|
EVENT SHAPES IN e+e- ANNIHILATION (1979) |
|
In this paper, we have presented several sets of observables for the analysis of the `shapes' of final states in events of various processes. We have considered mainly 
annihilation, for which no natural axis is defined in the final state. The 
(defined in eq. (1.1)) appear to be the most suitable class of observables for this case since they are rotational invariants whereas previous observables proposed require minimization to find a preferred axis in each event. An axis must be chosen to evaluate the , but their values are independent of the choice. In sect. 2, we showed that the information on the `shape' of an event contained in the may be reexpressed in a generalized `correlation function' 
for the relative density of particles at different angles. 
may be interpreted as an energy correlation function between two detectors an angle 
apart, in the limit that their areas tend to zero (see subsect. 2.6). In sect. 3, we showed that to order 
in QCD perturbation theory, the moments of the are infrared finite, and we gave arguments that this result will survive the inclusion of higher-order effects. ( is not, however, infrared finite.) If the quarks and gluons produced in the primary interaction are allowed to fragment into hadrons, then the resulting distributions in the are also infrared finite. The reason for this is that the fragmentation process acts somewhat like a detector with finite energy (and angular) resolution and properly averages over 2-jet events (
) and 3-jet events (
) that are `near' the two-jet kinematic configuration.
For the process 
, 
for even 
, and 
for odd . Energy-momentum conservation requires 
and 
. For final states consisting of three quarks and gluons, 
and 
are simple functions of the energies of the final particles. We gave analytic expressions for 
and 
for the processes 
GGG, where 
is a heavy vector meson containing a pair of heavy quarks, and (G). The latter process is defined to be a combination of and 
. According to QCD, only this combination should be present. The fact that and are finite for that case is an explicit demonstration of the infrared stability of the to order .
We used the Field-Feynman model for the fragmentation of quarks and gluons into hadrons. We found that for c.m. energies above about 
, the distributions of events in and allow good differentiation between the various mechanisms. The distribution was particularly powerful at distinguishing the processes (G) and GGG, while the distribution should provide a good measure of any pure processes. At least 
is required, however, before the and distributions for realistic hadronic events become similar to those obtained in the free quark and gluon approximation. Only for these energies can the and distributions be considered as direct quantitative tests of perturbative QCD; at lower energies, the details of the distributions calculated from QCD perturbation theory are blurred by the fragmentation of the quarks and gluons into hadrons. The agreement between the results for realistic hadron final states and for free quarks and gluons can be improved slightly by considering only the higher-momentum hadrons in each event. This procedure does not affect the discrimination between different types of events offered by the ; it merely gives results which are closer to the free quark and gluon approximation.
We also considered the production and decay of heavy leptons and of mesons containing heavy quarks. These processes give rise to events of a rather spherical `shape', which should be clearly distinguished from other types of event. Several possible mechanisms for the weak decays of heavy mesons were considered. It should be possible to discriminate between them on the basis of measurements of the `shapes' of events using the .
In sect. 6, we discussed a class of observables (
) which use the direction of the incoming beams as an axis and are, therefore, not rotationally invariant. There is no clean separation of and 
in the distributions and generally they do not provide very good discrimination between different types of events, although events involving heavy quark or lepton production could be distinguished by their use.
In sect. 7, we considered the extension of our analysis of event shapes in annihilation to final states in deep inelastic lepton-hadron scattering. The main difficulty in this case is the presence of fragments of the target in the final state. Nevertheless, it appears that the could be used to identify three-jet final states in deep inelastic scattering.
Finally, in sect. 8, we gave a very brief discussion of the analogues of the in two dimensions (the 
). These should be useful for analyses of final states in which a natural plane is defined. If this plane is chosen to be orthogonal to the virtual photon direction in the target nucleon's rest frame, then idealized two-jet events in deep inelastic lepton-nucleon scattering give 
, while three-jet ones give nonzero values for 
. The also promise to be useful in the analysis of hadron-hadron collisions involving the production of particles with large transverse momenta. The related variables 
, also discussed in sect. 7, appear to allow a more precise formulation of certain azimuthal angular correlations that have been previously proposed as tests of QCD.
We hope to present more detailed analyses of the `shapes' of events in deep inelastic lepton-nucleon scattering in a future publication.
We are grateful to R. D. Field and R.P. Feynman for the use of their jet-development computer program, and to the MATHLAB group of the MIT Laboratory for Computer Science for the use of MACSYMA.
