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### Thrust

The quantity thrust is defined by [Bra64]

 (286)

and the thrust axis is given by the vector for which maximum is attained. The allowed range is , with a 2-jet event corresponding to and an isotropic event to .

In passing, we note that this is not the only definition found in the literature. The definitions agree for events studied in the c.m. frame and where all particles are detected. However, a definition like

 (287)

(where is the step function, if , else ) gives different results than the one above if e.g. only charged particles are detected. It would even be possible to have ; to avoid such problems, often an extra fictitious particle is introduced to balance the total momentum [Bra79].

Eq. () may be rewritten as

 (288)

(This may also be viewed as applying eq. () to an event with particles, carrying the momenta and the momenta , thus automatically balancing the momentum.) To find the thrust value and axis this way, different possibilities would have to be tested. The reduction by a factor of 2 comes from being unchanged when all . Therefore this approach rapidly becomes prohibitive. Other exact methods exist, which `only' require about combinations to be tried.

In the implementation in PYTHIA, a faster alternative method is used, in which the thrust axis is iterated from a starting direction according to

 (289)

(where for and for ). It is easy to show that the related thrust value will never decrease, . In fact, the method normally converges in 2-4 iterations. Unfortunately, this convergence need not be towards the correct thrust axis but is occasionally only towards a local maximum of the thrust function [Bra79]. We know of no foolproof way around this complication, but the danger of an error may be lowered if several different starting axes are tried and found to agree. These are suitably constructed from the (by default 4) particles with the largest momenta in the event, and the starting directions constructed from these are tried in falling order of the corresponding absolute momentum values. When a predetermined number of the starting axes have given convergence towards the same (best) thrust axis this one is accepted.

In the plane perpendicular to the thrust axis, a major [MAR79] axis and value may be defined in just the same fashion as thrust, i.e.

 (290)

In a plane more efficient methods can be used to find an axis than in three dimensions [Wu79], but for simplicity we use the same method as above. Finally, a third axis, the minor axis, is defined perpendicular to the thrust and major ones, and a minor value is calculated just as thrust and major. The difference between major and minor is called oblateness, . The upper limit on oblateness depends on the thrust value in a not-so-simple way. In general corresponds to an event symmetrical around the thrust axis and high to a planar event.

As in the case of sphericity, a generalization to arbitrary momentum dependence may easily be obtained, here by replacing the in the formulae above by . This possibility is included, although so far it has not found any experimental use.

Next: Fox-Wolfram moments Up: Event Shapes Previous: Sphericity   Contents
Stephen_Mrenna 2012-10-24