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<chapter name="Event Analysis">

<h2>Event Analysis</h2>

<h3>Introduction</h3>

The routines in this section are intended to be used to analyze
event properties. As such they are not part of the main event
generation chain, but can be used in comparisons between Monte
Carlo events and real data. They are rather free-standing, but 
assume that input is provided in the PYTHIA 8 
<code>Event</code> format, and use a few basic facilities such 
as four-vectors.

<p/>
In addition to the methods presented here, there is also the 
possibility to make use of <aloc href="JetFinders">external 
jet finders </aloc>.

<h3>Sphericity</h3>

The standard sphericity tensor is
<eq>
    S^{ab} = (sum_i p_i^a p_i^b) / (sum_i p_i^2)
</eq>
where the <ei>sum i</ei> runs over the particles in the event,
<ei>a, b = x, y, z,</ei> and <ei>p</ei> without such an index is 
the absolute size of the three-momentum . This tensor can be 
diagonalized to find eigenvalues and eigenvectors.

<p/>
The above tensor can be generalized by introducing a power 
<ei>r</ei>, such that
<eq>
    S^{ab} = (sum_i p_i^a p_i^b p_i^{r-2}) / (sum_i p_i^r)
</eq>
In particular, <ei>r = 1</ei> gives a linear dependence on momenta 
and thus a collinear safe definition, unlike sphericity.

<p/>  
To do sphericity analyses you have to set up a <code>Sphericity</code>
instance, and then feed in events to it, one at a time. The results 
for the latest event are available as output from a few methods.

<method name="Sphericity::Sphericity(double power = 2., int select = 2)">
create a sphericity analysis object, where
<argument name="power" default="2."> 
is the power <ei>r</ei> defined above, i.e. 
<argoption value="2.">gives Spericity, and</argoption> 
<argoption value="1.">gives the linear form.</argoption>
</argument>
<argument name="select" default="2"> 
tells which particles are analyzed,
<argoption value="1">all final-state particles,</argoption>
<argoption value="2">all observable final-state particles, 
i.e. excluding neutrinos and other particles without strong or
electromagnetic interactions (the <code>isVisible()</code> 
particle method), and
</argoption>
<argoption value="3">only charged final-state particles.</argoption>
</argument>
</method>

<method name="bool Sphericity::analyze( const Event& event, 
ostream& os = cout)">
perform a sphericity analysis, where 
<argument name="event">is an object of the <code>Event</code> class, 
most likely the <code>pythia.event</code> one.
</argument>
<argument name="os" default="cout"> is the output stream for 
error messages. (The method does not rely on the <code>Info</code>
mchinery for error messages.)
</argument>
<br/>If the routine returns <code>false</code> the 
analysis failed, e.g. if too few particles are present to analyze.
</method>

<p/>
After the analysis has been performed, a few methods are available 
to return the result of the analysis of the latest event:

<method name="double Sphericity::sphericity()">
gives the sphericity (or equivalent if <ei>r</ei> is not 2),
</method>

<method name="double Sphericity::aplanarity()"> 
gives the aplanarity (with the same comment),
</method>

<method name="double Sphericity::eigenValue(int i)"> 
gives one of the three eigenvalues for <ei>i</ei> = 1, 2 or 3, in 
descending order,
</method>

<method name="Vec4 Sphericity::eventAxis(i)"> 
gives the matching normalized eigenvector, as a <code>Vec4</code> 
with vanishing time/energy component.
</method>

<method name="void Sphericity::list(ostream& os = cout)"> 
provides a listing of the above information.
</method>

<p/>
There is also one method that returns information accumulated for all
the events analyzed so far.

<method name="int Sphericity::nError()"> 
tells the number of times <code>analyze(...)</code> failed to analyze 
events, i.e. returned <code>false</code>.
</method>

<h3>Thrust</h3>

Thrust is obtained by varying the thrust axis so that the longitudinal
momentum component projected onto it is maximized, and thrust itself is 
then defined as the sum of absolute longitudinal momenta divided by
the sum of absolute momenta. The major axis is found correspondingly 
in the plane transverse to thrust, and the minor one is then defined 
to be transverse to both. Oblateness is the difference between the major 
and the minor values. 

<p/>
The calculation of thrust is more computer-time-intensive than e.g. 
linear sphericity, introduced above, and has no specific advantages except 
historical precedent. In the PYTHIA 6 implementation the search was 
speeded up at the price of then not being guaranteed to hit the absolute
maximum. The current implementation studies all possibilities, but at
the price of being slower, with time consumption for an event with
<ei>n</ei> particles growing like <ei>n^3</ei>.

<p/>  
To do thrust analyses you have to set up a <code>Thrust</code>
instance, and then feed in events to it, one at a time. The results 
for the latest event are available as output from a few methods.

<method name="Thrust::Thrust(int select = 2)">
create a thrust analysis object, where 
<argument name="select" default="2"> 
tells which particles are analyzed,
<argoption value="1">all final-state particles,</argoption>
<argoption value="2">all observable final-state particles, 
i.e. excluding neutrinos and other particles without strong or
electromagnetic interactions (the <code>isVisible()</code> 
particle method), and
</argoption>
<argoption value="3">only charged final-state particles.</argoption>
</argument>
</method>

<method name="bool Thrust::analyze( const Event& event, 
ostream& os = cout)">
perform a thrust analysis, where 
<argument name="event">is an object of the <code>Event</code> class, 
most likely the <code>pythia.event</code> one.
</argument>
<argument name="os" default="cout"> is the output stream for 
error messages. (The method does not rely on the <code>Info</code>
mchinery for error messages.)
</argument>
<br/>If the routine returns <code>false</code> the 
analysis failed, e.g. if too few particles are present to analyze.
</method>

<p/>
After the analysis has been performed, a few methods are available 
to return the result of the analysis of the latest event:

<method name="double Thrust::thrust()">
</method>
<methodmore name="double Thrust::tMajor()">
</methodmore>
<methodmore name="double Thrust::tMinor()">
</methodmore>
<methodmore name="double Thrust::oblateness()">
gives the thrust, major, minor and oblateness values, respectively, 
</methodmore>

<method name="Vec4 Thrust::eventAxis(int i)"> 
gives the matching normalized event-axis vectors, for <ei>i</ei> = 1, 2 or 3
corresponding to thrust, major or minor, as a <code>Vec4</code> with 
vanishing time/energy component.
</method>

<method name="void Thrust::list(ostream& os = cout)"> 
provides a listing of the above information.
</method>

<p/>
There is also one method that returns information accumulated for all
the events analyzed so far.

<method name="int Thrust::nError()"> 
tells the number of times <code>analyze(...)</code> failed to analyze 
events, i.e. returned <code>false</code>.
</method>

<h3>ClusterJet</h3>

<code>ClusterJet</code> (a.k.a. <code>LUCLUS</code> and 
<code>PYCLUS</code>) is a clustering algorithm of the type used for 
analyses of <ei>e^+e^-</ei> events, see the PYTHIA 6 manual. All 
visible particles in the events are clustered into jets. A few options 
are available for some well-known distance measures. Cutoff 
distances can either be given in terms of a scaled quadratic quantity 
like <ei>y = pT^2/E^2</ei> or an unscaled linear one like <ei>pT</ei>. 

<p/>  
To do jet finding analyses you have to set up a <code>ClusterJet</code>
instance, and then feed in events to it, one at a time. The results 
for the latest event are available as output from a few methods.

<method name="ClusterJet::ClusterJet(string measure = &quot;Lund&quot;, 
int select = 2, int massSet = 2, bool precluster = false, 
bool reassign = false)">
create a <code>ClusterJet</code> instance, where 
<argument name="measure" default="&quot;Lund&quot;">distance measure, 
to be provided as a character string (actually, only the first character 
is necessary)
<argoption value="&quot;Lund&quot;">the Lund <ei>pT</ei> distance,
</argoption>
<argoption value="&quot;JADE&quot;">the JADE mass distance, and
</argoption>
<argoption value="&quot;Durham&quot;">the Durham <ei>kT</ei> measure.
</argoption>
</argument>
<argument name="select" default="2"> 
tells which particles are analyzed,
<argoption value="1">all final-state particles,</argoption>
<argoption value="2">all observable final-state particles, 
i.e. excluding neutrinos and other particles without strong or
electromagnetic interactions (the <code>isVisible()</code> particle 
method), and
</argoption>
<argoption value="3">only charged final-state particles.</argoption>
</argument>
<argument name="massSet" default="2">masses assumed for the particles 
used in the analysis
<argoption value="0">all massless,</argoption>
<argoption value="1">photons are massless while all others are
assigned the <ei>pi+-</ei> mass, and
</argoption>
<argoption value="2">all given their correct masses.</argoption>
</argument>
<argument name="precluster" default="false">
perform or not a early preclustering step, where nearby particles
are lumped together so as to speed up the subsequent normal clustering.
</argument>
<argument name="reassign" default="false"> 
reassign all particles to the nearest jet each time after two jets
have been joined. 
</argument>
</method>

<method name="ClusterJet::analyze( const Event& event, double yScale, 
double pTscale, int nJetMin = 1, int nJetMax = 0, ostream& os = cout)"> 
performs a jet finding analysis, where 
<argument name="event">is an object of the <code>Event</code> class, 
most likely the <code>pythia.event</code> one.
</argument>
<argument name="yScale"> 
is the cutoff joining scale, below which jets are joined. Is given
in quadratic dimensionless quantities. Either <code>yScale</code>
or <code>pTscale</code> must be set nonvanishing, and the larger of 
the two dictates the actual value.
</argument>
<argument name="pTscale"> 
is the cutoff joining scale, below which jets are joined. Is given
in linear quantities, such as <ei>pT</ei> or <ei>m</ei> depending on
the measure used, but always in units of GeV. Either <code>yScale</code>
or <code>pTscale</code> must be set nonvanishing, and the larger of 
the two dictates the actual value.
</argument>
<argument name="nJetMin" default="1"> 
the minimum number of jets to be reconstructed. If used, it can override 
the <code>yScale</code> and <code>pTscale</code> values. 
</argument>
<argument name="nJetMax" default="0"> 
the maximum number of jets to be reconstructed. Is not used if below
<code>nJetMin</code>. If used, it can override the <code>yScale</code>
and <code>pTscale</code> values. Thus e.g. 
<code>nJetMin = nJetMax = 3</code> can be used to reconstruct exactly
3 jets.
</argument>
<argument name="os" default="cout"> is the output stream for 
error messages. (The method does not rely on the <code>Info</code>
mchinery for error messages.)
</argument>
<br/>If the routine returns <code>false</code> the analysis failed, 
e.g. because the number of particles was smaller than the minimum number
of jets requested.
</method>

<p/>
After the analysis has been performed, a few <code>ClusterJet</code> 
class methods are available to return the result of the analysis:

<method name="int ClusterJet::size()">
gives the number of jets found, with jets numbered 0 through 
<code>size() - 1</code>,
</method>

<method name="Vec4 ClusterJet::p(int i)">
gives a <code>Vec4</code> corresponding to the four-momentum defined by 
the sum of all the contributing particles to the <ei>i</ei>'th jet,
</method>

<method name="int ClusterJet::jetAssignment(int i)">
gives the index of the jet that the particle <ei>i</ei> of the event
record belongs to,
</method>

<method name="void ClusterJet::list(ostream& os = cout)">
provides a listing of the reconstructed jets.
</method>

<method name="int ClusterJet::distanceSize()">
the number of most recent clustering scales that have been stored
for readout with the next method. Normally this would be five, 
but less if fewer clustering steps occured.
</method>

<method name="double ClusterJet::distance(int i)">
clustering scales, with <code>distance(0)</code> being the most 
recent one, i.e. normally the highest, up to <code>distance(4)</code> 
being the fifth most recent. That is, with <ei>n</ei> being the final
number of jets, <code>ClusterJet::size()</code>, the scales at which
<ei>n+1</ei> jets become <ei>n</ei>, <ei>n+2</ei> become <ei>n+1</ei>,
and so on till <ei>n+5</ei> become <ei>n+4</ei>. Nonexisting clustering 
scales are returned as zero. The physical interpretation of a scale is 
as provided by the respective distance measure (Lund, JADE, Durham).
</method>

<p/>
There is also one method that returns information accumulated for all
the events analyzed so far.

<method name="int ClusterJet::nError()"> 
tells the number of times <code>analyze(...)</code> failed to analyze 
events, i.e. returned <code>false</code>.
</method>

<h3>CellJet</h3>

<code>CellJet</code> (a.k.a. <code>PYCELL</code>) is a simple cone jet 
finder in the UA1 spirit, see the PYTHIA 6 manual. It works in an 
<ei>(eta, phi, eT)</ei> space, where <ei>eta</ei> is pseudorapidity, 
<ei>phi</ei> azimuthal angle and <ei>eT</ei> transverse energy.
It will draw cones in <ei>R = sqrt(Delta-eta^2 + Delta-phi^2)</ei> 
around seed cells. If the total <ei>eT</ei> inside the cone exceeds 
the threshold, a jet is formed, and the cells are removed from further 
analysis. There are no split or merge procedures, so later-found jets 
may be missing some of the edge regions already used up by previous 
ones. Not all particles in the event are assigned to jets; leftovers 
may be viewed as belonging to beam remnants or the underlying event. 
It is not used by any experimental collaboration, but is closely 
related to the more recent and better theoretically motivated 
anti-<ei>kT</ei> algorithm <ref>Cac08</ref>.   

<p/>  
To do jet finding analyses you have to set up a <code>CellJet</code>
instance, and then feed in events to it, one at a time. Especially note 
that, if you want to use the options where energies are smeared in
order so emulate detector imperfections, you must hand in an external
random number generator, preferably the one residing in the 
<code>Pythia</code> class. The results for the latest event are 
available as output from a few methods.

<method name="CellJet::CellJet(double etaMax = 5., int nEta = 50, 
int nPhi = 32, int select = 2, int smear = 0, double resolution = 0.5, 
double upperCut = 2., double threshold = 0., Rndm* rndmPtr = 0)">
create a <code>CellJet</code> instance, where 
<argument name="etaMax" default="5."> 
the maximum +-pseudorapidity that the detector is assumed to cover.
</argument>
<argument name="nEta" default="50"> 
the number of equal-sized bins that the <ei>+-etaMax</ei> range 
is assumed to be divided into.
</argument>
<argument name="nPhi" default="32"> 
the number of equal-sized bins that the <ei>phi</ei> range 
<ei>+-pi</ei> is assumed to be divided into. 
</argument>
<argument name="select" default="2"> 
tells which particles are analyzed,
<argoption value="1">all final-state particles,</argoption>
<argoption value="2">all observable final-state particles, 
i.e. excluding neutrinos and other particles without strong or
electromagnetic interactions (the <code>isVisible()</code> particle 
method), 
and</argoption>
<argoption value="3">only charged final-state particles.</argoption>
</argument>
<argument name="smear" default="0">
strategy to smear the actual <ei>eT</ei> bin by bin, 
<argoption value="0">no smearing,</argoption>
<argoption value="1">smear the <ei>eT</ei> according to a Gaussian 
with width <ei>resolution * sqrt(eT)</ei>, with the Gaussian truncated 
at 0 and <ei>upperCut * eT</ei>,</argoption>
<argoption value="2">smear the <ei>e = eT * cosh(eta)</ei> according 
to a Gaussian with width <ei>resolution * sqrt(e)</ei>, with the 
Gaussian truncated at 0 and <ei>upperCut * e</ei>.</argoption>
</argument>
<argument name="resolution" default="0.5">
see above.
</argument>
<argument name="upperCut" default="2.">
see above.
</argument>
<argument name="threshold" default="0 GeV">
completely neglect all bins with an <ei>eT &lt; threshold</ei>.
</argument>
<argument name="rndmPtr" default="0">
the random-number generator used to select the smearing described
above. Must be handed in for smearing to be possible. If your 
<code>Pythia</code> class instance is named <code>pythia</code>,
then <code>&pythia.rndm</code> would be the logical choice.
</argument>
</class>

<method name="bool CellJet::analyze( const Event& event, 
double eTjetMin = 20., double coneRadius = 0.7, double eTseed = 1.5, 
ostream& os = cout)"> 
performs a jet finding analysis, where 
<argument name="event">is an object of the <code>Event</code> class, 
most likely the <code>pythia.event</code> one.
</argument>
<argument name="eTjetMin" default="20. GeV"> 
is the minimum transverse energy inside a cone for this to be 
accepted as a jet.
</argument>
<argument name="coneRadius" default="0.7"> 
 is the size of the cone in <ei>(eta, phi)</ei> space drawn around 
the geometric center of the jet.
</argument>
<argument name="eTseed" default="1.5 GeV"> 
the mimimum <ei>eT</ei> in a cell for this to be acceptable as 
the trial center of a jet. 
</argument>
<argument name="os" default="cout"> is the output stream for 
error messages. (The method does not rely on the <code>Info</code>
mchinery for error messages.)
</argument>
<br/>If the routine returns <code>false</code> the analysis failed, 
but currently this is not foreseen ever to happen.
</method>

<p/>
After the analysis has been performed, a few <code>CellJet</code> 
class methods are available to return the result of the analysis:

<method name="int CellJet::size()">
gives the number of jets found, with jets numbered 0 through 
<code>size() - 1</code>,
</method>

<method name="double CellJet::eT(i)">
gives the <ei>eT</ei> of the <ei>i</ei>'th jet, where jets have been
ordered with decreasing <ei>eT</ei> values,
</method>

<method name="double CellJet::etaCenter(int i)">
</method>
<methodmore name="double CellJet::phiCenter(int i)">
gives the <ei>eta</ei> and <ei>phi</ei> coordinates of the geometrical 
center of the <ei>i</ei>'th jet,
</methodmore>

<method name="double CellJet::etaWeighted(int i)">
</method>
<methodmore name="double CellJet::phiWeighted(int i)">
gives the <ei>eta</ei> and <ei>phi</ei> coordinates of the 
<ei>eT</ei>-weighted center of the <ei>i</ei>'th jet,
</methodmore>

<method name="int CellJet::multiplicity(int i)">
gives the number of particles clustered into the <ei>i</ei>'th jet,
</method>

<method name="Vec4 CellJet::pMassless(int i)">
gives a <code>Vec4</code> corresponding to the four-momentum defined 
by the <ei>eT</ei> and the weighted center of the <ei>i</ei>'th jet,
</method>

<method name="Vec4 CellJet::pMassive(int i)">
gives a <code>Vec4</code> corresponding to the four-momentum defined by 
the sum of all the contributing cells to the <ei>i</ei>'th jet, where 
each cell contributes a four-momentum as if all the <ei>eT</ei> is 
deposited in the center of the cell,
</method>

<method name="double CellJet::m(int i)">
gives the invariant mass of the <ei>i</ei>'th jet, defined by the 
<code>pMassive</code> above,
</method>

<method name="void CellJet::list()">
provides a listing of the above information (except <code>pMassless</code>, 
for reasons of space).
</method>

<p/>
There is also one method that returns information accumulated for all
the events analyzed so far.
<method name="int CellJet::nError()"> 
tells the number of times <code>analyze(...)</code> failed to analyze 
events, i.e. returned <code>false</code>.
</method>

</chapter>

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