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%% subsection 2.2 The jet-finding algorithm [slac-pub-7073-0-0-2-2 in slac-pub-7073-0-0-2: slac-pub-7073-0-0-2-3]
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\subsection{\usemenu{slac-pub-7073::context::slac-pub-7073-0-0-2-2}{The jet-finding algorithm}}\label{subsection::slac-pub-7073-0-0-2-2}
We now turn to the problem of the definition of the measurement
functions ${\cal S}_3$ and ${\cal S}_4$. As a preliminary, we need some
prescription defining the way in which unobservable partons
are eventually merged into physical jets. To define a jet in terms
of partons, it is customary to distinguish two separate steps:
the clustering algorithm, which decides whether a given set of partons
is mergeable into a jet, and the merging procedure, which defines
the jet momentum as a function of the parton momenta.
The clustering algorithm we choose
to use was introduced by Ellis and Soper~[\docLink{slac-pub-7073-0-0-8.tcx}[ESalg]{24}]. It is
a $k_{\sss T}$ algorithm specifically designed for hadron-hadron
collisions. It is formulated in terms of the transverse momenta
$k_{i{\sss T}}$ and of the lego plot distances $R_{ij}$ of
the final state partons
\beqn
d_i&=&k_{i{\sss T}}^2\,,
\\
R_{ij}&=&(\eta_i-\eta_j)^2+(\varphi_i-\varphi_j)^2\,,
\\
d_{ij}&=&min(k_{i{\sss T}}^2,k_{j{\sss T}}^2)\frac{R_{ij}}{D^2}\,,
\eeqn
where the constant $D$ is the jet-resolution parameter which value
is set at convenience; in practice,
$0.4 < D < 1.0$. The algorithm is defined by means of an
iterative procedure. One starts with an empty list of jets and a list
of protojets, the latter being in the first step by definition identical
to the partons. Then, the quantities $d_i$ and $ d_{ij}$ are evaluated
for all the protojets and the minimum among them is found. If this
minimum is $d_i$, then protojet $i$ is moved from the list
of protojets to the list of jets. Otherwise, if the minimum is
$d_{ij}$, then protojets $i$ and $j$ are merged into a protojet.
The four-momentum of the protojet is defined by means of the merging
procedure of ref.~[\docLink{slac-pub-7073-0-0-8.tcx}[D0report]{25}]: it is the sum of the four-momenta
of the two constituent protojets. The jet-finding procedure
is repeated as long as there are protojets around. When the list of
jets is completed, that is, the list of protojets is empty, all jets
with $k_{\sss T}$ below a certain threshold are dropped. We choose the
threshold to be a given fraction (which we denote by $f$) of the
greatest squared transverse momentum; this means that the threshold
has to be recomputed for every event.
Other jet-finding algorithms are obviously
possible~[\docLink{slac-pub-7073-0-0-8.tcx}[onejettheory]{2},\docLink{slac-pub-7073-0-0-8.tcx}[snowmass]{26}]. Nevertheless,
preliminary studies~[\docLink{slac-pub-7073-0-0-8.tcx}[D0report]{25}] indicate that, contrary
to $e^+e^-$ annihilation, in hadroproduction the prescription
described here is favoured by the data.