolution 2\\ +$A = $ & $12141(201)$ & $B = $ & $973(35)$ \\ +$\tau = $&$0.246(5) ns $&$\tau = $&$1.800(34) ns $\\ +$\mu = $&$16.998(2) ns $&$\mu = $&$16.597(9) ns$\\ +$\sigma = $&$0.208(2) ns$&$\sigma = $&$0.180(7) ns$\\ +\\ +&Gaussian & & \\ +$G =$ & $608(32)$\\ +$\mu = $&$17.279(23) ns$ & &\\ +$\sigma = $&$0.809(19) ns$ & & +\end{tabular} +\end{center} + +\begin{figure}[H] +\begin{center} +\includegraphics[width=0.8\textwidth]{picture/ResidualsPOM.png} +\caption{Residual} +\label{fig.res_pom_s} +\end{center} +\end{figure} + +This is again a very clean result. The value $\chi^2 = 1.081$ also confirmed a good fit.\\ +This time we see clearly a difference in lifetime. So we can recognize the para-positronium with a lifetime $\tau=0.246(5)ns$ and the ortho-positronium with $\tau = 1.800(34)ns$. This values are coherent with what we expected.\\ +\\ +At this point we want to mention that the goal of the measurement with the aluminium frame was to define the starting point of the process. The position should then be used as a constant parameter in the fit. In the end we decided that this wasn't necessary since the position of the gaussians in all three fits (figures \ref{fig.gau_alu}, \ref{fig.gau_alu_s} and \ref{fig.res_pom_s}) agrees with very little variation. Nevertheless we tried some fits while holding the parameters $\mu$ and $\sigma$ fixed. The result of these fits agreed on previous results, so we do not show the result here.\\ +Again, we want to remember that all the errors on the values are given by the least square method. diff --git a/Report/08_Bibliography.tex b/Report/08_Bibliography.tex new file mode 100644 index 0000000..9cbf0a3 --- /dev/null +++ b/Report/08_Bibliography.tex @@ -0,0 +1,12 @@ +\section{Bibliography} +\subsection{Sources} + +\begin{itemize} +\item Versuchsanleitung Positronenvernichtung, Andres van der Schaaf, 15.02.06 +\end{itemize} + +\subsection{Graphics} + +\begin{itemize} +\item Figure \ref{Na-decay}: \url{https://www.nucleonica.com/Application/ReducedDecaySchemes/Na22_RDS.png} +\end{itemize} \ No newline at end of file diff --git a/Report/375px-Standard_Model_of_Elementary_Particles.svg.png b/Report/375px-Standard_Model_of_Elementary_Particles.svg.png new file mode 100644 index 0000000..64c9730 --- /dev/null +++ b/Report/375px-Standard_Model_of_Elementary_Particles.svg.png Binary files differ diff --git a/Report/Links.txt b/Report/Links.txt new file mode 100644 index 0000000..8565fec --- /dev/null +++ b/Report/Links.txt @@ -0,0 +1,3 @@ +https://st.depositphotos.com/1232814/3851/v/950/depositphotos_38511129-stock-illustration-standard-model-of-elementary-particles.jpg +https://upload.wikimedia.org/wikipedia/commons/thumb/8/89/Beta_Negative_Decay.svg/200px-Beta_Negative_Decay.svg.png +https://upload.wikimedia.org/wikipedia/commons/1/13/Muon_Decay.svg \ No newline at end of file diff --git a/Report/beta_decay_feynman.png b/Report/beta_decay_feynman.png new file mode 100644 index 0000000..1e1bb7f --- /dev/null +++ b/Report/beta_decay_feynman.png Binary files differ diff --git a/Report/depositphotos_38511129-stock-illustration-standard-model-of-elementary-particles.jpg b/Report/depositphotos_38511129-stock-illustration-standard-model-of-elementary-particles.jpg new file mode 100644 index 0000000..82cc366 --- /dev/null +++ b/Report/depositphotos_38511129-stock-illustration-standard-model-of-elementary-particles.jpg Binary files differ diff --git a/Report/muon-decay-feynman.png b/Report/muon-decay-feynman.png new file mode 100644 index 0000000..c5c1208 --- /dev/null +++ b/Report/muon-decay-feynman.png Binary files differ diff --git a/Report/neutrino_oscillation.png b/Report/neutrino_oscillation.png new file mode 100644 index 0000000..ec0c31f --- /dev/null +++ b/Report/neutrino_oscillation.png Binary files differ diff --git a/beta_decay_feynman.png b/beta_decay_feynman.png deleted file mode 100644 index 1e1bb7f..0000000 --- a/beta_decay_feynman.png +++ /dev/null Binary files differ diff --git a/charged_lfv_neutrino_osc.png b/charged_lfv_neutrino_osc.png deleted file mode 100644 index 05e5985..0000000 --- a/charged_lfv_neutrino_osc.png +++ /dev/null @@ -1 +0,0 @@ -exit status 1 \ No newline at end of file diff --git a/muon-decay-feynman.png b/muon-decay-feynman.png deleted file mode 100644 index c5c1208..0000000 --- a/muon-decay-feynman.png +++ /dev/null Binary files differ diff --git a/neutrino_oscillation.png b/neutrino_oscillation.png deleted file mode 100644 index ec0c31f..0000000 --- a/neutrino_oscillation.png +++ /dev/null Binary files differ