# -*- mode: org; -*- #+TITLE: Estimation and modelling of Standard Model backgrounds in the search for \(W'\) gauge bosons with ATLAS (\mu channel) #+AUTHOR: Suvayu Ali #+EMAIL: Suvayu.Ali@cern.ch #+DATE: \today #+DESCRIPTION: #+KEYWORDS: #+LANGUAGE: en #+OPTIONS: H:4 num:4 toc:nil ::t |:t ^:t -:t f:t *:t <:nil #+OPTIONS: TeX:t LaTeX:t skip:nil d:nil todo:nil pri:nil tags:nil #+EXPORT_SELECT_TAGS: export #+EXPORT_EXCLUDE_TAGS: noexport #+STARTUP: content #+BIND: org-confirm-babel-evaluate nil #+BIND: org-export-latex-title-command "" #+LaTeX_CLASS: book #+LaTeX_CLASS_OPTIONS: [12pt,letterpaper,oneside] #+LaTeX_HEADER: \usepackage{amsfonts} #+LaTeX_HEADER: \usepackage{amsmath} #+LaTeX_HEADER: \usepackage{appendix} #+LaTeX_HEADER: \usepackage{varioref} #+LaTeX_HEADER: \usepackage[nokeyprefix]{refstyle} # more Physics specific packages # LaTeX Macros #+LaTeX_HEADER: \newcommand{\p}[1]{\phantom{#1}} #+LaTeX_HEADER: \newcommand{\modulus}[1]{\ensuremath{\lvert #1 \rvert}} # some more Physics specific macros # University stuff ##+LaTeX_HEADER: \usepackage[headings]{thesisstyle} # ... # #+LaTeX_HEADER: \include{abstract} # \include{preamble} * Introduction ** The Standard Model :SM: \label{chap:SMintro} The Standard Model is a quantum field theoretical approach to describe interactions in nature. The theory is primarily built upon symmetry arguments supported by experimental data. It describes the interactions between all fermions mediated by vector gauge bosons. These include the electromagnetic, weak and strong interactions. The formulation of the SM does not include gravity. There are various approaches to include gravity in BSM theories which are discussed elsewhere \cite{ArkaniHamed:1998rs,Randall:1999ee}. #+CAPTION: [Interactions and particles described by the SM.]{The interactions and the participating particles described by the Standard Model are tabulated below. Although the $Z^0$ is listed both as force carrier and a particle interacting by the weak force, it should be noted it does not couple with another $Z^0$.} #+LABEL: tbl:interactions |----------+-----------------+----------------+---------------------------------| | Charge | Force | Force carrier | Interacting particles | |----------+-----------------+----------------+---------------------------------| | Electric | Electromagnetic | \gamma | all charged fermions, \(W^\pm\) | | Weak | Weak | \(W^\pm, Z^0\) | all fermions, \(W^\pm, Z^0\) | | Colour | Strong | \(g\) | all quarks, anti-quarks, \(g\) | |----------+-----------------+----------------+---------------------------------| In the SM all particles carry various charges which describe all the interactions they undergo: the electric charge is responsible for all electromagnetic interactions, the weak charge for all weak interactions and the colour charge for all strong interactions. Since any quantum fields have bosonic exchange particles, all the interactions discussed earlier are associated with different spin-1 bosons as force carriers. This has been summarised in \Tabref{tbl:interactions}. *** Production and Decay of the \(W'\) Boson :pdf: \label{subsec:Wprod} #+CAPTION: A representative diagram for \(W'\) production at the LHC. #+LABEL: fig:LO-Wprime #+ATTR_LaTeX: width=0.6\textwidth [[file:figs/Wprime-s-channel.eps]] \noindent *Production:* The $W'$ boson is assumed to have a coupling constant that is the same as its lighter SM counterpart. This is known as the sequential SM $W'$ boson. The diagrams[fn:1] responsible for the production of $W'$ bosons at a hadron collider are known as charged current (CC) processes. These are flavour changing interactions occurring between up-like and down-like quarks (or anti-quarks) where the weak current involved is charged as the gauge bosons involved are charged. \Figref{fig:LO-Wprime} shows a diagram responsible for $W'$ production at the LHC. The production of a heavy $W'$ boson from a pp collision depends strongly on the momenta of the initial state particles. Since protons are composite particles made of (anti-)quarks and gluons, knowledge about the momentum fraction carried by the initial state particles is important. This is achieved by a /Parton Distribution Function/ (PDF) determined from deep inelastic scattering experiments with hadrons and leptons. In 1968--69, Bjorken proposed a dimensionless quantity /x/ which is independent of the momentum transfer during a hard scattering event. They exhibit a scaling behaviour in the asymptotic limit of very high energy collisions, known as /Bjorken scaling/. The discovery of scaling behaviour as predicted by Bjorken led to the modern formulation of the strong force as an asymptotically free quantum field theory \cite{Tung:2009}. The scaling was later shown to be only approximately true at low energies due to higher order QCD corrections (scaling violations). In 1969 Feynman proposed the /parton model/. It describes hadrons as composite particles made of its constituent quarks, anti-quarks and gluons \cite{PhysRevLett.23.1415}. The constituent quarks can be either valence or sea quarks. If the quark is one of the three quarks forming the hadron bound state and responsible for the quantum number, then it is called the /valence quark/. On the other hand if it is a virtual quark formed by splitting of gluons, it is called a /sea quark/. Modern QCD explains the interaction of hadrons in hard scattering experiments using this parton model. #+begin_src latex \begin{equation} \label{eq:partondistribfn} \frac{d\sigma}{dm_T} = \sum\limits_{a,b} \int dx_A f_{a/A}(x_A,\mu) \int dx_B f_{b/B}(x_B,\mu) \frac{d\hat{\sigma}}{dm_T} \end{equation} #+end_src The differential production cross section ($d\sigma/dm_T$) for observing certain events in a channel in a hadron collider is given by \Eqref{eq:partondistribfn}. $f_{a/A}$ and $f_{b/B}$ are the parton distribution functions mentioned earlier. They are the probability distributions of finding a parton of type /a/ (/b/) originating from a hadron of type /A/ (/B/) with the momentum fraction $x_A$ ($x_B$). $d\hat{\sigma}/dm_T$ is the parton level cross section given by perturbative QCD. The Bjorken scaling violations observed earlier could be incorporated into this picture by including corrections to the PDFs from perturbative QCD. These corrections introduce the dependence on the parameter \mu in the PDFs. The parameter \mu is related to the renormalisation of the strong coupling in the cross section calculation and the QCD corrections to the PDFs. They are often referred to separately as /renormalisation scale/ and /factorisation scale/ \cite{Soper:1996sn}. #+begin_src latex \begin{figure}[p] \centering \begin{tabular}{c} \includegraphics[width=0.7\textwidth,trim=0 150 0 150]{plots/PDF-upv.pdf} \\ \includegraphics[width=0.7\textwidth,trim=0 150 0 150]{plots/PDF-downv.pdf} \end{tabular} \caption[PDF for the proton at $Q^2 = 1$ TeV/c$^2$]{Proton PDFs showing the probability distributions for up (top) and down (bottom) quarks at $Q^2 = 1$ TeV/c$^2$. The valence quark PDF (black) is compared with the sea quark (red) and gluon (green) PDFs. A common choice for the factorisation scale when calculating a PDF set is to set it to $Q^2$ of the process \cite{website:HepData}.} \label{fig:PDF} \end{figure} #+end_src * Appendices :ignoreheading: :PROPERTIES: :VISIBILITY: folded :END: #+LaTeX: \newpage #+LaTeX: \appendix # Needs \usepackage{appendix} #+LaTeX: \addappheadtotoc #+LaTeX: \appendixpage ** Particle track parametrisation in ATLAS :trk: \label{app:trackparam} A particle track in ATLAS is represented by the perigee[fn:2] of the track. The perigee is specified by the five parameters: + d_0 - transverse impact parameter, the distance of the closest approach of helix to beam pipe, + z_0 - longitudinal impact parameter, the z value at the point of closest approach, + \phi_0 - azimuth angle of the momentum at point of closest approach, measured in the range $[-\pi,\pi)$, + \theta - polar angle in the range $[0,\pi]$, + q/p - charge over momentum magnitude. In ATLAS the /z/-axis is parallel to the magnetic field (which is parallel to the beam axis). \Figref{fig:trkperigee} shows the five parameters split into transverse and longitudinal planes. #+CAPTION: [Schematic showing all 5 track parameters]{The diagram shows the five parameters, the three longitudinal parameters in the x-y plane and two longitudinal parameters in the r-z plane.} #+LABEL: fig:trkperigee #+ATTR_LaTeX: width=0.6\textwidth placement=[p] [[file:figs/Perigee.png]] ** Limit formalism :limits: \label{app:limits} The limits are set with the observed events in a bin with $m_T$ > $m_{Tmin}$, where the lower bin threshold is determined by taking the half of the $W'$ mass. The expected number of events are given by \Eqref{eq:expevts}, where $\nbg$ is the expected number of events from our background model and $N_{sig}$ is the predicted number of signal events from \Eqref{eq:sigevts}. $L_{int}$ is the integrated luminosity of the recorded data and $\epsilon_{sig}$ is the event selection efficiency. #+begin_src latex \begin{align} \label{eq:expevts} N_{exp} &= N_{sig} + \nbg \\ \label{eq:sigevts} N_{sig} &= L_{int} \epsilon_{sig} \SB \end{align} #+end_src The likelihood function in \Eqref{eq:likelihood} is obtained from Poisson statistics for $N_{obs}$ observed events. Uncertainties are included as nuisance parameters in the analysis. They are assumed to follow a Gaussian probability distribution. In this analysis we consider uncertainties on $L_{int}$, $\epsilon_{sig}$ and $\nbg$ in our calculations. Taking the above into consideration we get the likelihood function shown in \Eqref{eq:llnuissance}. #+begin_src latex \begin{align} \label{eq:likelihood} \mathcal{L}(\SB) &= \frac{(L_{int}\SB + \nbg)^{N_{obs}}e^{-(L_{int}\epsilon_{sig}\SB+\nbg)}}{N_{obs}!} \\ \label{eq:llnuissance} \mathcal{L}(\SB,\theta_1,\dotsc\theta_N) &= \frac{(L_{int}\SB + \nbg)^{N_{obs}}e^{-(L_{int}\epsilon_{sig}\SB+\nbg)}}{N_{obs}!} \prod g_i(\theta_i) \\ g_i(\theta_i) &= \frac{1}{\sqrt{2\pi}\sigma_i}e^{-\frac{(\theta -\bar{\theta}_i)^2}{2\sigma_i^2}} \nonumber \end{align} #+end_src This likelihood function is used to calculate the log-likelihood ratio (LLR) test statistic shown in \Eqref{eq:LLR} \cite{Junk:1999kv}. The LLR is used to calculate /p/-values which correspond to confidence levels for the signal + background (CL_{s+b}) and background (CL_b) only hypotheses. These are evaluated by conducting pseudo-experiments and integrating over the corresponding LLR distributions. #+begin_src latex \begin{equation} \label{eq:LLR} LLR = -2 \ln \frac{\mathcal{L}(data\lvert s+b)}{\mathcal{L}(data\lvert b)} \end{equation} #+end_src The CL_s method finally uses the ratio of the /p/-values to determine the exclusion limits (\eqref[name=eq.~]{eq:CLs}). To set the limits at 95% confidence level, the signal cross section is increased until CL_s = 1 - \alpha where \alpha is 0.95. For a more complete discussion of the limit setting procedure see the referenced ATLAS internal note \cite{Adams:1317922}. #+begin_src latex \begin{equation} \label{eq:CLs} CL_s = \frac{CL_{s+b}}{CL_b} \end{equation} #+end_src * Bibliography :ignoreheading: #+LaTeX: \backmatter #+LaTeX: \newpage #+LaTeX: \addcontentsline{toc}{chapter}{\bibname} #+LaTeX: \bibliographystyle{ieeetr} #+LaTeX: \bibliography{master} * Footnotes [fn:1] Here and in the rest of this thesis /diagram/ refers to Feynman diagrams unless otherwise specified. [fn:2] The point where the track comes closest to the geometrical centre of the detector is called the perigee of the track. * COMMENT local setup # Local Variables: # org-export-allow-BIND: t # org-latex-to-pdf-process: ("pdflatex -interaction nonstopmode %b" "/usr/bin/bibtex %b" "pdflatex -interaction nonstopmode %b" "pdflatex -interaction nonstopmode %b") # End: