\chapter{Results}
	\label{chapter_results}
In this chapter, hadron-level cross sections for jet production in CC DIS are presented.  The cross sections represent an integrated luminosity of $180~\pb^{-1}$ of $e^-p$ data, and $179~\pb^{-1}$ of $e^+p$ data, using the ZEUS detector at the HERA collider.  The cross sections are grouped into four categories, depending on the lepton longitudinal-polarization and jet topology considered; Inclusive-jet polarized, Inclusive-jet unpolarized, Dijet unpolarized and trijet unpolarized.  The unpolarized cross sections represent extrapolations from polarized cross sections using the SM relation $\sigma^{e^{\pm}p}_{\mbox{\tiny CC}}(P) = (1\pm P)\sigma^{e^{\pm}p}_{\mbox{\tiny CC}}(0)$, where $P$ is the luminosity-averaged lepton longitudinal polarization.  The inclusive jet measurements represent 5335 negatively-polarized $e^-p$ jets, 870 negatively polarized $e^+p$ jets, 2122 positively-polarized $e^-p$ jets and 2284 positively polarized $e^+p$ jets.  The dijet samples represent 1117 $e^-p$ and 464 $e^+p$ events.  These events form a subset of the inclusive jet sample.  The trijet sample is a subset of the dijet sample, with 109 events for $e^-p$ data and 30 for $e^+p$.

In all figures and tables, measured cross sections are compared to SM predictions provided by the {\mepjet} 2.1~\cite{ref_mepjet} program.  The calculations from {\mepjet} were corrected from the electroweak born-level to represent cross sections for jets of hadrons with ISR/FSR via LO MC as described in section~\ref{section_theroretical_corrections}.  All calculations were for unpolarized leptons, but polarized predictions were produced by scaling by a factor of $1/(1\pm P)$ for $e^{\pm}p$ data.  Numerical tables for all figures presented here are given in appendix~\ref{chapter_data_tables}.

\section{Inclusive Jet Polarized Differential Cross Sections}

The inclusive-jet differential cross sections presented here represent samples of negatively-
(positively-) polarized electron beams with an integrated luminosity of $106.4$ $(73.6)~\pb^{-1}$ and luminosity-weighted average polarization of $P_{e^-}^{\rm neg}=-0.27\pm 0.01$ ($P_{e^-}^{\rm pos}=+0.29\pm 0.01$) were analyzed. For positrons, the samples analyzed were of $76.5$ and $102.1~\pb^{-1}$ with a luminosity-weighted average polarization of $P_{e^+}^{\rm neg}=-0.37^{+0.01}_{-0.02}$ and 
$P_{e^+}^{\rm pos}=+0.32\pm 0.01$, respectively.

The cross sections $\frac{d\sigma_{\mbox{\tiny jet}}}{d\etjet}$, for negatively- and positively-polarized $e^{-}p$ ($e^{+}p$) collisions are shown in Fig.~\ref{figure_polxs_et_e_1jet}a  (\ref{figure_polxs_et_p_1jet}b).  Jets were defined as $\etjet > 14~\GeV$ and $-1 < \rapjet < 2.5$.  Jets with $\etjet$ up to $120~\GeV$ are accessible within present statistics. The figures also show the ratio of the cross sections for negatively- and positively-polarized lepton beams.  This ratio is nearly constant within errors and in agreement with the measured polarization ratio, given by the luminosity averaged polarization during data taking as described in sections~\ref{section_description_of_sample} and~\ref{section_preselection}. This was $(1 - P_e^{\rm neg} )/(1 - P_e^{\rm pos})  = 1.79 \pm 0.05$ for $e^{-}p$ collisions and $(1 - P_e^{\rm neg} )/(1 - P_e^{\rm pos}) = 2.10+0.08-0.14$ for $e^{+}p$ collisions. The integrated polarized inclusive-jet cross sections, $\sigma_{\mbox{\tiny jets}}$, are given in Table~\ref{table_integrated_xs_pol}.  The measured integrated cross sections are in good agreement with the predictions of the SM as given by the {\mepjet} calculations.  The unpolarized inclusive-jet cross sections as functions of $Q^2$, $\rapjet$ and $x$ show similar levels of agreement with the predictions, and are included in appendix~\ref{chapter_additional_plots}.

\begin{figure}[htbp]
\vfill
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	\begin{picture} (18.0,8.5)
		\put(-.25,-1.0){\includegraphics[draft=false, width=.65\textwidth]
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	\end{picture}
	\caption[Polarized $e^-p$ and $e^+p$ $\etjet$ Inclusive-Jet Cross sections]{Polarized $e^-p$ and $e^+p$ $\etjet$ inclusive-jet cross sections}
	\label{figure_polxs_et_e_1jet}
	\label{figure_polxs_et_p_1jet}
\vfill
\end{figure}

\begin{table}
\begin{center}
    \begin{tabular}{|c||c|c|c|c|c|}
\hline
  lepton /  polarization
& $\sigma_{\rm jets}$ (pb)
& $\delta_{\rm stat}$ (pb)
& $\delta_{\rm syst}$ (pb)
& $\delta_{\rm ES}$ (pb)
& $\sigma_{\mbox{\tiny SM}}$ (pb)\\
\hline\hline
$P_{e^-}=-0.27\pm 0.01$ & $70.54$ & $0.97$ & $0.58$ & $_{-0.40}^{+0.43}$ & $69.17$\\
%\hline
$P_{e^-}=+0.29\pm 0.01$ & $40.53$ & $0.88$ & $0.45$ & $_{-0.23}^{+0.24}$ & $38.67$\\
\hline
$P_{e^+}=-0.37^{+0.01}_{-0.02}$ & $17.55$ & $0.60$ & $0.57$ & $\pm 0.11$ & $16.86$\\
%\hline
$P_{e^+}=+0.32\pm 0.01$ & $34.51$ & $0.72$ & $1.05$ & $_{-0.22}^{+0.23}$ & $35.33$\\
\hline
    \end{tabular}
\caption{
  Integrated polarized inclusive-jet cross-sections 
  $\sigma_{\rm jets}$ for jets of hadrons in the laboratory frame
  selected with the longitudinally invariant $k_{T}$ cluster
  algorithm, described in section~\ref{section_jet_reconstruction}. The statistical, uncorrelated systematic and energy-scale ({\rm ES}) uncertainties are shown separately. The uncertainty coming from the luminosity measurement is not listed. The predictions of the SM as given by the {\mepjet} calculations and are shown in the last column.}
 \label{table_integrated_xs_pol}
\end{center}
\end{table}
\section{Inclusive Jet Unpolarized Differential Cross Sections}



Figures~\ref{figure_upolxs_rap_1jet}a, ~\ref{figure_upolxs_x_1jet}a, ~\ref{figure_upolxs_et_1jet}a and ~\ref{figure_upolxs_q2_1jet}a show the unpolarized inclusive-jet differential cross sections as functions of $\rapjet$ , $x$, $\etjet$ and $Q^2$ in CC $e^{\pm} p$ DIS.  The measured cross sections represent the combination of data from left- and right-handed polarizations.  These were extrapolated to unpolarized measurements by $\sigma^{e^{\pm}p}_{\mbox{\tiny CC}}(P) = (1\pm P)\sigma^{e^{\pm}p}_{\mbox{\tiny CC}}(0)$, where $P$ is the luminosity-averaged lepton polarization.  Theoretical prediction curves from {\mepjet} calculations using the ZEUS-S PDF sets~\cite{pr:d67:012007} are also compared to the data.  The relative difference to the NLO calculations and the ratios of $e^{-} p$ cross sections to $e^{+} p$ are presented in figures~\ref{figure_upolxs_rap_1jet}b, ~\ref{figure_upolxs_x_1jet}b, ~\ref{figure_upolxs_et_1jet}b and ~\ref{figure_upolxs_q2_1jet}b.

The measured $\frac{d\sigma}{d\rapjet}$, shown in Figure~\ref{figure_upolxs_rap_1jet}a, has a maximum at $\rapjet \approx 1$, and is roughly similar in shape to $\frac{d\sigma}{dx}$, shown in Figure~\ref{figure_upolxs_x_1jet}a. This similarity in shape is because the center-of-mass frame of the $eq$ system is boosted in the electron direction relative to the lab system at low-$x$, and boosted in the proton direction at high-$x$.  Thus, jets at low-$x$ are likely to be scattered far in $\eta$ from the proton remnant, and close to the remnant at high-$x$~\cite{Klein:2008di}.  The values in $x$ accessible by the data are within the range $0.013 < x < 0.63$.  The {\mepjet} predictions give a reasonable description of the shape and normalization of the data. The measured ratio as a function of $x$, shown in Figure~\ref{figure_ratioxs_upol_x_1jet}b, increases as a function of $x$, because of the increased contribution of $u$-PDF relative to the $d$-PDF at high-$x$. The measured ratio as a function of $\rapjet$ is approximately $2$ across the entire range, which is expected since the integral of valence $u$ and $d$ quarks densities are 2 and 1 respectively~\cite{halzen_and_martin}. The ratio slightly increases as a function of $\rapjet$, due to its kinematic relationship with $x$.

\begin{figure}[htbp]
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	\end{picture}
	\caption[Unpolarized $e^-p$ and $e^+p$ $\rapjet$ Inclusive-Jet Cross sections]{Unpolarized $e^-p$ and $e^+p$ $\rapjet$ Inclusive-Jet Cross sections}
	\label{figure_upolxs_rap_1jet}
	\label{figure_ratioxs_upol_rap_1jet}
\vfill
\end{figure}

\begin{figure}[htbp]
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	\end{picture}
	\caption[Unpolarized $e^-p$ and $e^+p$ $x$ Inclusive-Jet Cross sections]{Unpolarized $e^-p$ and $e^+p$ $x$ Inclusive-Jet Cross sections}
	\label{figure_upolxs_x_1jet}
	\label{figure_ratioxs_upol_x_1jet}
\vfill
\end{figure}


\begin{figure}[htbp]
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	\end{picture}
	\caption[Unpolarized $e^-p$ and $e^+p$ $\etjet$ Inclusive-Jet Cross sections]{Unpolarized $e^-p$ and $e^+p$ $\etjet$ Inclusive-Jet Cross sections}
	\label{figure_upolxs_et_1jet}
	\label{figure_ratioxs_upol_et_1jet}
\vfill
\end{figure}

\begin{figure}[htbp]
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	\end{picture}
	\caption[Unpolarized $e^-p$ and $e^+p$ $Q^2$ Inclusive-Jet Cross sections]{Unpolarized $e^-p$ and $e^+p$ $Q^2$ Inclusive-Jet Cross sections}
	\label{figure_upolxs_q2_1jet}
	\label{figure_ratioxs_q2_rap_1jet}
\vfill
\end{figure}
 
The measured $\frac{d\sigma}{d\etjet}$ exhibits a fall-off of two (three) orders of magnitude in the $e^{-}p$ ($e^{+}p$) sample, which is similar to the behavior observed by the ZEUS and H1 collaborations in previous CC DIS jet production publications~\cite{epj:c19:429, epj:c31:149}. Values of $\etjet$ on the order of $120~{\GeV}$ are accessible with the present statistics. For $200 < Q^2 < 2000 ~{\GeV}^2$, the distributions display a weak dependence on $Q^2$.  The cross sections as functions of $\etjet$ and $Q^2$ show a less rapid fall-off below values of $\etjet\sim M_W$ and $Q^2\sim M_W^2$ than what is observed in NC DIS processes due to the massive $W$ propagator~\cite{np:b765:1}.  Similar shapes were observed in previous inclusive CC DIS~\cite{pl:b637:210,epj:c32:1, epj:c19:269} publications by the ZEUS and H1 collaborations.  The cross sections with respect to $Q^2$ ($\etjet$) shows a slightly different shape than the predictions, with the data above the predictions at low $Q^2$ ($\etjet$), and the data below the predictions at high $Q^2$ ($\etjet$).  This was also observed in the previous $e^+p$ ZEUS CC DIS jet production measurement~\cite{epj:c31:149}.  However, the measured ratios of $e^-p$ to $e^+p$ are in very good agreement with the predictions. 

The measured cross sections for the $e^{+}p$ sample decrease more rapidly as a function of $\etjet$ and $Q^2$ than for the $e^{-}p$ sample, which is visible in the ratios presented in  Figures~\ref{figure_upolxs_et_1jet}b and~\ref{figure_upolxs_q2_1jet}b.  The increase of $e^{-}p$ relative to $e^{+}p$ at high values is expected due to the increasing contribution from the valence-quark densities in the proton at high $x$~\cite{Nadolsky:2008cw}. The observed ratios of the measured cross sections as a function of $Q^2$ is similar to the ratio of $u$ and $d$ parton densities, showing increase as a function of $Q^2$, a rise of the ration of the cross sections as a function of $x$ is also observed, but to a lesser degree. 

Figures~\ref{fig9}a-d show the estimates of theoretical uncertainty from the terms beyond NLO, the parton-shower model and the uncertainty in the PDFs, separately for $e^-p$ and $e^+p$ collisions.  The methods used to produce these estimates are described in section~\ref{section_mepjet_uncertainty}. Calculations using the PDF sets CTEQ6~\cite{epj:c12:375, Pumplin:2002vw, Stump:JHEP0310} and MRST2001~\cite{pr:d50:6734} are also shown.  These PDF sets are widely used in high energy physics, and include data from HERA, the Tevatron, and fixed target experiments.  With the exception of $\frac{d\sigma_{\mbox{\tiny jets}}}{d\rapjet}$ in the range $-1 < \rapjet < 0$, the uncertainty coming from the PDFs is dominant for all inclusive-jet $e^\pm p$ CC cross sections. At high $\etjet$, $Q^2$ and $x$, the uncertainty in the predicted cross sections for positron beams is larger than for electron beams. This difference in the uncertainty due to the PDFs in the calculations for $e^-$ and $e^+$ beams can be attributed to the different flavor content probed: in $e^-p$ ($e^+p$) at high $x$ the $W^-$ ($W^+$) will couple predominantly to the $u$ ($d$) valence quark in the proton; at present, the uncertainty in the $d$ parton density is larger than that for the $u$ quark~\cite{Nadolsky:2008cw}. The predictions derived from the alternative PDF sets shows a wide spread in the predictions, especially for positron beams. These measurements, in a phase-space region where the other theoretical uncertainties are well under control, have the potential to constrain the flavor content of the proton if used together with other data in PDF extractions. A fast and accurate method to perform fits to extract the proton PDFs on data sets that included jet cross sections in NC DIS and photoproduction was recently developed by the ZEUS Collaboration~\cite{epj:c42:1}; the result was a sizable reduction of the uncertainty on the gluon density for $x\gtrsim 10^{-2}$ . Using the data presented here and extending such a method to jet cross sections in CC DIS may help to constrain the u and d valence quark distributions at high $x$. The integrated unpolarized inclusive-jet cross sections, $\sigma$-jets , are shown in Table~\ref{table_upol_int_xs}. The measured cross sections are in good agreement with the {\mepjet} predictions, also shown in the table using the alternative PDF sets.
\begin{figure}[p]
\vfill
\setlength{\unitlength}{1.0cm}
\begin{picture} (15.0,13.0)(.1,1.9)
\put (-1.0,7.1){\includegraphics[draft=false, width=10cm]
			{/home/hwolfe/thesis/figures/results/DESY-08-024_9a}}
\put (7.0,7.1){\includegraphics[draft=false, width=10cm]
			{/home/hwolfe/thesis/figures/results/DESY-08-024_9b}}
\put (-1.0,0.){\includegraphics[draft=false, width=10cm]
			{/home/hwolfe/thesis/figures/results/DESY-08-024_9c}}
\put (7.0,0.){\includegraphics[draft=false, width=10cm]
			{/home/hwolfe/thesis/figures/results/DESY-08-024_9d}}
\put (6.3,14.7){\bf\small (a)}
\put (14.2,14.7){\bf\small (b)}
\put (6.3,7.5){\bf\small (c)}
\put (14.2,7.5){\bf\small (d)}
\end{picture}
\caption
{\it 
Overview of theoretical uncertainties for the inclusive-jet cross
sections in CC DIS for jets with $\etjet>14~{\GeV}$ and $-1<\rapjet<2.5$
in the kinematic regime given by $Q^2>200~{\GeV}^2$ and $y<0.9$ as
functions of (a) $\rapjet$, (b) $\etjet$, (c) $Q^2$ and (d) $x$ in $e^-p$ (shaded areas) and $e^+p$ (hatched areas)
collisions. Shown are the relative uncertainties induced by the
variation of the renormalization scale $\mur$, the uncertainties on
the proton PDFs and hadronization model. Also shown are the relative
differences between the NLO QCD calculations using the CTEQ6 (dashed
lines) or MRST (dot-dashed lines) PDF sets to the calculations based
on the ZEUS sets.
}
\label{fig9}
\vfill
\end{figure}


\begin{table}[htbp]
\begin{center}
\scalebox{0.95}{
    \begin{tabular}{|c||c|c|c|c|c|c|c|}
\hline
  lepton/
& $\sigma_{\rm jets}$ (pb)
& $\delta_{\rm stat}$ (pb)
& $\delta_{\rm syst}$ (pb)
& $\delta_{\rm ES}$ (pb)
& \multicolumn{3}{c|}{QCD predictions (pb)}\\ \cline{6-8}
  jet multiplicity
& 
& 
& 
& 
& ZEUS-S & CTEQ6 & MRST \\
\hline\hline
$e^-$/inclusive jet & $56.18$ & $0.68$ & $0.53$ & $_{-0.32}^{+0.34}$ &
$54.47\pm 0.75$ & $54.05$ & $54.56$\\
$e^+$/inclusive jet & $26.88$ & $0.51$ & $0.82$ & $_{-0.17}^{+0.18}$ &
$26.77\pm 0.45$ & $25.85$ & $26.49$\\
\hline
$e^-$/dijet & $10.87$ & $0.34$ & $0.80$ & $_{-0.23}^{+0.24}$ &
$9.14\pm 0.35$ & $9.05$ & $9.26$\\
$e^+$/dijet & $5.83$ & $0.29$ & $0.45$ & $_{-0.12}^{+0.13}$ & 
$4.57\pm 0.19$ & $4.38$ & $4.55$\\
\hline
$e^-$/three jet & $1.52$ & $0.15$ & $0.09$ & $\pm 0.06$ &
$0.79\pm 0.22$ & $0.79$ & $0.82$\\
$e^+$/three jet & $0.563$ & $0.110$ & $0.037$ & $_{-0.022}^{+0.025}$ &
$0.397\pm 0.118$ & $0.386$ & $0.409$\\
\hline
    \end{tabular}}
 \caption{
   Integrated unpolarized jet cross-sections $\sigma_{\rm jets}$
   for jets of hadrons in the laboratory frame selected with the
   longitudinally invariant $k_{T}$ cluster algorithm. The statistical,
   uncorrelated systematic and energy-scale ({\rm ES}) uncertainties
   are shown separately. The predictions of QCD as given by the 
   {\mepjet} calculations using the ZEUS-S PDFs are shown at NLO
   for the inclusive-jet and dijet cross sections and at LO for the
   three-jet cross sections, together with the total theoretical
   uncertainty. Also shown are the total cross sections predicted by
   QCD using the CTEQ6 or MRST PDF sets.}
 \label{table_upol_int_xs}
\end{center}
\end{table}


\section{Dijet Unpolarized Differential Cross Sections}


Unpolarized dijet differential cross sections were measured in the kinematic regime $Q^2 >
200~{\GeV}^2$ and $y < 0.9$. The cross sections were determined for jets with $\etjeta > 14 ~{\GeV}$, $\etjetb > 5~{\GeV}$ and $-1 < \rapjet < 2.5$. Figures~\ref{figure_upolxs_rap_2jet}a, ~\ref{figure_upolxs_et_2jet}a, ~\ref{figure_upolxs_q2_2jet}a and~\ref{figure_upolxs_m_2jet}a show the unpolarized dijet differential cross sections as functions of $\rapjetbar$, $\etjetbar$, $Q^2$ and the dijet invariant mass, $\mdijet$ , where $\rapjetbar = (\rapjeta + \rapjetb )/2$ and $\etjetbar = (\etjeta + \etjetb )/2$ in CC $e^\pm p$ DIS. Both measured cross sections and NLO QCD predictions are presented.  Figures~\ref{figure_upolxs_rap_2jet}b, ~\ref{figure_upolxs_et_2jet}b, ~\ref{figure_upolxs_q2_2jet}b and~\ref{figure_upolxs_m_2jet}b show the relative differences of the measured and predicted cross sections, as well as the ratio of $e^-p$ to $e^-p$ cross sections.

The measured $\rapjetbar$ cross section has a maximum at $\rapjetbar \approx 1.25$. The measured cross section as a function of $\etjetbar$ exhibits a fall-off of two orders of magnitude for $\etjetbar \gtrsim 20~{\GeV}$. For $200< Q^2 < 2000~{\GeV}^2$, the distribution displays a weak dependence on $Q^2$ . Values of $\mdijet$ from 10 to $120~{\GeV}$ are accessible with the present statistics. The {\mepjet} predictions are compared to the measured $\mdijet$ cross sections in Fig.~\ref{figure_upolxs_m_2jet}a. Figure~\ref{figure_ratioxs_upol_m_2jet}b shows the relative difference to the predictions. The {\mepjet} predictions do not give an adequate description in shape and normalization of the measured differential cross sections over the entire phase space. In particular, for $\mdijet$, the data tend to be above the predictions for $\mdijet \gtrsim 70~{\GeV}$. The presented $\mathcal{O}(\alphas^2)$ calculations are NLO for dijet production, but higher order terms could contribute to the observed difference.  This is partially implied by the integrated unpolarized dijet cross sections, shown in Table~\ref{table_upol_int_xs}, in which a larger data cross section is observed than predicted by a factor of $\sim 19\%$ and $\sim 28\%$ for $e^-p$ and $e^+p$ data, respectively.

While the predictions show shape and normalization differences from the cross sections, they do provide a reasonable description of the ratios of the cross sections for $e^-p$ and $e^+p$ interactions. Dijet distributions include large contributions from boson-gluon-fusion (discussed in section~\ref{section_jet_production_in_dis}), which is sensitive to the gluon PDF~\cite{epj:c42:1}.  Understanding of the gluon PDF is critical to reducing the total uncertainty of measured cross sections at the Large Hadron Collider~\cite{Nadolsky:2008cw}, because a large fraction of particle production is derived from gluons in the range $x\lesssim 10^{-2}$. In order to use the differential dijet cross sections presented here in global PDF fits, the ratios of $e^-p$ to $e^-p$ cross sections present a means to attain sensitivity to the PDFs, while avoiding the discrepancies between the measured and predicted cross sections.

\begin{figure}[htbp]
\vfill
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	\begin{picture} (18.0,6.5)
		\put(-.25,-1.0){\includegraphics[draft=false, width=.65\textwidth]
%			{/home/hwolfe/thesis/paper_plots/cc14eta21ncol}}
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	\end{picture}
		\caption[Unpolarized $e^-p$ and $e^+p$ $\rapjetbar$ Dijet Cross sections]{Unpolarized $e^-p$ and $e^+p$ $\rapjetbar$ Dijet Cross sections}
	\label{figure_upolxs_rap_2jet}
	\label{figure_ratioxs_upol_rap_2jet}
\vfill
\end{figure}

\label{section_results_dijet}
\begin{figure}[htbp]
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	\end{picture}
		\caption[Unpolarized $e^-p$ and $e^+p$ $\etjetbar$ Dijet Cross sections]{Unpolarized $e^-p$ and $e^+p$ $\etjetbar$ Dijet Cross sections}
	\label{figure_upolxs_et_2jet}
	\label{figure_ratioxs_upol_et_2jet}
\vfill
\end{figure}

\begin{figure}[htbp]
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	\end{picture}
		\caption[Unpolarized $e^-p$ and $e^+p$ $Q^2$ Dijet Cross sections]{Unpolarized $e^-p$ and $e^+p$ $Q^2$ Dijet Cross sections}
	\label{figure_upolxs_q2_2jet}
	\label{figure_ratioxs_upol_q2_2jet}
\vfill
\end{figure}

\begin{figure}[htbp]
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	\end{picture}
		\caption[Unpolarized $e^-p$ and $e^+p$ $\mdijet$ Dijet Cross sections]{Unpolarized $e^-p$ and $e^+p$ $\mdijet$ Dijet Cross sections}
	\label{figure_upolxs_m_2jet}
	\label{figure_ratioxs_upol_m_2jet}
\vfill
\end{figure}


\section{Trijet Unpolarized Differential Cross Sections}

Three-jet cross sections are measured for the first time in CC DIS.  The cross sections are measured in the kinematic regime $Q^2>200~{\GeV}^2$  and $y<0.9$, for jets with $\etjeta>14$~{\GeV}, $\etjetb>5$~{\GeV}, $\etjetc>5$~{\GeV} and $-1<\rapjet<2.5$. Figure~\ref{fig12} shows a three-jet candidate event in the ZEUS detector: a distinct three-jet topology and large net transverse momentum are observed. 


\begin{figure}[p]
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	\put (0.4,14.1){\bf\small (a)}
	\put (6.7,14.1){\bf\small (b)}
	\put (9.5,8.0){\bf\small (c)}
	\end{picture}
\caption
{
Three-jet candidate event in CC DIS in the ZEUS detector. The event is viewed in the $x-y$ plane in (a), the $z-r$ view in (b), and $E_T$ of CAL deposition $\eta - \phi$ plane in (c).  In (a) and (b), reocnstructed tracks are superimposed over CTD hits, and CAL depositions are drawn proportional to the energy deposited.  In the (a), only the energy deposition in the barrel calorimeter is shown. in the The energy deposition in the CAL is proportional to the size and density of shading in the CAL cells. The Lego plot shows the CAL transverse energy deposition projected in the $\eta - \phi$ plane. 
}
\label{fig12}
\vfill
\end{figure}

Figure~\ref{figure_upolxs_rap_3jet}a presents the differential cross sections as functions of the average rapidity of the three-jet system, $\rapjetbar$, where $\rapjetbar=(\rapjeta+\rapjetb+\rapjetc)/3$.  The currently available QCD calculations are only lowest order and do not predict the normalization of the data. They were scaled by 1.92 and 1.42 for $e^-p$ and $e^+p$ collisions, respectively, so as to reproduce the measured integrated three-jet cross sections. The integrated cross sections for both data and predictions are presented in Table~\ref{table_upol_int_xs}.  The central values of calculations provide a far better description of the shape of the distributions compared to the dijet cross sections presented in section~\ref{section_results_dijet}, although statistical and theoretical errors are much larger.  The peak of the cross section is also near 1, as it was for the dijet systems.


\begin{figure}[htbp]
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	\end{picture}
		\caption[Unpolarized $e^-p$ and $e^+p$ $\rapjetbar$ Trijet Cross sections]{Unpolarized $e^-p$ and $e^+p$ $\rapjetbar$ Trijet Cross sections}
	\label{figure_upolxs_rap_3jet}
	\label{figure_ratioxs_upol_rap_3jet}
\vfill
\end{figure}

Figures~\ref{figure_upolxs_et_3jet} to~\ref{figure_upolxs_m_3jet} show the unpolarized three-jet 
$\etjetbar$, $Q^2$ and the three-jet invariant mass, $\mtrijet$, where and
$\etjetbar=(\etjeta+\etjetb+\etjetc)/3$. Values of $\mtrijet$ from 20 to $120$~{\GeV} are accessible with the present statistics.  The predictions of LO QCD are also compared to the data in the figures, and are scaled as described for the $\rapjetbar$ cross sections.  The scaled LO calculations give a good description of the shape of the data within statistical uncertainties. 

The ratio of $e^-p$ to $e^+p$ differential cross sections versus $\etjetbar$, presented in the lower part Figure~\ref{figure_upolxs_et_3jet}b rises from 1 to 4 over the measured range of $8~\GeV < \etjetbar < 29~\GeV$.  The ratios of inclusive, dijet and trijet differential cross sections versus $Q^2$, presented in Figure~\ref{figure_upolxs_q2_1jet}b, Figure~\ref{figure_upolxs_q2_2jet}b and Figure~\ref{figure_upolxs_q2_3jet}b, each show similar ranges of change over the range in $400 < Q^2< 7000$, but higher values of $Q^2$ were not statistically accessible in the trijet sample to make a comparison at higher values of $Q^2$.

\begin{figure}[htbp]
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	\end{picture}
		\caption[Unpolarized $e^-p$ and $e^+p$ $\etjetbar$ Trijet Cross sections]{Unpolarized $e^-p$ and $e^+p$ $\etjetbar$ Trijet Cross sections}
	\label{figure_upolxs_et_3jet}
	\label{figure_ratioxs_upol_et_3jet}
\vfill
\end{figure}

\begin{figure}[htbp]
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	\end{picture}
		\caption[Unpolarized $e^-p$ and $e^+p$ $Q^2$ Trijet Cross sections]{Unpolarized $e^-p$ and $e^+p$ $Q^2$ Trijet Cross sections}
	\label{figure_upolxs_q2_3jet}
	\label{figure_ratioxs_upol_q2_3jet}
\vfill
\end{figure}

\begin{figure}[htbp]
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	\end{picture}
		\caption[Unpolarized $e^-p$ and $e^+p$ $\mtrijet$ Trijet Cross sections]{Unpolarized $e^-p$ and $e^+p$ $\mtrijet$ Trijet Cross sections}
	\label{figure_upolxs_m_3jet}
	\label{figure_ratioxs_upol_m_3jet}
\vfill
\end{figure}



%One of these candidates is
%displayed in Fig.~\ref{fig13}: the fourth jet is clearly observed in
%the ZEUS detector.




The three-jet selected sample also contains 9 $e^-p$ and 2 $e^+p$ candidates with a fourth jet of transverse energy above 5~{\GeV} in the $\rapjet$ range $-1<\rapjet<2.5$. There are no calculations available at present that would predict the number of 4-jet events in the sample.  Assuming that the leading contribution differentiating dijet and trijet systems is single gluon emission, and that the same mechanism is the principle differentiating 3 and 4-jet samples, then the respective ratios should be similar.  The ratio of dijets to trijets is approximately 0.098 (0.065), and the ratio of 4-jets to trijets is 0.082 (0.067) for $e^-p$ ($e^+p$) data, which supports this reasoning.

%\section{Comparison With Previous Results}
