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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 2jet 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 24 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 notsosimple 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: FoxWolfram moments
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Stephen_Mrenna
20121024