\chapter{Charged Current Interactions in Deep Inelastic Scattering}
	\label{chapter_cc_dis}
\section{An Overview of $ep$ Scattering}
	Deep Inelastic Scattering (DIS) is the scattering of electrons on protons via the exchange of virtual $
\gamma$, $Z^0$, or $W$ bosons. Throughout this thesis, we will be primarily concerned with the following two processes:  $ep\rightarrow e+X$ and $ep\rightarrow \nu+X$, where $X$ denotes any outgoing state.  This is described in the two lowest order diagrams shown in figure~\ref{diagram_nc_cc_dis}.
\begin{figure}
  \begin{center}
	%\epsfig{file=figures/introduction/pdg_ratio_n_colors.ps,width=.8\textwidth}
	\includegraphics[draft=false, width=0.5\textwidth]{/home/hwolfe/thesis/figures/chapter_cc_dis/fig_DIS_intro}
	%http://pdg.lbl.gov/2007/reviews/hadronicrpp.pdf
  \end{center}
	\caption[Lowest oder diagrams for NC and CC DIS]{\fixspacing{caption}}
	\label{diagram_nc_cc_dis}
\end{figure}
	When the exchanged boson in this diagram is a $\gamma^*$ or $Z^0$, the process will be referred to as Neutral Current DIS (NC DIS), and when the exchanged boson is a $W^\pm$, will be called Charged Current DIS (CC DIS).  
\subsection{Kinematics}
	In this chapter, we will refer to the initial state electron simply as the incoming lepton, and the final state electron or neutrino simply as the outgoing lepton.  We label the 4-vectors of the incoming lepton, outgoing lepton, incoming proton and outgoing hadronic system as $k$, $k'$, $p$, and $p_X$, respectivley.  Electrons will be considored massless, and the mass of the proton is labled $M_p$.  The following are lorentz invariant quantities:
\begin{equation}
	q^2 = (k-k')^2 = -Q^2, Q^2 > 0
	\label{equation_def_q2}
\end{equation}
\begin{equation}
	s = (p+k)^2
	\label{equation_def_s}
\end{equation}
\begin{equation}
	W^2 = p_X^2
	\label{equation_def_W}
\end{equation}
\begin{equation}
	\nu = \frac{p\cdot q}{M_p} = \frac{1}{2M_p}{\left(W^2+Q^2 - M_p^2\right)}
	\label{equation_def_nu}
\end{equation}
$Q$, $s$, and $W$ are identified as the transfer momentum, center of mass energy (CME) of the $ep$ system, and invariant mass of the outgoing hadronic system.  In the rest frame of the proton, $\nu$ is the energy difference of the incoming and outgoing lepton.  We will also be using the following variables extensively:
\begin{equation}
	x = \frac{Q^2}{2p\cdot q} = \frac{Q^2}{2M_p\nu} = \frac{Q^2}{W^2+Q^2 - M_p^2}
	\label{equation_def_x}
\end{equation}
\begin{equation}
	y = \frac{p\cdot q}{p\cdot k}
	\label{equation_def_y}
\end{equation}
The Bjorken scaling variable $x$ is identified in the quark-parton model with the fraction of the momentum of the quark which participates in the hard scattering with that of the initial state proton.  The variable $y$ is called the inelasticity of the interaction.  Neglecting the masses of the incoming particles they are related by $Q^2 = sxz$.
\subsection{Cross Sections}
	The results of this thesis are differential cross sections, $\frac{d\sigma}{d\Omega}$ which are defined as the rate or number of outgoing particles in a given state divided by the rate or number of incoming particles.  Summing over all outgoing states results in the total cross section:
\begin{equation}
	\sigma = \frac{d\sigma}{d\Omega} = \frac{N}{L}
	\label{equation_def_total_cross section}
\end{equation}
	Where $N$ is the number of outgoing states of a given variety, and $L$ is the {\it luminosity}, a measure of the incoming flux.  This is related to the scattering amplitude by 
\begin{equation}
	d\sigma = \frac{1}{\mbox{incoming flux}}\frac{d^3k'}{dk'_0}\frac{1}{4}|\mathcal{M}|^2
	\label{equation_crosssection_amplitude_1}
\end{equation}
	Where the amplitude for leading order NC scattering is given by
\begin{equation}
	\mathcal{M_{\mbox{NC}}} = e^2\bar{u}(k',\lambda')\gamma^{\mu}u(k,\lambda) \frac{1}{q^2}\left< X \left|J_\mu^{\mbox{\tiny EM}}\right|p,\sigma\right>
	\label{equation_amplitude_em}
\end{equation}
	and for CC is given by
\begin{equation}
	\mathcal{M_{\mbox{CC}}} = \frac{\sqrt{2}G_F}{1+Q^2/M^2_W}\bar{u}(k',\lambda')\gamma^\mu\left(1-\gamma_5\right)u(k,\lambda) \frac{1}{q^2}\left< X \left|J_\mu^{\mbox{\footnotesize weak}}\right|p,\sigma\right>
	\label{equation_amplitude_weak}
\end{equation}
	The generality of the outgoing hadronic state introduces ambiguity, which we account for by summing over all these states.  we proceed by computing the spin-averaged cross sections via the introduction of a leptonic and hadronic tensor~\cite{halzen_and_martin}
\begin{equation}
	d\sigma \sim L_{\mu\nu}W^{\mu\nu}
	\label{equation_cross_section_tensor}
\end{equation}

\begin{equation}
	W^{\mu\nu} \equiv \sum_{N}\int_{X_N}\left< p, \sigma \left|J^{\mu\dagger}\right|X_N\right>\left< X_N \left|J^{\nu\phantom{\dagger}}\right|p,\sigma\right>
	\label{equation_definition_hadronic_tensor}
\end{equation}
Because the proton is a composite object governed by QCD outside the perturbative regime, we must write the hadronic tensor in a general form, and parameterize it through experimentally determined functions.  We begin by writing all possible Lorentz invariant terms composed of the metric tensor, $p$, and $q$ with preceding unitless scalar functions $W_i$.  The choices of coefficients comes from hindsight to more easily arrive at general notation.
\begin{equation}
	W^{\mu\nu} =  -W_1 g^{\mu\nu} + \frac{W_2}{M_p}p^{\mu}p^{\nu}+W_3i\epsilon^{\mu\nu\alpha\beta}p_\alpha q_\beta + \frac{W_4}{M_p}q^\mu q^\nu+\frac{W_5}{M_p}\left(p^\mu q^\nu+q^\mu p^\nu\right)
	\label{equation_hadronic_tensor_general}
\end{equation}
	Current conservation requires $q_\mu W^{\mu\nu} = q_\nu W^{\mu\nu} = 0$, which can be used to show 
\begin{equation}
	 W_4 = \frac{M_p}{q^2}W_1+\left(\frac{p\cdot q}{q^2}\right)^2 W_2 \mbox{ and } W_5 = -\frac{p\cdot q}{q^2}W_2
	\label{equation_hadronic_current_conservation}
\end{equation}
	Combining these yields
%\begin{eqnarray}
	%W^{\mu\nu} %&=&  %-W_1 g^{\mu\nu} + \frac{W_2}{M_p}p^{\mu}p^{\nu}+W_3i\epsilon^{\mu\nu\alpha\beta}p_\alpha q_\beta  +\frac{1}{M_p}q^\mu q^\nu\left(\frac{M_p}{q^2}W_1+\left(\frac{p\cdot q}{q^2}\right)^2  W_2\right) \nonumber \\ 
	%	&\ &+\frac{1}{M_p}\left(p^\mu q^\nu+q^\mu p^\nu\right)\left(-\frac{p\cdot q}{q^2}W_2\right)\\
	%&=&  W_1\left(\frac{q^\mu q^\nu}{q^2}-g^{\mu\nu}\right) +W_3i\epsilon^{\mu\nu\alpha\beta}p_\alpha q_\beta \nonumber \\ 
	%&\ &+ \frac{W_2}{M_p}\left(q^\mu q^\nu\left(\frac{p\cdot q}{q^2}\right)^2 +\left(p^\mu q^\nu+q^\mu %p^\nu\right)\left(-\frac{p\cdot q}{q^2}\right)+p^{\mu}p^{\nu}\right)%\\
	%&=&  W_1\left(\frac{q^\mu q^\nu}{q^2}-g^{\mu\nu}\right) +W_3i\epsilon^{\mu\nu\alpha\beta}p_\alpha q_\beta \nonumber \\ 
	%&\ &+ \frac{W_2}{M_p^2}\left(p^\mu -\frac{p\cdot q}{q^2} q^\mu\right)\left(p^\nu -\frac{p\cdot q}{q^2} q^\nu\right)
	%\label{equation_hadronic_tensor_general}
%\end{eqnarray}
\begin{equation}
	W^{\mu\nu} = W_1\left(\frac{q^\mu q^\nu}{q^2}-g^{\mu\nu}\right) + \frac{W_2}{M_p^2}\left(p^\mu -\frac{p\cdot q}{q^2} q^\mu\right)\left(p^\nu -\frac{p\cdot q}{q^2} q^\nu\right) +W_3i\epsilon^{\mu\nu\alpha\beta}p_\alpha q_\beta  
	\label{equation_hadronic_tensor_general}
\end{equation}
	The leptonic tensor can be computed exactly, and is 
\begin{equation}
	L_{\mu\nu}(e^-p\rightarrow \nu X) \equiv  \frac{1}{4}\left[\bar{\nu_e}(k')\gamma_\mu\left(1-\gamma_5\right)e(k)\right]\left[\bar{\nu_e}(k')\gamma_\nu\left(1-\gamma_5\right)e(k)\right]
	\label{equation_leptonic_tensor_eminus}
\end{equation}
	for electron scattering and 
\begin{equation}
	L_{\mu\nu}(e^+p\rightarrow \bar{\nu} X) \equiv  \frac{1}{4}\left[\bar{e}(k)\gamma_\mu\left(1-\gamma_5\right)\nu(k')\right]\left[\bar{e}(k)\gamma_\nu\left(1-\gamma_5\right)\nu(k')\right]
	\label{equation_leptonic_tensor_eplus}
\end{equation}
	for protons.  Because we neglect the mass of the incoming electron, we can write both of these expressions as
\begin{equation}
	L_{\mu\nu} =  2\left[k_{\mu}k'_\nu+k'_{\mu}k_{\nu}-g_{\mu\nu}k\cdot k' \pm i\epsilon_{\mu\nu\alpha\beta}k'^\alpha k^\beta\right],
	\label{equation_leptonic_tensor_simple}
\end{equation}
	where the term takes $+$ for electrons and $-$ for positrons.  
\subsection{Charged Current DIS}
\begin{figure}
  \begin{center}
	%\includegraphics[draft=false, width=0.5\textwidth]{/home/hwolfe/thesis/figures/chapter_cc_dis/fig_DIS_intro}
  \end{center}
	\caption[Feynman Diagram for CC DIS]{\fixspacing{MISSING: Feynman Diagram for CC DIS}}
	\label{diagram_cc_dis}
\end{figure}
\subsection{Neutral Current DIS}
\begin{figure}
  \begin{center}
	%\includegraphics[draft=false, width=0.5\textwidth]{/home/hwolfe/thesis/figures/chapter_cc_dis/fig_DIS_intro}
  \end{center}
	\caption[Feynman Diagram for NC DIS]{\fixspacing{MISSING: Feynman Diagram for NC DIS}}
	\label{diagram_nc_dis}
\end{figure}
\subsection{Photoproduction}
\begin{figure}
  \begin{center}
	%\includegraphics[draft=false, width=0.5\textwidth]{/home/hwolfe/thesis/figures/chapter_cc_dis/fig_DIS_intro}
  \end{center}
	\caption[Feynman Diagram for PHP]{\fixspacing{MISSING: Feynman Diagram for PHP}}
	\label{diagram_php}
\end{figure}

\subsection{Structure Functions and Parton Distribution Functions}
	The functions $W_1$, $W_2$ and $W_3$ can be replaced with {\it structure functions}
\begin{eqnarray}
	F_1^CC(x,Q^2) &=& M_PW_1\nonumber\\
	F_1^CC(x,Q^2) &=& \nu W_2\nonumber\\
	F_1^CC(x,Q^2) &=& \nu W_3,
	\label{equation_structure_functions}
\end{eqnarray}
	which allows us to write the differential cross section for CC DIS as 
\begin{equation}
	\frac{d^2\sigma(e^{\pm}p\rightarrow \nu X)}{dxdQ^2} = \frac{G_F^2}{4{\pi}x}\left(\frac{M_W^2}{M_W^2+Q^2}\right)
	\label{equation_leptonic_tensor_simple}
\end{equation}

\section{Jet Production in DIS}
	While isolated quarks and gluons cannot be directly observed, objects which are very correlated to these states can be.     Because the QCD coupling constant is large for small momentum transfers, QCD radiation is largely dominated by low-$p_t$ emission of gluons and $q\bar{q}$ pairs.  This implies that the QCD radiation will not commonly diverge greatly from the quark or gluon that initiated the process, creating nearly parrallell cones of hadrons in detectors.  These sprays of hadrons in this formation are referred to as jets.  The event shown in Figure ~\ref{figure_jade_3j} is from the JADE detector at the electron-positron collider PETRA, and displays three jets in the detector, which is viewed as the initial production of a $q\bar{q}$ pair, followed by a high-$p_t$ radiation of a gluon.  Events of this type helped establish QCD as a theory of quarks and gluons.
	
\begin{figure}
	\begin{center}
		%\epsfig{file=figures/introduction/pdg_ratio_n_colors.ps,width=.8\textwidth}
		\includegraphics[draft=false, width=1.\textwidth]{/home/hwolfe/thesis/figures/introduction/jade-3j}
		%http://pdg.lbl.gov/2007/reviews/hadronicrpp.pdf
	\end{center}
		\caption[Observation of a 3-Jet Event in the JADE Detector at PETRA]{
			\fixspacing{  
				Observation of a 3-Jet Event in $e^+e^-$ scattering in the JADE Detector at PETRA.   The beam line is perpendicular to the page.  ~\cite{ref_jade_3jet}.
			}
		}
	\label{figure_jade_3j}
\end{figure}
\subsection{Perturbative Quantum Chromodynamics}
	Because the strong coupling constant $\alpha_S(\mu)$ is much less than one for transfer momenta much greater than the scale  $\Lambda_{QCD}$ (about 200 MeV), it is possible to write matrix elements for hard processes as a Taylor expansion in $\alpha_S(\mu)$.  
	\begin{equation}
		\mathcal{M}(\alpha_S) = \sum_0^\infty {\mathcal{M}_n\times\alpha_S^n}
	\end{equation}
	Assuming convergence of the series, we can approximate the full matrix element by a truncated series, where {\it leading order} refers to the $n=0$ term, {\it next-to-leading order} refers to $n=1$, and so on.  In practice, matrix elements are not computed exactly in the electromagnetic or strong coupling, but are either computed to finite order, or are computed to all orders in some other expansion parameter. 

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