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| − | The sphericity tensor is the basis for calculation of Event Shape observables, a series of 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 1</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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| − | == 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]]:
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| − | [[File:http://mcplots.cern.ch/cache/plots/zhad-pToutSph-General-Purpose_MCs.Defaults-aleph1-charged-ee-91-.png|thumb|left|alt=The ALEPH measurement of out of plane transverse momentum.| ss]]
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| − | <image></image>
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