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Jet Physics

To relate the hadronic final state to the underlying hard partonic behaviour it is generally necessary to apply a jet algorithm. The JADE algorithm [11] has been used in the following analyses as it was, at the time, the only algorithm which allowed comparison to the NLO calculations (PROJET [12] and DISJET [13]). The JADE algorithm is a cluster algorithm based on the scaled invariant mass-squared

\begin{displaymath}
y_{ij}^{\rm JADE}=
\frac{2 E_i E_j(1-\cos\theta_{ij})}{W^2}\end{displaymath}

for any two objects i and j assuming that these objects are massless. W2 is the squared invariant mass of the hadronic final state and $\theta_{ij}$ is the angle between the two objects of energies Ei and Ej. The minimum yij of all possible combinations is found. If the value of this minimum yij is less than the variable cut-off parameter $\mbox{$y_{cut}$}$, the two objects i and j are merged into a new object by adding their four-momenta and the process is repeated until all $y_{ij}\gt\mbox{$y_{cut}$}$. The surviving objects are called jets which represent the underlying partonic structure that is dependent on $\alpha_s$.


  
Figure 3: Jet production rates Rj as a function of the jet resolution parameter ycut for Q2 in the range (a) 120<Q2<240 GeV2, (b) 240<Q2<720 GeV2, (c) 720<Q2<3600 GeV2, and (d) 120<Q2<3600 GeV2. Only statistical errors are shown. Two NLO QCD calculations, DISJET and PROJET, each with the value of $\Lambda_{\overline{MS}}$ obtained from the fit at ycut=0.02, are also shown.
\begin{figure}
\centerline{ 

\epsfig {file=DESY-95-182_2.eps,height=10.0cm}
}
\vspace*{-0.7cm}\end{figure}

Figures 3a-d show the ZEUS jet rates using data taken in 1994, R1+1, R2+1 and R3+1 as a function of ycut for data compared with the DISJET and PROJET NLO QCD calculations for three Q2 intervals 120<Q2<240 GeV2, 240<Q2<720 GeV2, 720<Q2<3600 GeV2, and the combined region 120<Q2<3600 GeV2. There is good agreement between the corrected 1+1 and 2+1 jet rates and the NLO QCD calculation over most of the range in ycut shown. Both programs agree well in their prediction of the jet-rate dependence as a function of ycut.


 
Figure 4: Left: Values and total error of $\alpha_s(M_Z)$ from various processes. The solid line indicates the world average and the band its total error. Right: $\alpha_s$(Q) from HERA (open symbols) and other processes with increasing Q (closed circles): $\Gamma_{\! \Upsilon}$ and $\sigma_{\rm \! had}/\sigma_{\rm \! tot}$, event shapes and $\Gamma_{\rm \! hadron}/\Gamma_{\rm \! lepton}$ in e+e-.  
\begin{figure}
\begin{center}
 
\epsfig {file=alpha_s.ps,width=10.0cm,bbllx=5pt,bblly=1pt,bburx=604pt,bbury=290pt}
\end{center}\vspace*{-0.5cm}\end{figure}

The values of $\alpha_s$(Q) extracted by the H1 [14] and ZEUS [15] collaboration as a function of Q are shown in Fig. 4. The value of was determined by varying the $\Lambda$ scale parameter in the QCD calculation until the best fit to the ratio R2+1 was obtained at a particular value of ycut. The measured decreases with increasing Q, consistent with the running of the strong coupling constant, with Q2 taken as the scale. In addition the figure shows the curves for $\Lambda_{\overline{MS}}^{(5)}$ = 100, 200, and 300 MeV. An extrapolation to $\alpha_s$(Mz) yields:

which are consistent with other values obtained from a large variety of different processes as shown Fig. 4 (see [16] for references). Even with the current statistics the HERA measurements are already competitive with those made elsewhere.

Recently two new, more flexible NLO calculations (MEPJET [17] and DISENT [18]) have become available allowing the experiments to analyze the data using any particular jet algorithm. The kT algorithm [19] is particularly suited for DIS as it allows factorization between the beam fragmentation and the hard process [20]. The ZEUS collaboration has reanalyzed [21] their 1994 data using this algorithm. The preliminary values of $\alpha_s$(Q) obtained in the three bins of Q are shown (with statistical errors only) in Fig. 4 and are consistent with the results obtained with the JADE algorithm.


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Next: Event Shapes Up: Hadronic Final States in Previous: DIS Kinematics

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