Event Analysis

1. Introduction
2. Sphericity
3. Thrust
4. ClusterJet
5. CellJet
6. SlowJet

Introduction

In addition to the methods presented here, there is also the possibility to make use of external jet finders .

Sphericity

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

To do sphericity analyses you have to set up a Sphericity 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.

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

bool Sphericity::analyze( const Event& event)  
perform a sphericity analysis, where
argument event : is an object of the Event class, most likely the pythia.event one.
If the routine returns false the analysis failed, e.g. if too few particles are present to analyze.

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

double Sphericity::sphericity()  
gives the sphericity (or equivalent if r is not 2),

double Sphericity::aplanarity()  
gives the aplanarity (with the same comment),

double Sphericity::eigenValue(int i)  
gives one of the three eigenvalues for i = 1, 2 or 3, in descending order,

Vec4 Sphericity::eventAxis(int i)  
gives the matching normalized eigenvector, as a Vec4 with vanishing time/energy component.

void Sphericity::list()  
provides a listing of the above information.

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

int Sphericity::nError()  
tells the number of times analyze(...) failed to analyze events, i.e. returned false.

Thrust

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 sped 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 n particles growing like n^3.

To do thrust analyses you have to set up a Thrust 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.

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

bool Thrust::analyze( const Event& event)  
perform a thrust analysis, where
argument event : is an object of the Event class, most likely the pythia.event one.
If the routine returns false the analysis failed, e.g. if too few particles are present to analyze.

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

double Thrust::thrust()  
double Thrust::tMajor()  
double Thrust::tMinor()  
double Thrust::oblateness()  
gives the thrust, major, minor and oblateness values, respectively,

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

void Thrust::list()  
provides a listing of the above information.

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

int Thrust::nError()  
tells the number of times analyze(...) failed to analyze events, i.e. returned false.

ClusterJet

Note that we have deliberately chosen not to include the e^+e^- equivalents of the Cambridge/Aachen and anti-kRT algorithms. These tend to be good at clustering the densely populated (in angle) cores of jets, but less successful for the sparsely populated transverse regions, where many jets may come to consist of a single low-momentum particle. In hadron collisions such jets could easily be disregarded, while in e^+e^- annihilation all particles derive back from the hard process.

To do jet finding analyses you have to set up a ClusterJet 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.

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

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

After the analysis has been performed, a few ClusterJet class methods are available to return the result of the analysis:

int ClusterJet::size()  
gives the number of jets found, with jets numbered 0 through size() - 1.

Vec4 ClusterJet::p(int i)  
gives a Vec4 corresponding to the four-momentum defined by the sum of all the contributing particles to the i'th jet.

int ClusterJet::mult(int i)  
the number of particles that have been clustered into the i'th jet.

int ClusterJet::jetAssignment(int i)  
gives the index of the jet that the particle i of the event record belongs to,

void ClusterJet::list()  
provides a listing of the reconstructed jets.

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 occurred.

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

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

int ClusterJet::nError()  
tells the number of times analyze(...) failed to analyze events, i.e. returned false.

CellJet

To do jet finding analyses you have to set up a CellJet 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 Pythia class. The results for the latest event are available as output from a few methods.

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

bool CellJet::analyze( const Event& event, double eTjetMin = 20., double coneRadius = 0.7, double eTseed = 1.5)  
performs a jet finding analysis, where
argument event : is an object of the Event class, most likely the pythia.event one.
argument eTjetMin (default = 20. GeV) : is the minimum transverse energy inside a cone for this to be accepted as a jet.
argument coneRadius (default = 0.7) : is the size of the cone in (eta, phi) space drawn around the geometric center of the jet.
argument eTseed (default = 1.5 GeV) : the minimum eT in a cell for this to be acceptable as the trial center of a jet.
If the routine returns false the analysis failed, but currently this is not foreseen ever to happen.

After the analysis has been performed, a few CellJet class methods are available to return the result of the analysis:

int CellJet::size()  
gives the number of jets found, with jets numbered 0 through size() - 1,

double CellJet::eT(int i)  
gives the eT of the i'th jet, where jets have been ordered with decreasing eT values,

double CellJet::etaCenter(int i)  
double CellJet::phiCenter(int i)  
gives the eta and phi coordinates of the geometrical center of the i'th jet,

double CellJet::etaWeighted(int i)  
double CellJet::phiWeighted(int i)  
gives the eta and phi coordinates of the eT-weighted center of the i'th jet,

int CellJet::multiplicity(int i)  
gives the number of particles clustered into the i'th jet,

Vec4 CellJet::pMassless(int i)  
gives a Vec4 corresponding to the four-momentum defined by the eT and the weighted center of the i'th jet,

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

double CellJet::m(int i)  
gives the invariant mass of the i'th jet, defined by the pMassive above,

void CellJet::list()  
provides a listing of the above information (except pMassless, for reasons of space).

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

int CellJet::nError()  
tells the number of times analyze(...) failed to analyze events, i.e. returned false.

SlowJet

The recent introduction of fjcore, containing the core functionality of FastJet in a very much smaller package, has changed the conditions. It now is possible (even encouraged by the authors) to distribute the two fjcore files as part of the PYTHIA package. Therefore the SlowJet class doubles as a convenient front end to fjcore, managing the conversion back and forth between PYTHIA and FastJet variables. Some applications may still benefit from using the native codes, but the default now is to let SlowJet call on fjcore for the jet finding.

The first step is to decide which particles should be included in the analysis, and with what four-momenta. The SlowJet constructor allows to pick a maximum pseudorapidity defined by the extent of the assumed detector, to pick among some standard options of which particles to analyze, and to allow for some standard mass assumptions, like that all charged particles have the pion mass. Obviously this is only a restricted set of possibilities.

Full flexibility can be obtained by deriving from the base class SlowJetHook to write your own include method. This will be presented with one final-state particle at a time, and should return true for those particles that should be analyzed. It is also possible to return modified four-momenta and masses, to take into account detector smearing effects or particle identity misassignments, but you must respect E^2 - p^2 = m^2.

Alternatively you can modify the event record itself, or a copy of it (if you want to keep the original intact). For instance, only final particles are considered in the analysis, i.e. particles with positive status code, so negative status code should then be assigned to those particles that you do not want to see analyzed. Again four-momenta and masses can be modified, subject to E^2 - p^2 = m^2.

The jet reconstructions is then based on sequential recombination with progressive removal, using the E recombination scheme. To be more specific, the algorithm works as follows.

1. Each particle to be analyzed defines an original cluster. It has a well-defined four-momentum and mass at input. From this information the triplet (pT, y, phi) is calculated, i.e. the transverse momentum, rapidity and azimuthal angle of the cluster.
2. Define distance measures of all clusters i to the beam
   d_iB = pT_i^2p
   and of all pairs (i,j) relative to each other
   d_ij = min( pT_i^2p, pT_j^2p) DeltaR_ij^2 / R^2
   where
   DeltaR_ij^2 = (y_i - y_j)^2 + (phi_i - phi_j)^2.
   The jet algorithm is determined by the user-selected p value, where p = -1 corresponds to the anti-kT one, p = 0 to the Cambridge/Aachen one and p = 1 to the kT one. Also R is chosen by the user, to give an approximate measure of the size of jets. However, note that jets need not have a circular shape in (y, phi) space, so R can not straight off be interpreted as a jet radius.
3. Find the smallest of all d_iB and d_ij.
4. If this is a d_iB then cluster i is removed from the clusters list and instead added to the jets list. Optionally, a pTjetMin requirement is imposed, where only clusters with pT > pTjetMin are added to the jets list. If so, some of the analyzed particles will not be part of any final jet.
5. If instead the smallest measure is a d_ij then the four-momenta of the i and j clusters are added to define a single new cluster. Convert this four-momentum to a new (pT, y, phi) triplet and update the list of d_iB and d_ij.
6. Return to step 3 until no clusters remain.

To do jet finding analyses you first have to set up a SlowJet instance, where the arguments of the constructor specifies the details of the subsequent analyses. Thereafter you can feed in events to it, one at a time, and have them analyzed by the analyze method. Information on the resulting jets can be extracted by a few different methods. The minimal procedure only requires one call per event to do the analysis. We will begin by presenting it, and only afterwards some extensions.

SlowJet::SlowJet(int power, double R, double pTjetMin = 0., double etaMax = 25., int select = 2, int massSet = 2, SlowJetHook* sjHookPtr = 0, bool useFJcore = true, bool useStandardR = true)  
create a SlowJet instance, where
argument power : tells (half) the power of the transverse-momentum dependence of the distance measure,
argumentoption -1 : the anti-kT algorithm,
argumentoption 0 : the Cambridge/Aachen algorithm, and
argumentoption 1 : the kT algorithm.
argument R : the R size parameter, which is crudely related to the radius of the jet cone in (y, phi) space around the center of the jet.
argument pTjetMin (default = 0.0 GeV) : the minimum transverse momentum required for a cluster to become a jet. By default all clusters become jets, and therefore all analyzed particles are assigned to a jet. For comparisons with perturbative QCD, however, it is only meaningful to consider jets with a significant pT.
argument etaMax (default = 25.) : the maximum +-pseudorapidity that the detector is assumed to cover. If you pick a value above 20 there is assumed to be full coverage (obviously only meaningful for theoretical studies).
argument select (default = 2) : tells which particles are analyzed,
argumentoption 1 : all final-state particles,
argumentoption 2 : all observable final-state particles, i.e. excluding neutrinos and other particles without strong or electromagnetic interactions (the isVisible() particle method), and
argumentoption 3 : only charged final-state particles.
argument massSet (default = 2) : masses assumed for the particles used in the analysis
argumentoption 0 : all massless,
argumentoption 1 : photons are massless while all others are assigned the pi+- mass, and
argumentoption 2 : all given their correct masses.
argument sjHookPtr (default = 0) : gives the possibility to send in your own selection routine for which particles should be part of the analysis; see further below on the SlowJetHook class. If this pointer is sent in nonzero, etaMax and massSet are disregarded, and select only gives the basic selection, to which the user can add further requirements.
argument useFJcore (default = on) : choice of code used for finding the jets. Does not affect the outcome of the analysis, but only the speed, and some more specialized options.
argumentoption on : use the fjcore package of FastJet 3.0.5.
argumentoption off : use the native SlowJet implementation, which gives a slower jet finding, but allows some extra options of step-by-step jet joining.
argument useStandardR (default = on) : definition of R distance between two jets. This switch is only meaningful for useFJcore = false; within the fjcore package the standard option below is always used.
argumentoption on : standard, as described above, DeltaR_ij^2 = (y_i - y_j)^2 + (phi_i - phi_j)^2.
argumentoption off : alternative, DeltaR_ij^2 = 2 (cosh(y_i - y_j) - cos(phi_i - phi_j)), which corresponds to the rim of the "deformed cone" giving a constant invariant mass between the two partons considered (for fixed masses and transverse momenta).

bool SlowJet::analyze( const Event& event)  
performs a jet finding analysis, where
argument event : is an object of the Event class, most likely the pythia.event one.
If the routine returns false the analysis failed, but currently this is not foreseen ever to happen.

After the analysis has been performed, a few SlowJet class methods are available to return the result of the analysis:

int SlowJet::sizeOrig()  
gives the original number of particles (and thus clusters) that the analysis starts with.

int SlowJet::sizeJet()  
gives the number of jets found, with jets numbered 0 through sizeJet() - 1, and ordered in terms of decreasing transverse momentum values w.r.t. the beam axis,

double SlowJet::pT(int i)  
gives the transverse momentum pT of the i'th jet,

double SlowJet::y(int i)  
double SlowJet::phi(int i)  
gives the rapidity y and azimuthal angle phi of the center of the i'th jet (defined by the vector sum of constituent four-momenta),

Vec4 SlowJet::p(int i)  
double SlowJet::m(int i)  
gives a Vec4 corresponding to the four-momentum sum of the particles assigned to the i'th jet, and the invariant mass of this four-vector,

int SlowJet::multiplicity(int i)  
gives the number of particles clustered into the i'th jet,

vector<int> SlowJet::constituents(int i)  
gives a list of the indices of the particles that have been clustered into the i'th jet,

vector<int> SlowJet::clusConstituents(int i)  
gives a list of the indices of the particles that have been clustered into the i'th cluster, at the current stage of the clustering process,

int SlowJet::jetAssignment(int i)  
gives the index of the jet that the particle i of the event record belongs to, or -1 if there is no jet containing particle i,

void SlowJet::removeJet(int i)  
removes the i'th jet,

void SlowJet::list()  
provides a listing of the basic jet information from above.

These are the basic methods. For more sophisticated usage it is possible to trace the clustering, one step at a time. It requires the native jet finding code, useFJcore = false in the constructor. If so, the setup method should be used to read in the event and find the initial smallest distance. Each subsequent doStep will then do one cluster joining and find the new smallest distance. You can therefore interrogate which clusters will be joined next before the joining actually is performed. Alternatively you can take several steps in one go, or take steps down to a predetermined number of jets plus remaining clusters.

bool SlowJet::setup( const Event& event)  
selects the particles to be analyzed, calculates initial distances, and finds the initial smallest distance.
argument event : is an object of the Event class, most likely the pythia.event one.
If the routine returns false the setup failed, but currently this is not foreseen ever to happen.

bool SlowJet::doStep()  
do the next step of the clustering. This can either be that two clusters are joined to one, or that a cluster is promoted to a jet (which is discarded if its pT value is below pTjetMin).
The routine will only return false if there are no clusters left, or if useFJcore = true.

bool SlowJet::doNSteps(int nStep)  
calls the doStep() method nStep times, if possible. Will return false if the list of clusters is emptied before then. The stored jet information is still perfectly fine; it is only the number of steps that is wrong. Will also return false if useFJcore = true.

bool SlowJet::stopAtN(int nStop)  
calls the doStep() method until a total of nStop jet and cluster objects remain. Will return false if this is not possible, specifically if the number of objects already is smaller than nStop when the method is called. The stored jet and cluster information is still perfectly fine; it only does not have the expected multiplicity. Will also return false if useFJcore = true.

int SlowJet::sizeAll()  
gives the total current number of jets and clusters. The jets are numbered 0 through sizeJet() - 1, while the clusters are numbered sizeJet() through sizeAll() - 1. (Internally jets and clusters are represented by two separate arrays, but are here presented in one flat range.) Note that the jets are ordered in decreasing pT, while the clusters are not ordered. When useFJcore = true there are no intermediate steps, and thus the number of clusters is zero (after jet finding).

With this extension, the methods double pT(int i), double y(int i), double phi(int i), Vec4 p(int i), double m(int i) and int multiplicity(int i) can be used as before. Furthermore, list() generalizes

void SlowJet::list(bool listAll = false)  
provides a listing of the above information.
argument listAll : lists both jets and clusters if true, else only jets.

Three further methods can be used to check what will happen next.

int SlowJet::iNext()  
int SlowJet::jNext()  
double SlowJet::dNext()  
if the next step is to join two clusters, then the methods give the (i,j, d_ij) values, if instead to promote a cluster to a jet then (i, -1, d_iB). If no clusters remain then (-1, -1, 0.). Note that the cluster numbers are offset as described above, i.e. they begin at sizeJet(), which of course easily could be subtracted off. Also note that the jet and cluster lists are (moderately) reshuffled in each new step. When useFJcore = true there are no intermediate steps, and thus these methods do not return meaningul information.

Finally, and separately, the SlowJetHook class can be used for a more smart selection of which particles to include in the analysis. For instance, isolated and/or high-pT muons, electrons and photons should presumably be identified separately at an early stage, and then not clustered to jets.

Technically, it works like with User Hooks. That is, PYTHIA contains the base class. You write a derived class. In the main program you create an instance of this class, and hand it in to SlowJet; in this case already as part of the constructor.

The following methods should be defined in your derived class.

SlowJetHook::SlowJetHook()  
virtual SlowJetHook::~SlowJetHook()  
the constructor and destructor need not do anything, and if so you need not write your own destructor.

virtual bool SlowJetHook::include(int iSel, const Event& event, Vec4& pSel, double& mSel)  
is the main method that you will need to write. It will be called once for each final-state particle in an event, subject to the value of the select switch in the SlowJet constructor. The value select = 2 may be convenient since then you do not need to remove e.g. neutrinos yourself, but use select = 1 for full control. The method should then return true if you want to see particle included in the jet clustering, and false if not.
argument iSel : is the index in the event record of the currently studied particle.
argument event : is an object of the Event class, most likely the pythia.event one, where the currently studied particle is found.
argument pSel : is at input the four-momentum of the currently studied particle. You can change the values, e.g. to take into account energy smearing in the detector, to define the initial cluster value, without corrupting the event record itself.
argument mSel : is at input the mass of the currently studied particle. You can change the value, e.g. to take into account particle misidentification, to define the initial cluster value, without corrupting the event record itself. Note that the changes of pSel and mSel must be coordinated such that E^2 - p^2 = m^2 holds.

It is also possible to define further methods of your own. One such could e.g. be called directly in the main program before the analyze method is called, to identify and bookkeep some event properties you may not want to reanalyze for each individual particle.