ch3-06-jets.tex

\section{Jet Finding}

\subsection{Clustering Algorithm}

STAR has implemented the midpoint-cone algorithm in accordance with the
recommendations of Reference \cite{Blazey:2000qt}. The algorithm proceeds by
assembling a $p_{T}$ ordered list of four momenta (``protojets") for each
event. Each protojet with $p_T>p_{T}^{seed}$ (``seeds") initiates a clustering
sequence. For each clustering sequence, protojets within an angular distance
$\Delta r = \sqrt{\Delta\phi^{2} + \Delta\eta ^{2}} < r_{cone}$ are selected
and their four momenta are added to define the four momentum of the cluster
via $p_{\mu}^{cluster} = \sum p_{\mu}^{i}$. If $p_{\mu}^{cluster}$ lies within
a distance $\epsilon$ of the initiating protojet, the clustering sequence
terminates. Otherwise the clustering sequence is iterated about the direction
$p_{\mu}^{cluster}$ until convergence is reached. Once stable configurations
are identified, the association is cataloged for later use. However, no
protojets are removed from the sample. Clustering continues until the list of
seeds is exhausted, yielding a list of redundant, overlapping, stable
clusters. Before disentangling the stable clusters, the algorithm first tests
for missed initiating directions by constructing a set of test seed directions
at the ``midpoint" between all possible pairs of neighboring clusters.
Specifically, locations at the midpoint of all cluster pairs separated by
distance $d < 2 \times r_{cone}$ are tested for stable cluster configurations.
Clusters formed around midpoint seeds are only retained if the resulting
cluster lies within $\epsilon$ of the midpoint seed, and no iteration is
performed.

After all midpoint seeds have been tested, the resulting list of stable
clusters is disentangled via a splitting/merging algorithm. The algorithm
proceeds by finding the highest $p_{T}$ cluster, referred to as the ``root"
cluster. Next, all clusters sharing protojets with the root cluster
("neighbors") are identified, and the neighbor with the largest $p_T$ is
selected. The root and neighbor jet are merged if
$\frac{p_{T}^{shared}}{p_{T}^{neighbor}}>f_{split-merge}$ where $0 <
f_{split-merge} < 1$. If this condition is not satisfied the clusters are
split such that each protojet is assigned to the closest of the two
overlapping clusters. After each split/merge, the cluster list is again sorted
by $p_T$, a new root cluster is chosen, and the splitting/merging continues
until no protojets are shared amongst clusters. It is important to note that
the split/merge step takes a list of clusters that are essentially circular,
while the final clusters are no longer necessarily circular. Finally, each of
the unique clusters is tagged as a ``jet", whose four momentum is the vector
sum of the constituent protojets.

\subsection{Application at STAR}

STAR applies the midpoint-cone algorithm to cluster charged particle tracks
and BEMC tower energies as follows. Charged particle protojets are constructed
from all primary TPC tracks satisfying the aforementioned cuts. A charged pion
mass is assumed when relating energy and momentum. Each BEMC tower energy
measurement is corrected for charged particle energy deposition by i)
identifying the number of TPC tracks projecting to the tower and ii)
subtracting the most probable MIP energy deposition for each of the projecting
tracks. After MIP subtraction, each tower energy measurement is converted to a
four momentum using knowledge of the highest-ranking primary vertex location
and assuming a photon mass. Protojets are then constructed for towers with
corrected $E_{T}>0.2 $GeV.

The midpoint-cone algorithm first sorts the protojets onto a grid of
$\Delta\eta=\Delta\phi=0.05$, where the properties of each grid ``cell" are
defined by the vector sum of its constituent protojets. This discretization
improves computational efficiency and minimizes potential biases arising from
the fact that the BEMC towers may measure energy from more than one particle.
Each cell maintains a list of its constituent protojets so that there is no
ultimate loss of information. The midpoint cone algorithm then operates on the
list of cells, and after clustering and splitting/merging conclude, each
cluster is then characterized by the vector sum of its constituent four
momenta. Jets with $p_{T}<5$ GeV/$c$ are discarded. The control parameters of
the clustering algorithm are listed in Table \ref{tbl:jetfinding-parameters}.
The restricted cone radius in the 2005 analysis was motivated by the limited
pseudorapidity acceptance of the partially installed BEMC.

\begin{table}
  \begin{center}
    % \begin{ruledtabular}
      \begin{tabular}{|c|c|c|}
        \hline
        Parameter & Value \\
        \hline
        cone radius  &   0.4 (2005), 0.7 (2006) \\
        seed $p_{T}$ threshold  &   0.5 GeV/$c$ \\
        $f_{split-merge}$  &  0.5  \\
        clustering convergence distance $\epsilon$  &  0.025  \\
        \hline
      \end{tabular}
    % \end{ruledtabular}    
  \end{center}
  \caption{Control parameters used in midpoint-cone clustering and described in detail in the text.}
  \label{tbl:jetfinding-parameters}
\end{table}