source: git/external/fastjet/contribs/Nsubjettiness/README@ 8497ac6

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Nsubjettiness Package
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The Nsubjettiness package is based on the physics described in:

   Identifying Boosted Objects with N-subjettiness.
   Jesse Thaler and Ken Van Tilburg.
   JHEP 1103:015 (2011), arXiv:1011.2268.

   Maximizing Boosted Top Identification by Minimizing N-subjettiness. 
   Jesse Thaler and Ken Van Tilburg. 
   JHEP 1202:093 (2012), arXiv:1108.2701.

New in v2.0 is the winner-take-all axis, which is described in:

   Jet Shapes with the Broadening Axis. 
   Andrew J. Larkoski, Duff Neill, and Jesse Thaler. 
   JHEP 1404:017 (2014), arXiv:1401.2158.

as well as in unpublished work by Gavin Salam.

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Core Classes
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There are various ways to access N-(sub)jettiness variables, described
in more detail below:

Nsubjettiness       [Nsubjettiness.hh]:
   A FunctionOfPseudoJet<double> interface to measure N-subjettiness
   (Recommended for most users)
NsubjettinessRatio  [Nsubjettiness.hh]:
   A FunctionOfPseudoJet<double> interface to measure ratios of
   two different N-subjettiness (i.e. tau3/tau2)
NjettinessPlugin    [NjettinessPlugin.hh]:
   A FastJet plugin for finding jets by minimizing N-jettiness
   (Recommended for advanced users)
Njettiness          [Njettiness.hh]:
   Access to the core Njettiness code.
   (Not recommended for users, since the interface might change)

The code assumes that you have FastJet 3.

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Basic Usage: Nsubjettiness and NsubjettinessRatio  [Nsubjettiness.hh]
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Most users will only need to use the Nsubjettiness class.  The basic
functionality is given by:

   Nsubjettiness nSub(N, AxesDefinition, MeasureDefinition)
   // N specifies the number of (sub) jets to measure
   // AxesDefinition is WTA_KT_Axes, OnePass_KT_Axes, etc.
   // MeasureDefinition is UnnormalizedMeasure(beta),
   //                      NormalizedMeasure(beta,R0), etc.
      
   // get tau value
   double tauN = nSub.result(PseudoJet);

Also available are ratios of N-subjettiness values
    NsubjettinessRatio nSubRatio(N, M, AxesDefinition,
                                 MeasureDefinition)
    // N and M give tau_N / tau_M, all other options the same

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AxesDefinition  [NjettinessDefinition.hh]
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N-(sub)jettiness requires choosing axes as well as a measure (see below). There
are a number of axes choices available to the user, though modes with a (*) are
recommended. Arguments in parentheses are parameters that the user must set.

Axes can be found using standard recursive clustering procedures.  New is the
option to use the "winner-take-all" recombination scheme:
(*) KT_Axes           // exclusive kt axes
    CA_Axes           // exclusive ca axes
    AntiKT_Axes(R0)   // inclusive hardest axes with antikt, R0 = radius
(*) WTA_KT_Axes       // exclusive kt with winner-take-all recombination
    WTA_CA_Axes       // exclusive ca with winner-take-all recombination

One can also run a minimization routine to find a (local) minimum of
N-(sub)jettiness:
(*) OnePass_KT_Axes          // one-pass minimization from kt starting point
    OnePass_CA_Axes          // one-pass min. from ca starting point
    OnePass_AntiKT(R0)       // one-pass min. from antikt starting point,R0=rad
(*) OnePass_WTA_KT_Axes      // one-pass min. from wta_kt starting point
    OnePass_WTA_CA_Axes      // one-pass min. from wta_ca starting point

In general, it is difficult to finding the global minimum, but this mode
attempts to do so
    MultiPass_Axes(Npass)    // axes that (attempt to) minimize N-subjettiness
                             // (NPass = 100 is typical)

Finally, one can set manual axes:
    Manual_Axes          // set your own axes with setAxes()
    OnePass_Manual_Axes  // one-pass minimization from manual starting point

For most cases, running with OnePass_KT_Axes or OnePass_WTA_KT_Axes gives
reasonable results (and the results are IRC safe).  Because it uses random
number seeds, MultiPass_Axes is not IRC safe (and the code is rather slow).  Note
that for the minimization routines, beta = 1.1 is faster than beta = 1, with
comparable performance.

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MeasureDefinition  [NjettinessDefinition.hh]
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At the moment, there are only a few measures.  Note that each one has a
different number of parameters.  The one indicated by (*)
is the one recommended for use by users new to Nsubjettiness.

The original N-subjettiness measures are:
    NormalizedMeasure(beta,R0)  //default normalized measure with
                                //parameters beta and R0 (dimensionless)
(*) UnnormalizedMeasure(beta)   //default unnormalized measure with just
                                //parameter beta (dimensionful)

There are also measures that incorporate a radial cutoff:
    NormalizedCutoffMeasure(beta,R0,Rcutoff) //normalized measure with
                                             //additional Rcutoff
    UnnormalizedCutoffMeasure(beta,Rcutoff)  //unnormalized measure with
                                             //additional Rcutoff

In beta testing are "geometric" measures where distances are measured using the
Lorentz dot product (N.B. the formula for the geometric measure is likely to
change since there should be separate beam and jet beta factors.)
    GeometricMeasure(beta)               //geometric measure with exponent beta
    GeometricCutoffMeasure(beta,Rcutoff) //geometric measure with Rcutoff

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A note on beta dependence
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The angular exponent in N-subjettiness is called beta.  The original
N-subjettiness paper advocated beta = 1, but it is now understood that different
beta values can be useful in different contexts.  The two main choices are:

beta = 1:  aka broadening/girth/width measure
   wta_kt_axes are approximately the same as minimizing beta = 1 measure

beta = 2:  aka thrust/mass measure
   kt_axes are approximately the same as minimizing beta = 2 measure

N.B. The minimization routines are only valid for 1 < beta < 3.

For quark/gluon discrimination with N = 1, beta~0.2 with wta_kt_axes appears
to be a good choice. 

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TauComponents  [MeasureFunction.hh]
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For most users, they will only need the value of N-subjettiness (i.e. tau)
itself.  For advanced users, they can access individual tau components (i.e. 
the individual numerator pieces, the denominator, etc.)  

   TauComponents tauComp = nSub.component_result(jet);
   vector<double> numer = tauComp.jet_pieces_numerator(); //tau for each subjet
   double denom = tauComp.denominator();  //normalization factor

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WinnerTakeAllRecombiner  [WinnerTakeAllRecombiner.hh]
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New for version 2.0 of Nsubjettiness are winner-take-all axes.  They are found
with the help of the WinnerTakeAllRecombiner. This class defines a new
recombination scheme for clustering particles. This scheme recombines two
PseudoJets into a PseudoJet with pT of the sum of the two input PseudoJet pTs
and direction of the harder PseudoJet.  This is a "recoil-free" recombination
scheme that guarantees that the axes is aligned with one of the input particles.
It is IRC safe.  Axes found with the standard E-scheme recombiner at similar to
the beta = 2 minimization, while winner-take-all is similar to the beta = 1
measure.

Note that the WinnerTakeAllRecombiner can be used outside of Nsubjettiness
itself for jet finding.  For example, the direction of anti-kT jets found
with the WinnerTakeAllRecombiner is particularly robust against soft jet
contamination.

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Advanced Usage:  NjettinessPlugin  [NjettinessPlugin.hh]
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The Njettiness FastJet plugin represents an exclusive jet finder (yielding a
fixed N number of jets). The algorithm finds N axes, and jets are simply the sum
of particles closest to a given axis (or unclustered if they are closest to the
beam).  The axes finding methods and measures are the same as for Nsubjettiness.

   NjettinessPlugin plugin(N, AxesDefinition, MeasureDefinition);
   JetDefinition def(&plugin);
   ClusterSequence cs(vector<PseudoJet>,def);
   vector<PseudoJet> jets = cs.inclusive_jets();

Note that despite being an exclusive jet algorithm, one finds the jets using the
inclusive_jets() call.

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Very Advanced Usage:  Njettiness  [Njettiness.hh]
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Most users will want to use the Nsubjettiness or NjettinessPlugin classes to
access N-(sub)jettiness information.  For direct access to the Njettiness class,
one can use Njettiness.hh directly. This class is still evolving, so users who
wish to extend its functionality should contact the authors first.

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Technical Details
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In general, the user will never need access to these header files.  Here is a
brief description about how they are used to help the calculation of
N-(sub)jettiness:

AxesFinder.hh:

The AxesFinder class (and derived classes) defines the axes used in the
calculation of N-(sub)jettiness.  These axes can be defined from the exclusive
jets from a kT or CA algorithm, the hardest jets from an anti-kT algorithm,
manually, or from minimization of N-jettiness.  In the future, the user will be
able to write their own axes finder, though currently the interface is still
evolving.  At the moment, the user should stick to the options allowed by
AxesDefinition.

MeasureFunction.hh:

The MeasureFunction class (and derived classes) defines the measure by which
N-(sub)jettiness is calculated. This measure is calculated between each
particle and its corresponding axis, and then summed and normalized to
produce N-(sub)jettiness. The default measure for this calculation is
pT*dR^beta, where dR is the rapidity-azimuth distance between the particle
and its axis, and beta is the angular exponent.  Again, in the future the user
will be able to write their own measures, but for the time being, only the
predefined MeasureDefinition values should be used.

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Known Issues
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-- The MultiPass_Axes mode gives different answers on different runs, since
   random numbers are used.
-- In rare cases, one pass minimization can give a larger value of Njettiness
   than without minimization.
-- Nsubjettiness is not thread safe, since there are mutables in Njettiness.