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| − | The sphericity tensor is the basis for calculation of sphericity and aplanarity, two observables particularly suited for <math>e^+ e^- \rightarrow Z^0/\gamma* \rightarrow q\bar{q}</math>, and routinely used in several [[LEP]] analyses.
| + | #REDIRECT [[Sphericity tensor]] |
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| − | == Definition ==
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| − | The sphericity tensor is defined as:
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| − | <math>S^{ab} = \frac{\sum_i p_i^a p_i^b}{\sum_i |p_i|^2}.</math>
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| − | Here <math>p_i</math> are the four-momenta of all particles in an event. Superscript ''a'' and ''b'' indicates spatial components, and the sphericity tensor can thus be represented as a 3-by-3 matrix. As such, three eigenvalues can be found. If they are ordered as <math>\lambda_1 \geq \lambda_2 \geq \lambda_3</math>, the sphericity is defined as:
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| − | <math>S = \frac{3}{2}(\lambda_2 + \lambda_3).</math>
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| − | The similar quantity aplanarity is defined as:
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| − | <math>A = \frac{3}{2}\lambda_1</math>
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| − | == Physical meaning ==
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| − | The eigenvector corresponding to <math>\lambda_1</math> is called the sphericity axis. ''S'' measures the amount of <math>p_\perp^2</math> with respect to that axis, and is constrained to values <math>1 \geq S \geq 0</math>. An event with sphericity 0 is a clean dijet event, and sphericity 1 signifies an isotropic event.
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| − | The eigenvectors corresponding to <math>\lambda_2</math> and <math>\lambda_3</math> spans a plane, the so-called sphericity plane. Aplanarity measures the <math>p_\perp</math> out of that plane, is constrained to <math>0.5 \geq A \geq 0</math>. Similarly to ''S'', ''A'' is used to signify the isotropicity of the event.
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| − | [[File:ptout-aleph-mcplots.png|thumb|left|alt=The ALEPH measurement of out of plane transverse momentum.|The ALEPH measurement of out of plane transverse momentum, compared to standard event generators. From [[MCPLOTS]]].]]
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| − | == Data and description ==
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| − | Sphericity and aplanarity was measured in all the LEP experiments, and the data are particularly important for tuning of [[parton showers]]. Measurements and comparisons to event generators can be found at MCplots for [http://mcplots.cern.ch/?query=plots,ee,zhad,S sphericity] and [http://mcplots.cern.ch/?query=plots,ee,zhad,A aplanarity].
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| − | The <math>p_\perp</math> out of the sphericity plane is a particularly interesting observable, as it has been problematic to describe for all the standard parton shower generators, see ''e.g.'' the measurement by [[ALEPH]] in the figure.
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| − | == In Rivet ==
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| − | In the [[Rivet]] framework, projections are in place to calculate the sphericity tensor, its eigenvalues and corresponding eigenvectors. Below example code is shown for calculating them all. The example is a full Rivet analysis in itself, which can be compiled with <code>rivet-buildplugin RivetSphericityExample.so SphericityExample.cc --std=c++11</code>. The example just calculates the quantities. One should then fill histograms as needed, etc.
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| − | <syntaxhighlight lang="cpp">
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| − | // -*- C++ -*-
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| − | #include "Rivet/Analysis.hh"
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| − | #include "Rivet/Projections/ChargedFinalState.hh"
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| − | #include "Rivet/Projections/Sphericity.hh"
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| − | namespace Rivet {
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| − | /// @brief Sphericity example calculation
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| − | class SphericityExample : public Analysis {
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| − | public:
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| − | /// Constructor
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| − | SphericityExample()
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| − | : Analysis("SphericityExample")
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| − | { }
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| − | /// @name Analysis methods
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| − | //@{
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| − | void init() {
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| − | // Projection containing charged particles of the event
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| − | const ChargedFinalState cfs;
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| − | addProjection(cfs, "CFS");
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| − | // The sphericity projection
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| − | addProjection(Sphericity(cfs), "Sphericity");
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| − | }
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| − | /// Perform the per-event analysis
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| − | void analyze(const Event& event) {
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| − | // The charged final state particles
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| − | const ChargedFinalState& cfs = applyProjection<ChargedFinalState>(event, "CFS");
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| − | // Apply the projection
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| − | const Sphericity& sphericity = applyProjection<Sphericity>(event, "Sphericity");
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| − | // Calculate the sphericity
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| − | const double sph = sphericity.sphericity();
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| − | // Calculate the aplanarity
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| − | const double apl = sphericity.aplanarity();
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| − | // We now loop over the charged particles
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| − | for(const Particle& p : cfs.particles()){
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| − | // We calculated the particle pT wrt. the axes of the sphericity plane
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| − | // First pTout
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| − | const double pTout = dot(p.p3(),sphericity.sphericityMinorAxis());
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| − | // Then pTin
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| − | const double pTin = dot(p.p3(),sphericity.sphericityMajorAxis());
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| − | }
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| − | }
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| − | void finalize() {
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| − | }
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| − | //@}
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| − | };
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| − | // The hook for the plugin system
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| − | DECLARE_RIVET_PLUGIN(SphericityExample);
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| − | }
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| − | </syntaxhighlight>
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