\chapter{Introduction}
	\label{chapter_introduction}

	In this thesis, measurements of charged current jet production in polarized electron-proton scattering and comparisons to theoretical predictions are presented.  The results presented are beneficial to the understanding of fundamental interactions and dynamics of the proton.  The measurements presented were performed using the ZEUS detector at the HERA collider, which was part of the DESY accelerator complex in Hamburg, Germany.  The predictions were derived from a theory called the Standard Model of Particle Physics (SM), which is briefly outlined in this chapter.  The parts of the SM most relevant to this analysis and the methods used to compute the predictions are described in detail in chapter \ref{chapter_cc_dis}, and the experimental conditions are described in chapter \ref{chapter_experimental_section}.  The methods used to process, simulate, and select data from the detector are described in chapters \ref{chapter_event_reconstruction}, \ref{chapter_event_simulation} and \ref{chapter_event_selection}, respectively.   The techniques used to analyze the data are presented in chapter \ref{chapter_analysis_method}, and the subsequent results are presented in chapter \ref{chapter_results}.    The importance of this work and potential extensions of this study are discussed in chapter \ref{chapter_conclusions}.

\section{Particles of The Standard Model}
	\label{section_particles_of_the_sm}
	At present, the most complete and general description of the content and behavior of matter is the Standard Model.  It is a highly successful theory, with a range of validity covering several orders of magnitude in energy, and predicted quantities which agree with observation to many significant figures.  The SM describes the properties of matter as the dynamics and interactions of particles which are fundamental in the sense that they have no internal structure, and are indistinguishable among a given variety.

	The particles of the SM can be organized into two major categories based on intrinsic angular momentum: particles which possess integral spin are referred to as {\it bosons}, while particles which possess half-integral spin are referred to as {\it fermions}.  The elementary bosons of the SM mediate interactions of three kinds, and come in four types:
	\begin{enumerate}
	 \item 	photons, $\gamma$, which mediate the electromagnetic interaction
	 \item gluons, $g$, which mediate the {\it strong} interaction
	  \item $W$ \& $Z$ bosons, which mediate the {\it weak} interaction
	\end{enumerate}
	Gravitational interactions are not described by the SM, but are widely accepted to be negligible in the conditions described in this work.

	The fermions of the SM are classified into two groups: {\it quarks}, which possess the {\it color charge} of the strong interaction, and {\it leptons}, which do not.  Protons, neutrons, and other {\it hadrons} are described in the standard model as composite objects, comprised of quarks and gluons.  Quarks and gluons are referred to collectively as {\it partons} when describing the content of hadrons.  Quarks each possess spin $\pm$1/2, a color of {\it red}, {\it green} or {\it blue}, and an electric charge of +2/3 or -1/3.  Quarks with electric charge 2/3 and -1/3 are referred to as {\it up-type}, and {\it down-type}, respectively.  Up-type quarks appear in three {\it flavors}: {\it up} ($u$), {\it charm} ($c$) and {\it top} ($t$), as do down-types: {\it down} ($d$), {\it strange} ($s$) and {\it bottom} ($b$).  

	Leptons possess spin $\pm$1/2, no color charge, and electric charge of -1 or 0.  Leptons with zero mass are referred to as {\it neutrinos}, and charged leptons include the electron ($e^-$), muon ($\mu$), and the tau ($\tau$).  Neutrinos interact solely via the weak interaction, which associates one neutrino with each charged lepton.  Neutrinos receive names from their weak interaction partners and are called the electron- ($\nu_e$), muon- ($\nu_\mu$), and the tau neutrino ($\nu_\tau$).  This pairing partitions the leptons into three {\it generations} of {\it doublets}, where two generations differs only by the mass of the particles of each type.  The weak interaction likewise partitions quarks into three generations of doublets by associating pairs of up- and down-type quarks, specifically $u-d$, $s-c$ and $t-b$.  The up type and down type quarks if each generation also differ only in their masses.  

	For each fermion, the SM additionally describes one antifermion.  Fermions and their antipartners differ only by electromagnetic charge, with the charge of antipartners being equal but opposite. The possible color charges of anti-quarks are the anti-colors {\it antired}, {\it antigreen} or {\it antiblue}.  The antiparticle of the electron is the {\it positron} ($e^+$), and is identical to the electron in every way except for having charge +1.  All other antiparticles are usually referred to  identically as their partners, but with a prefix of anti-, and are symbolically denoted with an overline.  For example, the antiparticle of the blue up is the anti-blue anti-up ($\overline{u}$).

	The weak interaction additionally partitions fermions into two categories based on their {\it helicity}, which is the sign of the projection of a particles spin along its direction of motion.  Particles with positive helicity are said to be {\it right-handed}, while particles with negative helicity are called {\it left-handed}.  The a particle's transition under the weak interactions is indicated by the quantum number {\it weak isospin} ($T_3$).  Left-handed charged leptons and left-handed up-type quarks have $T_3=1/2$, while left-handed neutrinos and left-handed down-type quarks have $T_3=-1/2$.  Right-handed fermions and left-handed antifermions have $T_3=0$.  They form weak isospin singlet states, and do not participate in the weak interaction.  {\it Weak hypercharge}, $Y_{\tiny W}$ is a number determined for the EM charge and weak isospin of each particle by $Y_{\tiny W} = 2(Q-T_3)$, where $Q$ is the EM charge.  Table \ref{table_fermion_properties} summarizes the most important quantum numbers for the left-handed fermions.  It is intersting to note that the electromagnetic, weak and  strong couplings for left-handed anti-neutrinos are all zero.  The do not participate in any SM interaction, but are included in the table for symmetry.

\begin{table}[htbp]
  \begin{center}
     	\begin{tabular}{c|c|c|c|c|c|c|c|} 
		{}
 		& \begin{sideways}Generation\end{sideways} 
 		& \begin{sideways}Name\end{sideways} 
 		& \begin{sideways}Symbol\end{sideways}
 		&\begin{sideways}\begin{tabular}{l}Electric \\ Charge \end{tabular}\end{sideways}
 		&\begin{sideways}\begin{tabular}{l}Weak \\ Isospin \end{tabular}\end{sideways}
 		&\begin{sideways}\begin{tabular}{l}Weak \\ Hypercharge \end{tabular}\end{sideways}
 		&\begin{sideways}Mass\end{sideways}\\
      \hline
	\multirow{12}{*}{\begin{sideways}Leptons\end{sideways}}  
	& 	\multirow{4}{*}{I} 
			&	Electron & $e^-$& 	-1& 	-1/2& 	-1& 	\multirow{2}{*}{511 keV }\\
      \cline{3-7}
		&	&	Positron & $e^+$& 	+1& 	0& 	+2 & \\
      \cline{3-8}
		&	&	Electron Neutrino & $\nu_e$& 	0& 	+1/2& 	-1& 	\multirow{2}{*}{$<$ 2 eV} \\
      \cline{3-7}
		&	&	Electron Antineutrino & $\overline{\nu_e}$& 	0& 	0& 	0&\\
      \cline{2-8}
	&	\multirow{4}{*}{II} 
			&	Muon & $\mu^-$& 	-1& 	-1/2& 	-1& 	\multirow{2}{*}{106 MeV} \\
	\cline{3-7}
		&	&	Antimuon &$\mu^+$& 	+1& 	0& 	+2& 	\\
      \cline{3-8}
		&	&	Muon Neutrino & $\nu_\mu$& 	0& 	+1/2& 	-1& 	\multirow{2}{*}{$<$ 2 eV}  \\
	\cline{3-7}
		&	&	Muon Antineutrino & $\overline{\nu_\mu}$& 	0& 	0& 	0&\\
      \cline{2-8}
	&	\multirow{4}{*}{III} 
			&	Tau & $\tau^-$& 	-1& 	-1/2& 	-1& 	\multirow{2}{*}{1.78 GeV}\\
	\cline{3-7}
		&	&	Antitau& $\tau^+$& 	+1& 	0& 	+2& \\
      \cline{3-8}
		&	&	Tau Neutrino & $\nu_\tau$& 	0& 	+1/2& 	-1& 	\multirow{2}{*}{$<$ 2 eV}  \\
	\cline{3-7}
		&	&	Tau Antineutrino & $\overline{\nu_\tau}$& 	0& 	0& 	0&\\
	\hline
	\hline
	\multirow{12}{*}{	\begin{sideways}Quarks\end{sideways}}
 	&	\multirow{4}{*}{I} 
			&	Up Quark& $u$& 	+2/3& 	+1/2& 	+1/3& 	\multirow{2}{*}{$\sim$ 3 MeV} \\
      \cline{3-7}
		&	&	Up Antiquark  & $\overline{u}$& 	-2/3& 	0& 	-4/3& 	\\
      \cline{3-8}
		&	&	Down  Quark& $d$& 	-1/3& 	-1/2& 	+1/3& 	\multirow{2}{*}{$\sim$ 6 MeV}\\
      \cline{3-7}
		&	&	Down Antiquark & $\overline{d}$& 	+1/3& 	0& 	+2/3& \\
      \cline{2-8}
	&	\multirow{4}{*}{II} 
			&	Charm Quark & $c$& 	+2/3& 	+1/2& 	+1/3& 	\multirow{2}{*}{$\sim$ 1.3 GeV}\\
	\cline{3-7}
		&	&	Charm  Antiquark& $\overline{c}$& 	-2/3& 	0& 	-4/3& \\
      \cline{3-8}
		&	&	Strange  Quark& $s$& 	-1/3& 	-1/2& 	+1/3& 	\multirow{2}{*}{$\sim$ 100 MeV}\\
	\cline{3-7}
		&	&	Strange Antiquark & $\overline{s}$& 	+1/3& 	0& 	+2/3& 	\\
      \cline{2-8}
	&	\multirow{4}{*}{III} 
			&	Bottom Quark & $b$& 	-1/3& 	-1/2& 	+1/3& 	\multirow{2}{*}{$\sim$ 4.2 GeV}\\
	\cline{3-7}
		&	&	Bottom Antiquark& $\overline{b}$& 	+1/3& 	0& 	+2/3& 	\\
      \cline{3-8}
		&	&	Top  Quark& $t$& 	+2/3& 	+1/2& 	+1/3& \multirow{2}{*}{$\sim$	171 GeV}\\
	\cline{3-7}
		&	&	Top Antiquark & $\overline{t}$ & 	-2/3& 	0& 	-4/3& 	\\
	\hline    \end{tabular}
  \end{center}
	\label{table_fermion_properties}
\caption[Properties of Left-Handed Fermions]{\fixspacing{Properties of Left-Handed Fermions. }}
\end{table}

\section{The Standard Model Lagrangian and its Symmetries}
	In the SM, the particles listed in section~\ref{section_particles_of_the_sm} are the quanta of {\it fields}, $\phi(x)$, which are functions that take complex vector values at each point in space-time.  These functions are determined by the minimization of a scalar {\it action}, $S$.  The action is the space-time integral over the {\it lagrangian density}, $\mathcal{L}$, which is a functional of the fields and their {\it covariant derivatives} $D_\mu\phi$.  This is written as
\begin{equation}
\label{eq_action}
	S = \int d^4x\mathcal{L}(\phi,D_\mu\phi).
\end{equation}
	The covariant derivatives take the form $D_\mu \equiv \partial_\mu - igA_\mu$, were $g$ is the coupling strength of the interactions, and $A_\mu$ are {\it gauge potential fields}, analogous to the electromagnetic vector potential in classical electrodynamics. 
	
	The SM lagrangian is a member of a class of lagrangians which is invariant under local phase rotations, $\Lambda(x)$ of its fields $\phi \rightarrow e^{ig\Lambda}\phi$, and corresponding transformations of the gauge fields $A_\mu\rightarrow A_\mu - 1/g\partial_\mu\Lambda$.  The value of $\Lambda(x)$ is defined independently at each point in space time, and takes matrix values with dimension appropriate to the interaction in question.  Such transformations are called {\it gauge transformations}, Lagrangians that are invariant under these transformations are called {\it gauge invariant}, and theories with gauge invariant lagrangians are called {\it gauge theories}.  Gauge transformations form mathematical groups under composition, and provide the application of group theoretic technology to SM computations.  The principle point of interest here is that because gauge transformations must be compatible with the potential fields, and the potential fields determine the interaction, the gauge symmetry group of the Lagrangian determines the interactions and {\it vice-versa}.

	The symmetry group for electromagnetic interactions is $U(1)$, which can be represented as the multiplicative group of complex numbers of magnitude 1. The resultant field theory is referred to as Quantum Electrodynamics (QED).  The symmetry group of the weak interaction is $SU(2)$, which can be represented as a two-dimensional matrix group generated by traceless Hermitian matrices, such as the three Pauli matrices.  In the SM, the electromagnetic and weak interactions are unified at high energies to form the gauge group $U(1)_Y\times SU(2)_L$, where $Y=2(Q+I_z)$ denotes {\it hypercharge}, and $I_z$ denotes {\it isospin}.  This gauge is referred to as the {\it electroweak gauge} and its four generators correspond to four independent gauge fields, which at low energies form fields for the neutral ($\gamma$, $Z^0$) and charged ($W^+$, $W^-$) electromagnetic and weak bosons.

	The symmetry group for the strong interaction is $SU(3)$, which can be represented as the three dimensional matrix group generated by traceless Hermitian matrices, such as the eight Gell-Mann matrices.  From eight generators arise eight independent gluon color fields.  The resultant field theory is referred to as Quantum Chromodynamics (QCD).  While the U(1) group is Abelian, the SU(2) and SU(3) groups are not.  As a consequence, photons do not interact with other photons, allowing for the principle of superposition.  Weak and strong fields, on the otherhand, are self-interacting, and their dynamics are non-linear.

\section{Perturbative Field Theory and Renormalization}
	For a given observable quantity in the SM, it is often infeasible to perform an exact computation.  Instead, series expansion is often perfromed which can be truncated to finite order.  Often, this expansion parameter is the coupling strength of a process.
$$
	f(\alpha) = \sum_{n=0}^\infty f_{n}\alpha^n
$$
	In this case, each $f_{n}$ can be represented as a Feynman diagram, where the order $n$ corresponds to the number of interaction vertices in the diagram. If the coupling strength is less than 1, then the series has the potential to converge.   {\it Leading-order} (LO) diagrams denote those terms if lowest order in $n$, while {\it next-to-leading-order} (NLO) denote the next highest, and so forth. Although any observable must have a finite amplitude, individual Feynman diagrams are often not finite.  Truncated series must then be treated with a process called renormalization in order to produce a finite prediction.  

	When comparing measurements with these renormalized predictions from truncated series, the measurements are reinterpreted not in terms of the parameters of the lagrangian, called the {\it bare parameters}, but instead different parameters, called {\it renormalized parameters}.  Renormalized parameters are dependent upon how the measurement is made, and this is parameterized in terms of a {\it renormalization scale} ($\mur$).  

	The result is that the coupling strength of a renormalized theory is not a constant, but varies with scale $\alpha\rightarrow\alpha(\mur)$. This dependence is called {\it running} of the coupling, and agrees well with observed phenomena.  The physical interpretation of $\mur$ is dependent on the particular process being studied, but can be thought of as the energy of the probe used, or the inverse distance that the probe can resolve or penetrate. 

	Perturbative QCD has been used in this thesis to produce theoretical predictions, as described in sections~\ref{section_mepjet} to~\ref{hadronization_corrections}.  Leading order calculations are also at the core of the event simulations which are described in chapter~\ref{chapter_event_simulation}.

\section{Confinement and Asymptotic Freedom in QCD}
	\label{section_confinement}
\begin{figure}
  \begin{center}
	\includegraphics[draft=false, width=1.\textwidth]{/home/hwolfe/thesis/figures/introduction/fig_coupling}
  \end{center}
	\caption[Behavior of the QED and QCD Couplings]{
		\fixspacing{Behavior of the QED and QCD Couplings}
	}
	\label{figure_running}
\end{figure}

	Figure~\ref{figure_running} depicts the trend of running for the QED and QCD coupling constants, $\alphaem$ and $\alphas$.  Of greatest contrast between the two interactions is that $\alphaem$ asymptotically approaches a finite constant of approximately 1/137 for soft interactions, while the $\alphas$ increases, and diverges at a scale of $\lambda_{QCD}$.  The value of $\lambda_{QCD}$ depends on the renormalization procedure, but is usually on the order of $200~\MeV$.  The increase of $\alphas$ at low scales is responsible for the feature of {\it confinement} which describes why colored partons are not observed in nature, but instead are confined to color-singlet hadronic states.  At large scales, $\alphas$ asymptotically approaches zero.  When a particle with energy much greater than $1~\GeV$ is used to probe the proton, the value of $\alphas$ diminishes such that the interactions between partons becomes negligible.  The partons then appear to be non-interacting, which is called {\it asymptotic freedom}.

\section{Parton Density Functions}
	\label{section_pdf_intro}
	At scales below about $1~\GeV$, the strength of the strong force is so great that partons are confined in hadrons.  Perturbative calculations in the strong coupling, $\alphas$, are no longer possible because it is on the order of unity or greater, and the series no is no longer assumed to be convergent.  In order to describe the behavior of hadronic scattering within the SM, it is at present necessary to use phenomenological methods to describe their quark and gluon content.  This is described in terms of probability distributions, called {\it Parton Density Functions} (PDFs).  These functions are experimentally determined, but are universally applicable.  This universality means that the PDFs extracted at two experiments should be equivalent, and the PDFs determined at one experiment can be used to make other types of predictions at another.  Given a hadron with momentum $P$ and a probe with momentum transfer $Q^2$, the PDF $q(x,Q^2)$ expresses the probability of observing a parton of type $q$ in a hadron which carries a momentum $xP$. 
\section{Deep Inelastic Scattering}
\begin{figure}[htb]
\vfill
	\setlength{\unitlength}{1.0cm}
	\begin{picture} (18.0,4.5)
		\put(2.0,-0.5){\includegraphics[draft=false, width=.3\textwidth]
			{/home/hwolfe/thesis/figures/chapter_cc_dis/lo_ncdis}}
		\put(10.0,-0.5){\includegraphics[draft=false, width=.3\textwidth]
			{/home/hwolfe/thesis/figures/chapter_cc_dis/lo_ccdis}}
	\put (1.0,4.0){\bf (a)}
	\put(9.0,4.0){\bf (b)}
	\end{picture}
	\caption[L.O. Feynman Diagrams for NC and CC DIS]{L.O. Feynman Diagrams for NC and CC DIS}
	\label{figure_qpm}
\vfill
\end{figure}

	Asymptotic freedom described in section~\ref{section_confinement} allows the scattering of point-like electrons on composite protons at high center-of-mass (CME) energies to be described in terms of the scattering of electrons off of independent quarks and gluons.  This is accomplished by convolving predictions of parton-electron scattering in pQCD with the PDFs mentioned in section~\ref{section_pdf_intro}.  At leading order in the the electroweak coupling, $\alphaem$, and constant order in $\alphas$, $ep$ scattering proceeds via the quark parton model process, with Feynman diagrams shown in figure~\ref{figure_qpm}.  

	Figure~\ref{figure_qpm}(a) represents the Born-level exchange of virtual photons or $Z^0$ bosons between electrons and quarks. When this is generalized to $ep$ scattering at high energies, the process is referred to as Neutral Current (NC) Deep Inelastic Scattering (DIS).  Heres the term deep refers to the resolving power of the probe, and inelastic refers to an increase in the Lorentz-invariant mass of the final state hadronic system.  In NC DIS, the incoming electron is deflected, and appears in the final state.  Figure~\ref{figure_qpm}(b) represents the Born-level exchange of charged $W$ bosons between electrons and quarks.  When this process is generalized to $ep$ scattering, it is referred to as Charged Current (CC) DIS.  In this process, the incoming electron does not appear in the final state, but is iso-spin rotated into a neutrino of the same generation.  The initial-state up-type (down-type) quark likewise undergoes change of isospin, and is replaced in the final state by a down-type (up-type) quark of the same generation. The process is identical for $e^-p$ and $e^+p$ scattering, with the exception that the exchanged boson is either a  $W^-$ or $W^+$, in order to conserve electromagnetic charge at the upper EW vertex.

	Other processes occur in $ep$ scattering which were sources of background for this analysis.  One of the largest was from Photoproduction (PHP), in which a photon with nearly zero invariant mass is exchanged between the $e$ and $p$, which causes very little change in the electron's initial momentum.  The soft photon can still interact inelastically with the proton, however.  The ways that NC DIS and PHP can contaminate CC DIS samples and the means taken to remove these events are briefly discussed in section~\ref{section_baby_analysis}, and discussed in detail in chapter~\ref{chapter_event_selection}.
\section{Jet Production in DIS}
	\label{section_jet_production_in_dis}
\begin{figure}
  \begin{center}
	\includegraphics[draft=false, width=0.6\textwidth]{/home/hwolfe/thesis/figures/chapter_cc_dis/lo_jp_ccdis}
  \end{center}
	\caption[Schematic of LO jet Production in CC DIS]{\fixspacing{Schematic Diagram of Leading-order Jet Production in CC DIS.}}
	\label{figure_jet_production}
\end{figure}
	Due to confinement, color non-singlet states such as the outgoing quark in Figures~\ref{figure_qpm}(a) and (b) are not observable quantities.  Instead, partons dissociated from a hadron will undergo a process referred to as hadronization, in which observable color-singlet hadrons are produced.  Because this process is soft, only small momentum transfers take place, and final-state momenta of the resultant hadrons will be approximately parallel, which is schematically represented in figure~\ref{figure_jet_production}.  This group of hadrons, referred to collectively as a {\it jet}, will closely resemble the kinematic properties of the initiating parton.  This makes jets a powerful tool to study parton dynamics, and therefore a tool to understand the parton content of hadrons~\cite{ref_se95}.  
\begin{figure}
  \begin{center}
	\includegraphics[draft=false, width=0.6\textwidth]{/home/hwolfe/thesis/figures/chapter_cc_dis/bgf_jp_ccdis}
  \end{center}
	\caption[Schematic of BGF Dijet Production in CC DIS]{\fixspacing{Schematic Diagram of Boson-Gluon Fusion CC DIS.}}
	\label{figure_bgf_jet_production}
\end{figure}

	At higher order in $\alphas$, the Feynman diagrams for $eq$ scattering can have many partons in the final state.  These partons will undergo hadronization separately, and will produce multiple separated jets in the final state.  Events with at least one jet are referred to as {\it inclusive-jet production events}, while events with two, three or four are called {\it dijet-},  {\it trijet-}(3-jet) and  {\it quadrajet-}(4-jet) events, respectively.  Collectively, these are referred to as {\it multi-jet states}.

	An example process which can produce dijets is represented schematically in figure~\ref{figure_bgf_jet_production}, in which a gluon from the proton interacts with the $W$ via a quark pair.  This is referred to as {\it boson gluon fusion} (BGF).  Measureables from this process are sensitive to the gluon content of the proton.  Because the two jets are connected via a QCD vertex with the gluon, relationships between jets in the final state are an excellent means to test the predictions of QCD.  Jet production is a non-perturbative process, and is phenomenologically modeled in two phases: QCD radiation and fragmentation.  In the former, the initiating parton creates several new ones.  In the latter, the partons produced in the radiative stage are combined into color less on-shell hadrons.  Distributions of  simulated jets can then be compared to distributions of reconstructed jets from a detector, as was performed in this thesis.



\section{A Brief Overview Of The HERA Collider and The ZEUS Detector}
	The HERA collider was the world's first and only $e^{\pm}p$ collider, and produced $e^-p$ and $e^+p$ collisions during separate periods from 1992 to 2007.  Previous $ep$ experiments collided electrons with fixed targets like helium, but These experiments could only explore center of mass energies up to a few $\GeV$~\cite{slac_ep_inelastic}.  The HERA collider consisted of two nearly-evacuated, nearly-circular storage rings with counter-cycling beams. HERA provided electron or positron beams with an energy of $27.5~\GeV$, and collided these with proton beams of $920~\GeV$ to produce events with CMEs of approximately $300~\GeV$.  
\begin{figure}
  \begin{center}
	\includegraphics[draft=false, width=1.\textwidth]{/home/hwolfe/thesis/figures/chapter3/zeus3D}
  \end{center}
	\caption[A Diagram of The ZEUS Detector]{\fixspacing{A Diagram of The ZEUS Detector}}
	\label{diagram_zeus_detector}
\end{figure}

	As HERA produced $ep$ collisions, resultant particles were recorded in the ZEUS detector.  The ZEUS detector was constructed around one of HERA's interaction points, and was roughly cylindrical in shape.  Because interactions at HERA did not occur in the $ep$ center-of-mass system, the ZEUS detector was built with more material in the direction the protons were traveling towards.  Figure~\ref{diagram_zeus_detector} shows a cutaway diagram of the ZEUS detector, so that the HERA beampipe is visable, and the viewer is facing the direction of lepton entry into the interaction.  The standard ZEUS polar coordinate system places the nominal interaction point at the origin and the proton direction as the $\theta = 0$ line. ZEUS was composed of several independent detectors, each of which performed slightly different forms of measurement.  Of central importance to this thesis is the Central Tracking Detector (CTD), and Calorimeter (CAL).    The CTD covered a polar region of $30^\circ$ to $150^\circ$, and recorded the path of charged particles in a magnetic field, to infer their masses, momenta and charges.  The CAL covered a polar angle range of $2.2^\circ$ to $176.5^\circ$, and recorded the energy and  position of electromagnetic- and strongly-interacting particles as they were absorbed by very dense metal.  The absorber material was was arranged in units called {\it cells}, with a much coarser granularity than the CTD.  When signals were recorded in the detector, a trigger system was used to select which events would be readout, and events were reconstructed and stored for analysis.
\begin{figure}[htbp]
\vfill
	\setlength{\unitlength}{1.0cm}
	\begin{picture} (18.0,18.0)
		\put(-3.5,18.0){\centerline{\includegraphics[draft=false,
		width=.45\textwidth, angle=270]
			{/home/hwolfe/thesis/figures/event_displays/r56818e40902_zr}}}
		\put(5.5,18.0){\centerline{\includegraphics[draft=false,
		width=.45\textwidth, angle=270]
			{/home/hwolfe/thesis/figures/event_displays/r56818e40902_xy}}}
		\put(1.0,0.0){\centerline{\includegraphics[draft=false,
		width=1.\textwidth, ]
			{/home/hwolfe/thesis/figures/event_displays/r56818e40902_et}}}
	\end{picture}
	\caption[Event Display Single Jet CC DIS]{\it Event Display Single Jet CC DIS}
	\label{figure_event_display_1jet}
\vfill
\end{figure}

\begin{figure}[htbp]
\vfill
	\setlength{\unitlength}{1.0cm}
	\begin{picture} (18.0,18.0)
		\put(-3.5,18.0){\centerline{\includegraphics[draft=false,
		width=.45\textwidth, angle=270]
			{/home/hwolfe/thesis/figures/event_displays/r53440e129382_zr}}}
		\put(5.5,18.0){\centerline{\includegraphics[draft=false,
		width=.45\textwidth, angle=270]
			{/home/hwolfe/thesis/figures/event_displays/r53440e129382_xy}}}
		\put(1.0,0.0){\centerline{\includegraphics[draft=false,
		width=1.\textwidth, ]
			{/home/hwolfe/thesis/figures/event_displays/r53440e129382_et}}}
	\end{picture}
	\caption[Event Display Dijet CC DIS]{\it Event Display of Dijet CC DIS}
	\label{figure_event_display_2jet}
\vfill
\end{figure}

\begin{figure}[htbp]
\vfill
	\setlength{\unitlength}{1.0cm}
	\begin{picture} (18.0,18.0)
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		width=.45\textwidth, angle=270]
			{/home/hwolfe/thesis/figures/event_displays/r53331e14117_zr}}}
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		width=.45\textwidth, angle=270]
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	\caption[Event Display Dijet NC DIS]{\it Event Display of Dijet NC DIS}
	\label{figure_event_display_nc2jet}
\vfill
\end{figure}


	In figures~\ref{figure_event_display_1jet} and~\ref{figure_event_display_2jet}, CC DIS candidate events are shown in the ZEUS CAL, CTD, and Forward Tracking Detectors.  In figure~\ref{figure_event_display_1jet}, a single cluster of transverse energy is registered in the CAL, and tracks in the CTD lead from near the nominal vertex towards it.  We identify this deposit with a jet.  From the $x-y$ view, it can be seen that the energy deposits and tracks were not balanced in the direction transverse to the beamline, implying that something escaped detection in the direction opposite the jet.  Because neutrinos only interact  weakly, they are not expected to interact with the ZEUS detector and are therefore the most likely candidate for this type of signature.

	In figure~\ref{figure_event_display_2jet}, we see two isolated deposits in the CAL, and two groups of particle tracks in the CTD.  We label this event a dijet, which could have originated in a manner similar to figure~\ref{figure_bgf_jet_production}.  The plot of transverse energy in the ZEUS CAL at the bottom of figure~\ref{figure_event_display_2jet} shows the separation of the two jets the CAL.  These two candidate events can be compared with the ND DIS Dijet candidate event displayed in figure ~\ref{figure_event_display_nc2jet}.  Here, two broad, isolated hadronic deposits can be seen, and are counterbalanced in $\phi$ with the presence of a high-$E_T$ REMC deposit, which is comprised from nearly one REMC cell.  There is one isolated CTD track leading to this deposit, which strengthens its candidacy as a scattered electron.

\section{Methods Used In This Analysis}
	\label{section_baby_analysis}
	Once the data had been readout and stored from the ZEUS detector, software further reconstructed information about the interaction and CC DIS events were selected over background.  Jets were reconstructed from clusters of CAL cells, and information from the CTD was used to determine the location of the primary interaction.  Non $ep$ background, such as cosmic rays and interactions between protons and residual gas in the beampipe were removed by comparing information of the CAL and CTD.  

	Because neutrinos interact only via the weak force, they were neither recorded by the CTD or CAL.  Thus, a principle method of rejecting NC DIS is to reject events when an isolated electron was observed in the final state.  In CC DIS neutrinos carry momentum from the event without detection, events therefore appear to have a nonzero net momentum transverse to the beampipe, $\ptmiss$.  In PHP events the initial state electron is only slightly deflected in the interaction.  The electron may escape detection down the beampipe, and with it carry transverse momentum.  Some PHP events therefore have no recorded electron in the final state, and may have a non-zero net $\ptmiss$.  The recorded $\ptmiss$ from PHP may be larger than NC DIS, but is relatively small to the $E_T$ produced in the inelastic $\gamma p$ interaction.  In CC DIS, many events have $\ptmiss \approx E_T$, which makes $\ptmiss/E_T$ is a powerful quantity to differentiate the two.  Many other selection cuts are used to remove these events, as discussed later in chapter~\ref{chapter_event_selection}.

	Once events were reconstructed and selected, a simulation of $ep$ interactions and the ZEUS detector was used to correct the values obtained to remove the effects of the detector itself.  Rather than describing jets of calorimeter cells after particles passed through the detector, the corrected events described jets of hadrons.  NLO calculations convolved with PDF functions were also corrected from describing jets of partons to describing jets of hadrons.  The resulting distributions could then be compared to understand the strengths and weaknesses of the NLO calculations, and the PDF sets that were used.
	
	Several variations were also performed on the analysis procedure, so that the uncertainties due to the reconstruction, simulations, and methods could be understood.  Different PDF sets were also used, and their results were compared to understand the uncertainty on the measurements from the choice of PDFs used.

\section{Overview Of Results And Implications Of This Analysis}
	CC DIS at high $Q^2$ has been measured previously by ZEUS and H1 under a variety of conditions.  Inclusive CC DIS has been measured with and without longitudinally-polarized lepton beams~\cite{Aktas:2005ju, Chekanov:2003vw}.  Scattering structure functions for the proton, which are directly related to PDFs were measured.  Using polarized lepton beams, a search for the presence of right-handed weak interactions was performed, and no evidence for these effects was found.  The ZEUS collaboration has also performed studies of jet production in $e^+p$ CC DIS~\cite{Chekanov:2003jd}, in which the substructure of jets was studied and used to extract $\alphas$.  The H1 collaboration has published a study of Dijet production in $e^+p$ CC DIS~\ref{Adloff:2000hm},  in which CC Dijet cross sections were compared to NC Dijet distributions in the same phase space.  The QCD characteristics of the jets were shoen to be independent of the EW scattering process. 
	
	All of the PDFs for the proton are nonzero, but two of the largest functions are the $u$, and $d$, which respectively comprise approximately 2/3 and 1/3 of the proton at low energies.  In $e^+p$ CC DIS, a $W^+$ boson is exchanged, which, only interacts with $d$, $\bar{u}$, $s$ and $\bar{c}$ quarks in the proton at lowest order in $\alphas$.  In $e^-p$ CC DIS, a $W^-$ boson is exchanged, which only interacts with $\bar{d}$, $u$, $\bar{s}$ and $c$ quarks at lowerst order in $\alphas$.  This means that the $e^-p$ and $e^+p$ offer very different information about the flavor content of the proton.  The measurements presented in chapter~\ref{chapter_results} show a strong dependence on the PDF set used, most specifically a strong dependence on the $d$ PDF.  At present the $d$ PDF has much larger experimental uncertainties than the $u$, so it is believed that this data can be used in conjunction with other measurements to improve the uncertainty of the $d$ PDF.  This knowledge is critical to extracting meaningful results at any hadron collider, because the uncertainties due to PDFs are often relatively large.  Because the Large Hadron Collider will begin producing collisions in the near future, it is hoped that these results will be used by PDF extraction groups in future fits.