\chapter{Theoretical Predictions}
	\label{chapter_nlo_calc}

\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~\ref{section_isr_fsr}.  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 parameterizes 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 and Parton Density Functions}
	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, and the soft QCD processes in the proton.  The PDFs used must be evolved from some originating scale to the factorization scale at which they are used.  This evolution is accomplished via the DGLAP equations [MISSING:  Explain DGLAP]
\section{Scale Dependence and Uncertainty}
	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 for $\mur \equiv Q/2$ and $\mur \equiv 2Q$.  The optimal amount of adjustment used in this technique is not universally agreed upon, but a factor of 2 is common practice for ZEUS publications.

	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 $\muf$.  %see page 8 of Mirkes
	This scale was also set to $Q$, and modified by a factor of 2 up and down to quantify uncertainty based on this choice. 
\section{PDFs and PDF Uncertainty}
	To produce $ep$ predictions, hard scattering functions must be convolved with PDFs.  Calculations in this thesis were performed using the ZEUS-S\cite{pr:d67:012007} parameterization 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 multiple different sets of PDFs, each of which contains different experimental input.  To understand the theoretical uncertainty involved in the choice of parameterization, alternative calculations were performed using the CTEQ6\cite{Pumplin:2002vw, Stump:JHEP0310} and MRST~\cite{ref_mrst}.  
\section{Systematic Uncertainty Estimate for the Calculations}
	\label{section_mepjet_uncertainty}
	No alternative $\mathcal{O}({\alphas}^2)$ calculations exist 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 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\%.    
\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 described as representing the {\it parton level}.  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 said to describe the {\it hadron level}.  This was perfromed using the event simulations described in chapter ~\ref{chapter_event_simulation}, and the techniques used to accomplish this are described in detail in chapter~\ref{chapter_analysis_method}.  


