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### Sphericity

The sphericity tensor is defined as [Bjo70]

 (280)

where corresponds to the , and components. By standard diagonalization of one may find three eigenvalues , with . The sphericity of the event is then defined as
 (281)

so that . Sphericity is essentially a measure of the summed with respect to the event axis; a 2-jet event corresponds to and an isotropic event to .

The aplanarity , with definition , is constrained to the range . It measures the transverse momentum component out of the event plane: a planar event has and an isotropic one .

Eigenvectors can be found that correspond to the three eigenvalues of the sphericity tensor. The one is called the sphericity axis (or event axis, if it is clear from the context that sphericity has been used), while the sphericity event plane is spanned by and .

The sphericity tensor is quadratic in particle momenta. This means that the sphericity value is changed if one particle is split up into two collinear ones which share the original momentum. Thus sphericity is not an infrared safe quantity in QCD perturbation theory. A useful generalization of the sphericity tensor is

 (282)

where is the power of the momentum dependence. While thus corresponds to sphericity, corresponds to linear measures calculable in perturbation theory [Par78]:
 (283)

Eigenvalues and eigenvectors may be defined exactly as before, and therefore also equivalents of and . These have no standard names; we may call them linearized sphericity and linearized aplanarity . Quantities derived from the linear matrix that are standard in the literature are instead the combinations [Ell81]

 (284) (285)

Each of these is constrained to be in the range between 0 and 1. Typically, is used to measure the 3-jet structure and the 4-jet one, since is vanishing for a perfect 2-jet event and is vanishing for a planar event. The measure is related to the second Fox-Wolfram moment (see below), .

Noninteger values may also be used, and corresponding generalized sphericity and aplanarity measures calculated. While perturbative arguments favour , we know that the fragmentation `noise', e.g. from transverse momentum fluctuations, is proportionately larger for low-momentum particles, and so should be better for experimental event axis determinations. The use of too large an value, on the other hand, puts all the emphasis on a few high-momentum particles, and therefore involves a loss of information. It should then come as no surprise that intermediate values, of around 1.5, gives the best performance for event axis determinations in 2-jet events, where the theoretical meaning of the event axis is well-defined. The gain in accuracy compared with the more conventional choices or is rather modest, however.

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Stephen Mrenna 2007-10-30