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\subsection{On building better cone jet algorithms%
        \protect\footnote{Contributing authors: S.D. Ellis, J. Huston,
                                                 M. T\"onnesmann}}

% S.D. Ellis
% Department of Physics
% University of Washington
% Seattle, WA 98195
% USA
%
% J. Huston
% Department of Physics & Astronomy
% Michigan State University
% East Lansing, MI 48824
% USA
%
% M. T\"onnesmann
% Max-Planck-Institut f\"ur Physik
% (Werner-Heisenberg-Institut)
% F\"ohringer Ring 6
% 80805 M\"unchen
% Germany

% Abstract:
%     We discuss recent progress in understanding the issues essential to the
%     development of better cone jet algorithms.

An important facet of preparations \cite{Blazey:2000qt,Catani:2000zg} for Run II
at the Tevatron, and for future data taking at the LHC, has been the study of
ways in which to improve jet algorithms. These algorithms are employed to map
final states, both in QCD perturbation theory and in the data, onto jets. The
motivating idea is that these jets are the surrogates for the underlying
energetic partons. In principle, we can connect the observed final states, in
all of their complexity, with the perturbative final states, which are easier to
interpret and to analyze theoretically. Of necessity these jet algorithms should
be robust under the impact of both higher order perturbative and
non-perturbative physics and the effects introduced by the detectors themselves.
The quantitative goal is a precision of order 1\,\% in the mapping between
theory and experiment. In this note we will provide a brief summary of recent
progress towards this goal. A more complete discussion of our results will be
provided elsewhere \cite{Ellis:2002inprep}. Here we will focus on cone jet
algorithms, which have formed the basis of jet studies at hadron colliders.

As a starting point we take the Snowmass Algorithm \cite{Huth:1990mi}, which was
defined by a collaboration of theorists and experimentalists and formed the
basis of the jet algorithms used by the CDF and D\O\ collaborations during Run I
at the Tevatron. Clearly jets are to be composed of either hadrons or partons
that are, in some sense, nearby each other. The cone jet defines nearness in an
intuitive geometric fashion: jets are composed of hadrons or partons whose
3-momenta lie within a cone defined by a circle in $(\eta, \phi)$. These are
essentially the usual angular variables, where
$\eta = \ln\,(\cot\,\theta/2)$ is the pseudorapidity and $\phi$ is the azimuthal
angle. This idea of being nearby in angle can be contrasted with an algorithm
based on being nearby in transverse momentum as illustrated by the so-called
$k_T$ Algorithm \cite{Ellis:1993tq,Catani:1992zp,Catani:1993hr} that has been
widely used at $e^+e^-$ and $ep$ colliders. We also expect the jets to be
aligned with the most energetic particles in the final state. This expectation
is realized in the Snowmass Algorithm by defining an acceptable jet in terms of
a ``stable'' cone such that the geometric center of the cone is identical to the
$E_T$ weighted centroid. Thus, if we think of a sum over final state partons or
hadrons defined by an index $k$ and in the direction $(\eta_k, \phi_k)$, a jet
($J$) of cone radius $R$ is defined by the following set of equations:

\begin{displaymath}
k\in J: (\phi_k - \phi_J)^2 + (\eta_k - \eta_J)^2\leq R^2,
\end{displaymath}
\begin{eqnarray}
E_{T,J} & = & \sum_{k\in J}E_{T,k},\nonumber\\
\phi_J  & = & \sum_{k\in J}\frac{E_{T,k}\,\phi_k}{E_{T,J}},\label{cone}\\
\eta_J  & = & \sum_{k\in J}\frac{E_{T,k}\,\eta_k}{E_{T,J}}.\nonumber
\end{eqnarray}

In these expressions $E_T$ is the transverse energy ($|\overrightarrow{p_T}|$
for a massless 4-vector). It is important to recognize that jet algorithms
involve two distinct steps. The first step is to identify the ``members'' of the
jet, i.e., the calorimeter towers or the partons that make up the stable cone
that becomes the jet. The second step involves constructing the kinematic
properties that will characterize the jet, i.e., determine into which bin the
jet will be placed. In the original Snowmass Algorithm the $E_T$ weighted
variables defined in Eq.~(\ref{cone}) are used both to identify and bin the jet.

In a theoretical calculation one integrates over the phase space corresponding
to parton configurations that satisfy the stability conditions. In the
experimental case one searches for sets of final state particles (and
calorimeter towers) in each event that satisfy the constraint. In practice
\cite{Blazey:2000qt,Catani:2000zg} the experimental implementation of the cone
algorithm has involved the use of various short cuts to minimize the search
time. In particular, Run I algorithms made use of seeds. Thus one looks for
stable cones only in the neighborhood of calorimeter cells, the seed cells,
where the deposited energy exceeds a predefined limit. Starting with such a seed
cell, one makes a list of the particles (towers) within a distance $R$ of the
seed and calculates the centroid for the particles in the list (calculated as in
Eq.~(\ref{cone})). If the calculated centroid is consistent with the initial
cone center, a stable cone has been identified. If not, the calculated centroid
is used as the center of a new cone with a new list of particles inside and the
calculation of the centroid is repeated. This process is iterated, with the cone
center migrating with each repetition, until a stable cone is identified or
until the cone centroid has migrated out of the fiducial volume of the detector.
When all of the stable cones in an event have been identified, there will
typically be some overlap between cones. This situation must be addressed by a
splitting/merging routine in the jet algorithm. This feature was not foreseen in
the original Snowmass Algorithm. Normally this involves the definition of a
parameter $f_\mathrm{merge}$, typically with values in the range
$0.5\leq f_\mathrm{merge}\leq 0.75$, such that, if the overlap transverse energy
fraction (the transverse energy in the overlap region divided by the smaller of
the total energies in the two overlapping cones) is greater than
$f_\mathrm{merge}$, the two cones are merged to make a single jet. If this
constraint is not met, the calorimeter towers/hadrons in the overlap region are
individually assigned to the cone whose center is closer. This situation yields
2 final jets.

The essential challenge in the use of jet algorithms is to understand the
differences between the experimentally applied algorithms and the theoretically
applied ones and hence understand the uncertainties. This is the primary concern
of this paper. It has been known for some time that the use of seeds in the
experimental algorithms means that certain configurations kept by the
theoretical algorithm are likely to be missed by the experimental one
\cite{Ellis:1992qq,Ellis:1993ik,Abbott:1997fc}. At higher orders in perturbation
theory the seed definition also introduces an undesirable (logarithmic)
dependence on the seed $E_T$ cut (the minimum $E_T$ required to be treated as a
seed cell) \cite{Seymour:1998kj}. Various alternative algorithms are described
in Ref.~\cite{Blazey:2000qt} for addressing this issue, including the Midpoint
Algorithm and the Seedless Algorithm. In the last year it has also been
recognized that other final state configurations are likely to be missed in the
data, compared to the theoretical result. In this paper we will explain these
new developments and present possible solutions. To see that there is a problem,
we apply representative jet algorithms to data sets that were generated with the
HERWIG Monte Carlo \cite{Marchesini:1992ch,Corcella:2000bw,Corcella:1999qn} and
then run through a CDF detector simulation. As a reference we include in our
analysis the JetClu Algorithm \cite{Abe:1992ui}, which is the algorithm used by
CDF in Run I. It employs both seeds and a property called ``ratcheting''. This
latter term labels the fact that the Run I CDF algorithm (unlike the
corresponding D\O\ algorithm) was defined so that calorimeter towers initially
found in a cone around a seed continue to be associated with that cone, even as
the center of the cone migrates due to the iteration of the cone algorithm. Thus
the final ``footprint'' of the cone is not necessarily a circle in
$(\eta, \phi)$ (even before the effects of splitting/merging). Since the cone is
``tied'' to the initial seed towers, this feature makes it unlikely that cones
will migrate very far before becoming stable. We describe results from JetClu
both with and without this ratcheting feature. The second cone algorithm studied
is the Midpoint Algorithm that, like the JetClu Algorithm, starts with seeds to
find stable cones (but without ratcheting). The Midpoint Algorithm then adds a
cone at the midpoint in $(\eta, \phi)$ between all identified pairs of stable
cones separated by less than $2R$ and iterates this cone to test for stability.
This step is meant to ensure that no stable ``mid-cones'' are missed, compared
to the theoretical result, due to the use of seeds. Following the recommendation
of the Run II Workshop, we actually use 4-vector kinematics for the Midpoint
Algorithm and place the cone at the midpoint in $(y, \phi)$, where $y$ is the
true rapidity. The third cone algorithm is the Seedless Algorithm that places an
initial trial cone at every point on a regular lattice in $(y, \phi)$, which is
approximately as fine-grained as the detector. It is not so much that this
algorithm lacks seeds, but rather that the algorithm puts seed cones
``everywhere''. The Seedless Algorithm can be streamlined by imposing the
constraint that a given trial cone is removed from the analysis if the center of
the cone migrates outside of its original lattice cell during the iteration
process. The streamlined version still samples every lattice cell for stable
cone locations, but is less computationally intensive. Our experience with the
streamlined version of this algorithm suggests that there can be problems
finding stable cones with centers located very close to cell boundaries. This
technical difficulty is easily addressed by enlarging the distance that a trial
cone must migrate before being discarded. For example, if this distance is
60\,\% of the lattice cell width instead of the default value of 50\,\%, the
problem essentially disappears with only a tiny impact on the required time for
analysis. In the JetClu Algorithm the value $f_\mathrm{merge} = 0.75$ was used
(as in the Run I analyses), while for the other two cone algorithms the value
$f_\mathrm{merge} = 0.5$ was used as suggested in Ref.~\cite{Blazey:2000qt}.
Finally, for completeness, we include in our analysis a sample $k_T$ Algorithm.

Starting with a sample of 250,000 events, which were generated with HERWIG 6.1
and run through a CDF detector simulation and which were required to have at
least 1 initial parton with $E_T > 200\,$GeV, we applied the various algorithms
to find jets with $R = 0.7$ in the central region ($|\eta| < 1$). We then
identified the corresponding jets from each algorithm by finding jet centers
differing by $\Delta R < 0.1$. The plots in Fig.~\ref{deltaET} indicate the
average difference in $E_T$ for these jets as a function of the jet $E_T$. (We
believe that some features of the indicated structure, in particular the
``knees'' near $E_T = 150\,$GeV, are artifacts of the event selection process.)

\begin{figure}[ptb]
\begin{center}
\includegraphics[width=\textwidth]{conejetalgs1.eps}
\caption{Difference of $E_T$ for matched jets found with various jet algorithms
and compared to the JetClu CDF Run I algorithm. The events studied were
generated with HERWIG 6.1 and run through the CDF detector simulation.}
\label{deltaET}
\end{center}
\end{figure}

From these results we can draw several conclusions. First, the $k_T$ Algorithm
identifies jets with $E_T$ values similar to those found by JetClu, finding
slightly more energetic jets at small $E_T$ and somewhat less energetic jets at
large $E_T$. We will not discuss this algorithm further here except to note that
D\O\ has applied it in a study of Run I data \cite{Abazov:2001ak} and in that
analysis the $k_T$ Algorithm jets seem to exhibit slightly \emph{larger} $E_T$
than expected from NLO perturbation theory. The cone algorithms, including the
JetClu Algorithm without ratcheting, which is labeled JetCluNR, identify jets
with approximately 0.5\,\% to 1\,\% \emph{smaller} $E_T$ values than those
identified by the JetClu Algorithm (with ratcheting), with a corresponding
approximately 5\,\% smaller jet cross section at a given $E_T$ value. We believe
that this systematic shortfall can be understood as resulting from the smearing
effects of perturbative showering and non-perturbative hadronization.

To provide insight into the issues raised by Fig.~\ref{deltaET} we now discuss a
simple, but informative analytic picture. It will serve to illustrate the impact
of showering and hadronization on the operation of jet algorithms. We consider
the scalar function $F(\overrightarrow{r})$ defined as a function of the
2-dimensional variable $\overrightarrow{r} = (\eta, \phi)$ by the integral over
the transverse energy distribution of either the partons or the
hadrons/calorimeter towers in the final state with the indicated weight
function,

\begin{eqnarray}
F(\overrightarrow{r}) & = &
\frac{1}{2}\int d^2\rho\;
E_T(\overrightarrow{\rho})\cdot
\left(R^2 - (\overrightarrow{\rho} - \overrightarrow{r})^2\right)\cdot
\Theta\left(R^2 - (\overrightarrow{\rho} - \overrightarrow{r})^2\right)
\label{scalar}\\
& = &
\frac{1}{2}\sum_i E_{T,i}\cdot
\left(R^2 - (\overrightarrow{\rho_i} - \overrightarrow{r})^2\right)\cdot
\Theta\left(R^2 - (\overrightarrow{\rho_i} - \overrightarrow{r})^2\right).
\nonumber
\end{eqnarray}

The second expression arises from replacing the continuous energy distribution
with a discrete set, $i = 1\,...\,N$, of delta functions, representing the
contributions of either a configuration of partons or a set of calorimeter
towers (and hadrons). Each parton direction or the location of the center of
each calorimeter tower is defined in $\eta$, $\phi$ by
$\rho_i = (\eta_i, \phi_i)$, while the parton/calorimeter cell has a transverse
energy (or $E_T$) content given by $E_{T,i}$. This function is clearly related
to the energy in a cone of size $R$ containing the towers whose centers lie
within a circle of radius $R$ around the point $\overrightarrow{r}$. More
importantly it carries information about the locations of ``stable'' cones. The
points of equality between the $E_T$ weighted centroid and the geometric center
of the cone correspond precisely to the maxima of $F$. The gradient of this
function has the form (note that the delta function arising from the derivative
of the theta function cannot contribute as it is multiplied by a factor equal to
its argument)

\begin{equation}
\overrightarrow{\nabla}F(\overrightarrow{r}) = \sum_i E_{T,i}\cdot
(\overrightarrow{\rho_i} - \overrightarrow{r})\cdot
\Theta\left(R^2 - (\overrightarrow{\rho_i} - \overrightarrow{r})^2\right).
\label{gradient}
\end{equation}

This expression vanishes at points where the weighted centroid coincides with
the geometric center, i.e., at points of stability (and at minima of $F$, points
of extreme instability). The corresponding expression for the energy in the cone
centered at $\overrightarrow{r}$ is

\begin{equation}
E_C(\overrightarrow{r}) = \sum_i E_{T,i}\cdot
\Theta\left(R^2 - (\overrightarrow{\rho_i}-\overrightarrow{r})^2\right).
\label{energy}
\end{equation}

\begin{figure}[ptb]
\begin{center}
\includegraphics[width=\textwidth,bb=20 60 576 435,clip]{conejetalgs2.eps}
\caption{2-Parton distribution: a) transverse energy distribution; b)
distributions $F(r)$ and $E_C(r)$ in the perturbative limit of no smearing.}
\label{edist2}
\end{center}
\end{figure}

To more easily develop our understanding of these equations consider a
simplified scenario (containing all of the interesting effects) involving 2
partons separated in just one angular dimension
$\overrightarrow{\rho}\rightarrow\rho$ ($\overrightarrow{r}\rightarrow r$) with
$\rho_2 - \rho_1 = d$. It is sufficient to specify the energies of the 2 partons
simply by their ratio, $z = E_2/E_1\leq 1$. Now we can study what sorts of 2
parton configurations yield stable cones in this 2-D phase space specified by
$0\leq z\leq 1$, $0\leq d\leq 2R$ (beyond $2R$ the 2 partons are surely in
different cones). As a specific example consider the case $\rho_1 = 0$,
$\rho_2 = d = 1.0$ and $z = 0.7$ with $R = 0.7$ (the typical experimental
value). The underlying energy distribution is illustrated in Fig.~\ref{edist2}a,
representing a delta function at $\rho = 0$ (with scaled weight 1) and another
at $\rho = 1.0$ (with scaled weight 0.7). This simple distribution leads to the
functions $F(r)$ and $E_C(r)$ indicated in Fig.~\ref{edist2}b. In going from the
true energy distribution to the distribution $E_C(r)$ the energy is effectively
smeared over a range given by $R$. In $F(r)$ the distribution is further shaped
by the quadratic factor $R^2 - (\rho_i - r)^2$. We see that $F(r)$ exhibits 3
local maxima corresponding to the expected stable cones around the two original
delta functions ($r_1 = 0$, $r_2 = 1$), plus a third stable cone in the middle
($r_3 = zd/(1 + z) = 0.41$ in the current case). This middle cone includes the
contributions from both partons as indicated by the magnitude of the middle peak
in the function $E_C(r)$. Note further that the middle cone is found at a
location where there is initially no energy in Fig.~\ref{edist2}a, and thus no
seeds. One naively expects that such a configuration is not identified as a
stable cone by the experimental implementations of the cone algorithm that use
seeds simply because they do not look for it. Note also that, since both partons
are entirely within the center cone, the overlap fractions are unity and the
usual merging/splitting routine will lead to a single jet containing all of the
initial energy ($1 + z$). This is precisely how this configuration was treated
in the NLO perturbative analysis of the Snowmass Algorithm
\cite{Ellis:1992en,Ellis:1990ek,Ellis:1989vm,Ellis:1989hv}, i.e., only the
leading jet, the middle cone, was kept).

Similar reasoning leads to Fig.~\ref{perthy4}a, which indicates the various 2
parton configurations found by the perturbative cone algorithm. For $d < R$ one
finds a single stable cone and a single jet containing both partons. For
$R < d < (1+z)R$ one finds 3 stable cones that merge to 1 jet, again with all of
the energy. For $d > (1+z)R$ we find 2 stable cones and 2 jets, each containing
one parton, of scaled energies $1$ and $z$. Thus, except in the far right region
of the graph, the 2 partons are always merged to form a single jet.

\begin{figure}[ptb]
\begin{center}
\includegraphics[width=\textwidth,bb=20 55 576 430,clip]{conejetalgs3.eps}
\caption{Perturbation Theory Structure: a) $R_{sep} = 2$; b) $R_{sep} = 1.3$.}
\label{perthy4}
\end{center}
\end{figure}

We expect that the impact of seeds in experimental algorithms can be (crudely)
simulated in the NLO calculations
\cite{Ellis:1992qq,Ellis:1993ik,Abbott:1997fc} by including a parameter
$R_{sep}$ such that stable cones containing 2 partons are not allowed for
partons separated by $d > R_{sep}\cdot R$. As a result cones are no longer
merged in this kinematic region. In the present language this situation is
illustrated in Fig.~\ref{perthy4}b corresponding to $R_{sep} = 1.3$,
$R\cdot R_{sep} = 0.91$. This specific value for $R_{sep}$ was chosen
\cite{Ellis:1992qq,Ellis:1993ik,Abbott:1997fc} to yield reasonable agreement
with the Run I data. The conversion of much the ``3 cones $\rightarrow$ 1 jet''
region to ``2 cones $\rightarrow$ 2 jets'' has the impact of lowering the
average $E_T$ of the leading jet and hence the jet cross section at a fixed
$E_{T,J}$. Parton configurations that naively produced jets with energy
characterized by $1 + z$ now correspond to jets of maximum energy 1. This is
just the expected impact of a jet algorithm with seeds. Note that with this
value of $R_{sep}$ the specific parton configuration in Fig.~\ref{edist2}a will
yield 2 jets (and not 1 merged jet) in the theoretical calculation. As mentioned
earlier this issue is to be addressed by the Midpoint and Seedless Algorithms in
Run II. However, as indicated in Fig.~\ref{deltaET}, neither of these two
algorithms reproduces the results of JetClu. Further, they both identify jets
that are similar to JetClu \emph{without} ratcheting. Thus we expect that there
is more to this story.

As suggested earlier, a major difference between the perturbative level, with a
small number of partons, and the experimental level of multiple hadrons is the
smearing that results from perturbative showering and nonperturbative
hadronization. For the present discussion the primary impact is that the
starting energy distribution will be smeared out in the variable $r$. We can
simulate this effect in our simple model using gaussian smearing, i.e., we
replace the delta functions in Eq.~(\ref{scalar}) with gaussians of width
$\sigma$. (Since this corresponds to smearing in an angular variable, we would
expect $\sigma$ to be a decreasing function of $E_T$, i.e., more energetic jets
are narrower. We also note that this naive picture does not include the expected
color coherence in the products of the showering/hadronization process.) The
first impact of this smearing is that some of the energy initially associated
with the partons now lies outside of the cones centered on the partons. This
effect, typically referred to as ``splashout'' in the literature, is
(exponentially) small in this model for $\sigma < R$. Here we will focus on less
well known but phenomenologically more relevant impacts of splashout. The
distributions corresponding to Fig.~\ref{edist2}b, but now with $\sigma = 0.10$
(instead of $\sigma = 0$), are exhibited in Fig.~\ref{smearpt5}a.

\begin{figure}[ptb]
\begin{center}
\includegraphics[width=\textwidth,bb=20 60 576 440,clip]{conejetalgs4.eps}
\caption{The distributions $F(r)$ and $E_C(r)$ for smearing width a) 0.1; b)
0.25.}
\label{smearpt5}
\end{center}
\end{figure}

With the initial energy distribution smeared by $\sigma$, the distribution
$F(r)$ is now even more smeared and, in fact, we see that the middle stable cone
(the maximum in the middle of Fig.~\ref{edist2}b) has been washed out by the
increased smearing. Thus the cone algorithm applied to data (where such smearing
is present) may not find the middle cone that is present in perturbation theory,
not only due to the use of seeds but also due to this new variety of splashout
correction, which renders this cone unstable. Since, as a result of this
splashout correction, the middle cone is not stable, this problem is \emph{not}
addressed by either the Midpoint Algorithm or the Seedless Algorithm. Both
algorithms may look in the correct place, but they look for stable cones. This
point is presumably part of the explanation for why both of these algorithms
disagree with the JetClu results in Fig.~\ref{deltaET}.

Our studies also suggest a further impact of the smearing of
showering/hadronization that was previously unappreciated. This new effect is
illustrated in Fig.~\ref{smearpt5}b, which shows $F(r)$, still for $z = 0.7$ and
$d = 1.0$, but now for $\sigma = 0.25$. With the increased smearing the second
stable cone, corresponding to the second parton, has now also been washed out,
i.e., the right hand local maximum has also disappeared. This situation is
exhibited in the case of ``data'' by the lego plot in Fig.~\ref{lego} indicating
the jets found by the Midpoint Algorithm in a specific Monte Carlo event. The
Midpoint Algorithm does not identify the energetic towers (shaded in black) to
the right of the energetic central jet as either part of that jet or as a
separate jet, i.e., these obviously relevant towers are not found to be in a
stable cone. The iteration of any cone containing these towers invariably
migrates to the nearby higher $E_T$ towers.

\begin{figure}[ptb]
\begin{center}
\includegraphics[width=0.9\textwidth,bb=0 35 567 340,clip]{conejetalgs5.eps}
\caption{Result of applying the Midpoint Algorithm to a specific Monte Carlo
event in the CDF detector.}
\label{lego}
\end{center}
\end{figure}

In summary, we have found that the impact of smearing and splashout is expected
to be much more important than simply the leaking of energy out of the cone.
Certain stable cone configurations, present at the perturbative level, can
disappear from the analysis of real data due to the effects of showering and
hadronization. This situation leads to corrections to the final jet yields that
are relevant to our goal of 1\,\% precision in the mapping between perturbation
theory and experiment. Compared to the perturbative analysis of the 2-parton
configuration, both the middle stable cone and the stable cone centered on the
lower energy parton can be washed out by smearing. Further, this situation is
not addressed by either the Midpoint Algorithm or the Seedless Algorithm. One
possibility for addressing the missing middle cone would be to eliminate the
stability requirement for the added midpoint cone in the Midpoint Algorithm.
However, if there is enough smearing to eliminate also the second (lower energy)
cone, even this scenario will not help, as we do not find two cones to put a
third cone between. There is, in fact, a rather simple ``fix'' that can be
applied to the Midpoint Algorithm to address this latter form of the splashout
correction. We can simply use 2 values for the cone radius $R$, one during the
search for the stable cones and the second during the calculation of the jet
properties. As a simple example, the 3rd curve in Fig.~\ref{smearpt5}b
corresponds to using $R/\sqrt{2} = 0.495$ during the stable cone discovery phase
and $R = 0.7$ in the jet construction phase. Thus the $R/\sqrt{2}$ value is used
only during iteration; the cone size is set to $R$ right after the stable cones
have been identified and the larger cone size is employed during the
splitting/merging phase. By comparing Figs.~\ref{smearpt5}b and \ref{edist2}b
we see that the two outer stable cones in the perturbative case are in
essentially the same locations as in the smeared case using the smaller cone
during discovery. The improved agreement between the JetClu results and those of
the Midpoint Algorithm with the last ``fix'' (using the smaller $R$ value during
discovering but still requiring cones to be stable) are indicated in
Fig.~\ref{deltaETfix}.

\begin{figure}[ptb]
\begin{center}
\includegraphics[width=0.6\textwidth]{conejetalgs6.eps}
\caption{The difference in the $E_T$ of identified central jets for the JetClu
and Midpoint Algorithms, both with and without the ``fix'' discussed in the
text. The events studied were generated with HERWIG 6.1 and run through the CDF
detector simulation.}
\label{deltaETfix}
\end{center}
\end{figure}

Clearly most, but not all, of the differences between the jets found by the
JetClu and Midpoint Algorithms are removed in the fixed version of the latter.
The small $R$ ``fix'' suggested for the Midpoint Algorithm can also be employed
for the Seedless Algorithm but, like the Midpoint Algorithm, it will still miss
the middle (now unstable) cone.

Before closing this brief summary of our results, we should say a few more words
about the Run I CDF algorithm that we used as a reference. In particular, while
ratcheting is difficult to simulate in perturbation theory, we can attempt to
clarify how it fits into the current discussion. As noted above, the JetClu
Algorithm is defined so that calorimeter towers initially found around a seed
stay with that cone, even as the center of the cone migrates due to the
iteration of the cone algorithm. For the simple scenario illustrated in
Fig.~\ref{edist2}a we assume that the locations of the partons are identified as
seeds, even when smearing is present. To include both ratcheting and the way it
influences the progress of the stable cone search, we must define 2 scalar
functions of the form of Eq.~(\ref{scalar}), one to simulate the search for a
stable cone starting at $\rho = 0$ and the second for the search starting at
$\rho = 1.0$. The former function is defined to include the energy within the
range $-R\leq\rho\leq +R$ independent of the value of $r$, while the second
function is defined to always include the energy in the range
$1.0 - R\leq\rho\leq 1.0 + R$. Analyzing the two functions defined in this way
suggests, as expected, that the search that begins at the higher energy seed
will always find a stable cone at the location of that seed, independent of the
amount of smearing. (If the smearing is small, there is also a stable cone at
the middle location but the search will terminate after finding the initial,
nearby stable cone.) The more surprising result arises from analyzing the second
function, which characterizes the search for a stable cone seeded by the lower
energy parton. In the presence of a small amount of smearing this function
indicates stable cones at both the location of the lower energy parton and at
the middle location. Thus the corresponding search finds a stable cone at the
position of the seed and again will terminate before finding the second stable
cone. When the smearing is large enough to wash out the stable cone at the
second seed, the effect of ratcheting is to ensure that the search still finds
a stable cone at the middle location suggested by the perturbative result,
$r_{3} = z\rho/(1 + z)$ (with a precision given by
$\sigma\cdot e^{-(R/\sigma)^2}$). This result suggests that the JetClu Algorithm
with ratcheting always identifies either stable cones at the location of the
seeds or finds a stable cone in the middle that can lead to merging (in the case
of large smearing). It is presumably just these last configurations that lead to
the remaining difference between the JetClu Algorithm results and those of the
``fixed'' Midpoint Algorithm illustrated in Fig.~\ref{deltaETfix}. We find that
the jets found by the JetClu Algorithm have the largest $E_T$ values of any of
the cone jet algorithms, although the JetClu Algorithm still does not address
the full range of splashout corrections.

In conclusion, we have found that the corrections due to the splashout effects
of showering and hadronization result in unexpected differences between cone jet
algorithms applied to perturbative final states and applied to (simulated) data.
With a better understanding of these effects, we have defined steps that serve
to improve the experimental cone algorithms and minimize these corrections.
Further studies are required to meet the goal of 1\,\% agreement between
theoretical and experimental applications of cone algorithms.

%\begin{thebibliography}{99}
%\bibitem{workshops}
%G. C. Blazey \textit{et al.},
%\textit{Run II Jet Physics: Proceedings of the Run II QCD and Weak Boson Physics
%Workshop},
%hep-ex/0005012;\\
%P. Aurenche \textit{et al.},
%\textit{The QCD and Standard Model Working Group: Summary Report from Les
%Houches},
%hep-ph/0005114.
%
%\bibitem{longpaper}
%S.D. Ellis, J. Huston and M. T\"onnesmann, \textit{in preparation}.
%
%\bibitem{Snowmass}
%J. E. Huth \textit{et al.},
%in \textit{Proceedings of Research Directions for the Decade: Snowmass 1990},
%July 1990,
%edited by E. L. Berger (World Scientific, Singapore, 1992) p.~134.
%
%\bibitem{ktcluster}
%S.D. Ellis and D. Soper,
%Phys.~Rev.~\textbf{D48}, 3160 (1993);\\
%S. Catani, Yu.L. Dokshitzer and B.R. Webber,
%Phys.~Lett.~\textbf{B285}, 291 (1992);\\
%S. Catani, Yu.L. Dokshitzer, M.H. Seymour and B.R. Webber,
%Nucl.~Phys.~\textbf{B406}, 187 (1993).
%
%\bibitem{SDERsep}
%S.D. Ellis, Z. Kunszt and D. Soper,
%Phys.~Rev.~Lett.~\textbf{69}, 3615 (1992);\\
%S.D. Ellis,
%in Proceedings of the \textit{28th Rencontres de Moriond: QCD\ and High Energy
%Hadronic Interactions}, March 1993, p.~235;\\
%B. Abbott \textit{et al.},
%Fermilab Pub-97-242-E (1997).
%
%\bibitem{Seymour}
%M.H. Seymour, Nucl.~Phys.~\textbf{B513}, 269 (1998).
%
%\bibitem{Herwig}
%G. Marchesini, B.R. Webber, G. Abbiendi, I.G. Knowles, M.H. Seymour, L. Stanco,
%Computer Phys.~Commun.~\textbf{67}, 465 (1992);\\
%G. Corcella, I.G. Knowles, G. Marchesini, S. Moretti, K. Odagiri, P. Richardson,
%M.H. Seymour, B.R. Webber,
%JHEP \textbf{01}, 010 (2001) [hep-ph/9912396].
%
%\bibitem{JetClu}
%F. Abe \textit{et al.} (CDF Collaboration),
%Phys.~Rev.~\textbf{D45}, 1448 (1992).
%
%\bibitem{D0}
%V.M. Abazov \textit{et al.} (D\O\ Collaboration),
%hep-ex/0106032.
%
%\bibitem{EKS}
%S.D. Ellis, Z. Kunszt and D. Soper,\\
%Phys.~Rev.~Lett.~\textbf{69}, 1496 (1992);
%Phys.~Rev.~Lett.~\textbf{64}, 2121 (1990);
%Phys.~Rev.~\textbf{D40}, 2188 (1989);
%Phys.~Rev.~Lett.~\textbf{62}, 726 (1989).
%\end{thebibliography}

