\chapter{Theoretical Predictions}
	\label{chapter_cc_dis}
\section{Kinematics of $ep$ Scattering}
\begin{figure}[htb]
  \begin{center}
	\includegraphics[draft=false, width=0.5\textwidth]{\figdir/chapter_cc_dis/fig_DIS_intro}
  \end{center} 
	\caption[General Schematic of $ep$ Scattering]{\fixspacing{General Schematic of $ep$ Scattering.  The electron and proton interact via the spacelike exchange of a $\gamma$, $W$ or $Z$ with virtuality $Q^2$.  The outgoing hadronic system is labeled $X$.}}
	\label{diagram_ep_scatter}
\end{figure}
	Figure~\ref{diagram_ep_scatter} shows a very general $ep$ interaction, for the purposes of defining kinematic variables. To simplify discussion, the initial state electron or positron will simply be referred to as the incoming lepton, and the final state electron, positron or neutrino will simply be referred to 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 $p_X$, respectively.  Electrons will be considered massless thoughout this work, 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.  The following variables will be used 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 at the born level 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 referred to as the inelasticity of the interaction.  Neglecting the masses of the incoming particles they are related by $Q^2 = sxy$.
\section{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 number of incoming particles.  Summing over all possible outgoing states results in the {\it total-} or {\it integrated cross section}:
\begin{equation}
	\sigma = \int_{\Omega}\frac{d\sigma}{d\Omega} = \frac{N}{L}
	\label{equation_def_total_cross section}
\end{equation}
	Where $N$ is the number of outgoing states and $L$ is the {\it integrated luminosity}, a measure of the total incoming flux.  

The differential cross section for CC DIS versus $x$ and $Q^2$ can be written as~\cite{halzen:1984:quarks},
\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)\left[Y_+F^{CC}_2  -y^2F_L^{CC}\mp Y_-xF_3^{CC}\right],
	\label{equation_ccdis_xs_wstruct}
\end{equation}
	where $G_F$ is the Fermi coupling constant, $M_W$ is the mass of the $W$ boson, and $Y_\pm = 1\pm (1-y)^2$.  The scalar functions $F_n^{CC}(x,Q^2)$ are called {\it structure functions}, and parametrize the flavor content of the proton and soft QCD interactions involved in $ep$ scattering.  These are not computable in pQCD, and must be experimentally determined.  $F_L^{CC}$ is very small relative to the other two structure functions.  These are related to the PDFs by 
\begin{eqnarray}
 	F_2(e^{-}p\rightarrow \nu X)) & = & x\left(u+\bar{d}+c+\bar{s}\right) \approx x u\\
 	xF_3(e^{-}p\rightarrow \nu X)) & = & x\left(u-\bar{d}+c-\bar{s}\right) \approx x u\\
 	F_2(e^{+}p\rightarrow \bar{\nu} X)) & = & x\left(\bar{u}+d+\bar{c}+s\right) \approx xd\\
 	xF_3(e^{+}p\rightarrow \bar{\nu} X)) & = & x\left(-\bar{u}+d-\bar{c}+s\right) \approx xd,
\end{eqnarray}
	where we have neglected $b$ and $t$ PDFs, because we assume them to be statistically negligible at the scales probed here. Gluons have likewise been neglected in the above, which is only correct to lowest order in $\alphas$.  

	The differential cross section in terms of $x$ and $y$ can be derived by~\cite{Eidelman:2004wy}
\begin{equation}
 	\frac{d^2\sigma}{dxdy} = x\left(s-M_{P}^2\right)\frac{d^2\sigma}{dxdQ^2}.
\end{equation}



\section{The MEPJET Program}
	\label{section_mepjet}
	The pQCD calculations in this thesis were produced with the {\mepjet}~2.1~\cite{pl:380:205, ref_mepjet} program, which was written for DIS by E.~Mirkes. D.~Zeppenfeld and S.~Willfahrt.  It is capable of computing partonic cross sections in DIS via exchange of photons, $Z$, and $W$ bosons at $\mathcal{O}({\alphas})$ or $\mathcal{O}({\alphas}^2)$.   For this analysis, only $W$ exchange was calculated, with the mass of the $W$ boson set to $80.40~\GeV$.  The number of active quark flavors was set to 5, and $\Lambda^{(5)}_{\tiny \msbar}$ was set to $226~\MeV$, which corresponded to a strong coupling constant $\alphas(M_Z) = 0.118$.  $\alphaem$, $G_F$, and $M_Z$ were set to the same values as chosen for the MC described in section
\begin{eqnarray}
	\alphaem &:& 1/137.035999\nonumber\\
	\gfermi &:& 1.664\times10^{-5}~{\GeV}^{-2}\nonumber\\
	M_Z &:& 91.1876~\GeV
\end{eqnarray}.  
	For the inclusive-jet calculations,  $\mathcal{O}({\alphas})$ predictions were produced, which are NLO.  For dijet (three-jet ) cross sections, $\mathcal{O}({\alphas}^2)$ predictions were produced, which were NLO (LO).  
\section{Phase Space Slicing Method}
	The {\mepjet} program uses the phase space slicing method~\cite{ref_phase_space_slicing} to cancel infrared and collinear divergences.  A parameter $s_{\mbox{min}}$ was introduced, which acts as a minimum resolving power to separate partons, and effectively separates finite and infinite portions of the phase space.  Integrals in regions where particles $i$ and $j$ satisfied $s_{ij} \cong 2p_i\cdot p_j < s_{min}$ were calculated analytically.  Some divergences were matched and exactly cancelled with virtual contribution divergences, and others were subtracted as described in section ~\ref{section_renormalization_scheme}.  It has been shown that the final results are independent of the unphysical parameter $s_{\mbox{min}}$ when $s_{\mbox{min}}$ is less than or on the order of $0.1~\GeV$~\cite{ref_mepjet}.  The remaining phase space, where the integrals were free of these divergences, were left to be calculated numerically by MC integration techniques.  Leaving the finite portion of the phase space integral for the user provides a flexible system for custom modification of phase space cuts.  
\section{$\msbar$ Renormalization Scheme }
	\label{section_renormalization_scheme}
	In {\mepjet}, the one-loop divergences in the diagrams are handled in the $\msbar$ renormalization scheme.  The scheme begins by introducing a dimensional regulator for loop momentum, by transforming the 4-dimensional integral over virtual particle momenta into $4-2\epsilon$ dimensions.  This parametrizes the divergences in $\epsilon$, but artificially introduces additional constant terms.  The name stands for modified minimal subtraction, because the earlier method of subtracting only divergent parts~\cite{ref_thooft_msbar} was later modified by additionally subtracting the constant terms that were introduced~\cite{ref_bardeen_msbar}.    
\section{Factorization Scale and Evolution of Parton Density Functions}
	\label{section_pdfs_specific}
	In addition to the choice of the renormalization scale at which the matrix elements are evaluated, an additional scale choice is made called the {\it factorization scale}.  This scale is the delineation between the hard scattering computed in pQCD, and the soft QCD processes in the proton.  Just as observable quantities should not depend on $\mur$, they should not depend on $\muf$, either.  The  Dokschitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP)~\cite{np:b126:298,sovjnp:15:438,sovjnp:20:94,jetp:46:641} equations are a set of $2(n_f+1)$ coupled integro-differential equations which require that each structure function be independent of $\muf$.   In doing so, the DGLAP equations describe how the PDFs change with $Q^2$ and $\muf$, allowing PDFs measured in one process to be used in predictions for another.  The PDFs in this analysis come from the Coordinated Theoretical-Experimental Project on QCD (CTEQ5D, CTEQ6D)~\cite{epj:c12:375, Pumplin:2002vw, Stump:JHEP0310}, Martin Roberts Stirling Thorne (MRST2001)~\cite{pr:d50:6734}, and The ZEUS Collaboration~\cite{epj:c42:1}.  

\section{Scale Dependence and Uncertainty}
	\label{section_scale_dependence}
	In {\mepjet}, ultraviolet divergences were renormalized and absorbed into the bare coupling constant, introducing a dependence of the renormalized coupling $\alphas(\mur)$.   For this analysis, $\mur$ was set to $Q$ for the central values extracted from the calculations.  The dependence on $\mur$ would cancel for calculations performed at all orders in $\alphas$, but for our fixed order calculations, some residual scale dependence persists.   To quantify the uncertainty introduced by our choice, the calculations were also performed using the canonical adjustments $\mur \equiv Q/2$ and $\mur \equiv 2Q$.  The optimal amount of adjustment used in this technique is not universally agreed upon, but is common practice for High Energy Experimental Physics~\cite{Klein:2005fa}.

	In {\mepjet}, initial state collinear divergences were factorized into the bare parton densities,  introducing a dependence of the final cross sections on the factorization scale $\muf$~\cite{ref_mepjet}. 	This scale was also set to $Q$, and modified by a factor of 2 up and down to quantify uncertainty based on this choice. %see page 8 of Mirkes
\section{PDFs and PDF Uncertainty}
	To produce $ep$ scattering predictions, $eq$ and $eg$ hard scattering functions must be convolved with PDFs.  Calculations in this thesis were performed using the ZEUS-S~\cite{pr:d67:012007} parametrization of the proton PDFs.  PDFs must be experimentally determined, and contain both theoretical and experimental uncertainties.  To understand the experimental uncertainty of the PDFs effect on results, calculations were performed using the 22 alternative PDFs from the ZEUS-S set, each of which contains different experimental input.  This method takes into account the statistical and correlated systematic experimental uncertainties of each data set used in the determination.  To understand the theoretical uncertainty involved in the choice of parametrization, alternative calculations were performed using the CTEQ6 and MRST sets.
\section{Systematic Uncertainty Estimate for the Calculations}
	\label{section_mepjet_uncertainty}
	No alternative $\mathcal{O}({\alphas}^2)$ calculations exists for CC DIS processes at this time.  It is therefore not possible to produce a direct estimate of the uncertainty introduced by our choice of {\mepjet}.  However, because {\mepjet} also produces NC DIS calculations, and because alternative NLO programs for these processes do exist, we can compare NC DIS predictions, and consider the difference an estimate of the uncertainty introduced from the use of {\mepjet}.   It  was observed in \cite{ref_mepjet_fucked} that while {\mepjet} and the alternative program \textsc{Disent}~\cite{np:b485:291} agree for inclusive NC DIS jet cross sections on the order of 1\%, the relative disagreement for NC DIS dijet cross sections is on the order of $5-8$\%.
\section{Correction of the Calculations to the Hadron Level}
	\label{hadronization_corrections}
	The calculations described above produce cross sections of jets of partons.  These cross sections are referred to as {\it parton level} cross sections.  In order to compare these predictions from cross sections which were measured, these cross sections were corrected to describe jets of hadrons.  The resultant cross sections are referred to as {\it hadron level} cross sections.  This was performed using the event simulations described in chapter ~\ref{chapter_event_simulation}, and the techniques used to accomplish this are describedin section~\ref{section_theroretical_corrections} in detail.  


