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EVENT SHAPES IN e+e- ANNIHILATION (1979) |
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There is no natural plane defined in an 
annihilation event. In other processes the directions of initial or final particles often define a plane, and it may be useful to analyze the `shapes' of events projected onto this plane. For this purpose, define a set of observables
where the sum runs over all particles or pairs of particles in each event, and the angles 
are measured with respect to an axis chosen in the plane, 
. (
are the magnitudes of the particles' momenta projected into the plane .) The values of the 
will be independent of the axis used, so long as it lies in .
The occurrence of the 
in the definition (1.1) of the `shape' parameters 
is a consequence of the fact that these functions form bases for the irreducible representations of the three-dimensional rotation group. It is the functions 
which form bases for the representations (which are all trivial) of the two-dimensional rotation group, U(1), and so appear in the definition (8.1) of `shape' parameters for two-dimensional projections of events.
To discuss the properties of the , let us consider a continuous distribution 
around a circle. In this case the are given by
The obtained using an axis rotated by an angle 
from the original axis are then given by
thus proving the invariance of the under rotations in the plane . If another plane is chosen, then their values will clearly change.
If is taken to be the amplitude of an electric current as a function of time, with period 
, then is simply the power in the 
th harmonic of the signal.
The correlation function for the may be defined in analogy to the function 
(eq. (2.7)):
Defining the Fourier coefficients for the function as
the correlation function becomes
This is the analogue for 
of the series (2.10) for .
If all particles in the final state of an event are included in the sum (8.1), then
by momentum conservation. The value of 
depends on the plane chosen. It will, therefore, usually be convenient to consider the observables 
.
As an example of the use of the , consider the process 
. will be maximal if the 
and 
momenta lie in the plane (). All the will be zero if the and momenta are perpendicular to . Except in this case it is clear that
In general, if consists of a set of 
delta functions of equal weight at equally-spaced points around a circle, then
where 
is any integer. If is uniform in 
, then 
for all 
.
In deep inelastic lepton-nucleon scattering, the directions of the incoming and outgoing lepton in the target nucleon rest frame define a plane ( A much better analysis of deep inelastic lepton-nucleon scattering event may be made using the ). The hadronic final state may be analyzed by projecting all the hadron momenta on to this plane, and then evaluating the . 
is taken to be the c.m. energy in the 
collision. For a two-jet production process, as illustrated in fig. 49, the will be approximately 1 for even and zero for odd (assuming the fragments of the target to behave roughly like a point particle). So long as the analysis is made in the nucleon rest frame, `Fermi motion' of the struck quark within the target nucleon will have little effect on the values of the for the event. A typical three-jet event is shown in fig. 50. Such a process will give 
for all 
. The precise value will depend on assumptions about the structure of the target fragments, and on the momentum distribution of the primary quark or gluon within the nucleon. It will be infrared stable (see sect. 3).
[ Figure 39 ]
A typical two-jet event in deep inelastic scattering pictured in the target nucleon rest frame. The momenta of the
final hadrons are projected into the plane shown for the calculation of the .
[ Figure 40 ]
A typical three-jet event in deep inelastic scattering pictured in the target nucleon rest frame. The projections of
the momenta of the final particles into the plane used for the calculations of the are shown as dashed lines.
in the plane () orthogonal to the virtual photon (or 
) direction. This arrangement for a two-jet event is shown in fig. 51. Treating the target fragments as a point particle, and using the free quark approximation, one finds that for all in this case. `Fermi motion' of the initial quark within the target nucleon, the fragmentation of the outgoing quark into hadrons, and allowing for realistic target fragments will all serve to modify this result. Note, however, that a shower of hadrons which has cylindrical symmetry when projected onto will make no contribution to the . The arrangement for a three-jet event is shown in fig. 52. The target fragments will be directed approximately perpendicular to , and so will not contribute to the for the event. The projections of the momenta of the outgoing quark and gluon (in some events, outgoing and ) on to will be equal and opposite, by momentum conservation. For three jet events, therefore, 
for odd and 
for even . In idealized three-jet events, 
for even , and for odd . The shapes of the distributions in 
are similar to those in 
shown in fig. 23. 
is typically 
at 
, rising smoothly to about 
for 
. Once again, 
and longitudinal photons make an insignificant contribution.
[ Figure 41 ]
A typical two-jet event in deep-inelastic scattering. In this case, the plane n is taken orthogonal to the virtual
photon (or ) direction, and the projections of the momenta of the final hadrons on to (shown dashed) are distributed roughly symmetrically about the 
direction, so that they give a small value of .
taken orthogonal to the virtual photon (or ) direction.
The may be computed by measuring the angles of the hadron momenta projected on to the plane with respect to any axis in . However, in deep inelastic scattering, a natural axis in is defined by the projection of the incoming and outgoing lepton (
and 
) directions on to . In this case, one may define a set of variables analogous to the 
(see sect. 6):
where 
is the angle which particle 
makes with the fixed axis in the plane . For a two-jet event 
. Three-jet events will have a distribution of values of 
; the precise form depends on the detailed assumptions made about the inclusion of effects associated with hadrons. Note that because the photon (or ) has spin 1, 
for 
. 
is simply the 
summed over all hadrons in the event relative to the projection of the lepton direction.
Previous authors have considered the possibility of detecting three-jet production in deep inelastic N scattering by measuring the angular distribution of a single particle in the final state [21]. The discussion given in sect. 3 indicates that this procedure should not lead to infrared stable results, since processes involving the emission of particles of indefinitely low energy are distinguished from those in which no particles are emitted. To order 
, it turns out that the single-particle angular distributions are infrared finite, but this property will not survive in higher orders. This suggests that only angular distributions weighted with momenta and summed over all final particles as in the definition of the should be considered. These will be infrared stable.
We hope to discuss the use of the and in the analysis of final states from deep inelastic lepton-nucleon scattering in a future publication.
The may also be useful in an analysis of the structure of final states in high-energy hadron-hadron (
) collisions involving the production of jets of hadrons with large transverse momenta (
). We shall work in the c.m.s. for the colliding hadrons. We consider the for a plane orthogonal to the directions of the initial hadrons, as illustrated in fig. 53. The fragments of the initial hadrons should form a halo of particles around the axis. Fig. 53 shows a typical event involving the production of two jets of hadrons with high transverse momenta, as expected from QCD models for large processes [22]. QCD also predicts that in some events, three high transverse momentum hadron jets should be produced, arising, for example, from the subprocess qq
qqG. Measurements of the could be used to disentangle the different types of event. In an experiment with an azimuthally symmetric trigger, the halo of low particles from the fragmentation of the beam and target will be distributed symmetrically about the axis. Because of their isotropy, these unwanted particles will not contribute significantly to the values of the for an event. Care must be taken if a trigger is used which requires a particle or a jet at a particular azimuthal angle. In that case, the beam and target fragments tend to be directed oppositely to the trigger particles, and will therefore affect the values of the [23]. The application of a transverse momentum cut on the particles used in the calculation of the would, however, remove their effect. (This cut is the two-dimensional analogue of the total momentum cut 
discussed in sect. 5.4.) Final states containing two high transverse momentum jets will give ( even) and ( odd), while those in which more jets are produced will give rise to a wider distribution of values.
orthogonal to the directions of the incoming particles in their c.m.s.
It appears that the should provide a powerful tool for the
analysis of high hadron events. QCD makes definite predictions for the distributions of events in the .
