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Experiments (1) have shown that at high center-of-mass energies () the final states in hadrons usually consist predominantly of two jets of hadrons presumably resulting from the process . Quantum chromodynamics (QCD) explains this basic two-jet structure, (2) but predicts that one of the outgoing quarks should sometimes emit a gluon , tending to lead to three-jet final states.

Previous attempts (3) to discriminate between two- and three-jet events concentrated on finding a ``jet axis'' by minimization, and then measuring the collimation of particles with respect to it. Instead, one may use observables which directly characterize the ``shape'' of each event. Since there is no natural axis defined in the final state of annihilation, it is convenient to consider rotationally invariant observables. A set of such observables is given by are the usual spherical harmonics and the Legendre polynomials]

where the indices and run over the hadrons which are produced in the event, and is the angle between particles and . When the first for the is used, one must choose a particular set of axes to evaluate the angles of their momenta, but the values of the deduced will be independent of the choice. Energy-momentum conservation requires and . In principle, all the other carry independent information. (4) In practice, however, one need only consider the lower-order ; in this paper we concentrate on and .

The information contained in the may also be expressed by the ``autocorrelation function''

where is a continuous distribution of momentum and are operators in the rotation group. For particle events, we define the two-detector energy correlation (5)

where are energies incident on detectors covering the regions of total solid angle . We form the rotationally invariant observable by averaging over all possible positions for the detectors, while maintaining their relative orientation. In annihilation events, this may be achieved (apart from correlations with the beam axis and polarization) by averaging over events. In the limit , becomes a function solely of the angle between the two point detectors, and is identical to . may clearly be generalized to a correlation between detectors . However, unlike the case of the , there are infrared difficulties when the are calculated in QCD perturbation theory. (6)

The ability of the to distinguish between different processes is illustrated in Fig. 1. Final states of the process have for even and for odd . In contrast, the process gives events with a wide distribution of values, corresponding to a range of shapes. For example, the dependence of on the fractional energies of the final quarks and gluons in this case is given by

Each kinematic configuration, labeled by the , leads to an event of a different shape, and each is characterized by a particular value of .

[ Figure 1 ] The distributions in and for the processes (dotted lines), (full lines), and heavy resonance (dashed lines). The process alone yields an infinite total cross section, but when added to calculated through the combination of processes [denoted by ] gives a finite cross section. We have taken for the distribution.

The do not discriminate between final states differing by the inclusion of a very low-energy particle or by the replacement of one particle by two collinear particles with the same total momentum. It is believed that these properties are sufficient to ensure that calculations involving the are infrared finite in QCD perturbation theory. (3,6)

A convenient measure of the event shapes due to different processes is provided by the mean . For the sum of the process and calculated to lowest order in the QCD coupling constant , we have

so that a center-of-mass energy GeV, .

QCD suggests that heavy vector mesons (such as ) should decay to three gluons. Figure 1 shows that the and distributions due to this process are very different from those for . The flatter distribution for the decay is reflected in a lower :

Our results above were obtained by making the idealization that final states consist of free quarks and gluons. In reality, one must consider the ``fragmentation'' of these quarks and gluons into hadrons, although at sufficiently high energy the values of the should be the same whether they are calculated from Eq. (1) using the momenta of the actual hadrons in each event, or of their parent quarks and gluons. In order to estimate the shapes of realistic events at finite energy, one must go beyond the realms of present QCD theory and adopt an essentially phenomenological model for the generation of complete hadronic final states by the fragmentation of quarks and gluons. We use the model developed by Field and Feynman, (7) which agrees with available data. (8)

QCD predicts that, away from resonances, annihilation should be dominated by the processes and . The processes can give rise to final states containing either two or three jets of hadrons. Two-jet events occur when some of the quarks and gluons have low energy or are nearly collinear (9) and they cannot be distinguished from events by measurements on the hadron final state. Only when and [calculated through ] are added is the jet-production cross section infrared finite. We denote this combination of processes by .

[ Figure 2 ] The distributions predicted for hadronic events resulting from the processes (dotted lines), (full lines), and heavy resonance (dashed lines), at various center-of-mass energies .

In Fig. 2, we present the distributions for realistic hadronic events resulting from , , and ( is a heavy resonance.) The modifications to the results in Fig. 1 due to the fragmentation of the quarks and gluons into hadrons are striking. (They also occur for the higher and for other observables designed to identify three-jet events) Nevertheless, above GeV, the distributions for the different types of events are clearly distinguished. By GeV, the predictions are similar to those obtained in the idealization of free quarks and gluons (Fig. 1). and distributions are particularly effective at distinguishing and events, while distributions are very sensitive to the presence of any pure component. The distributions for realistic events may be made more similar to the idealized ones of Fig. 1 by using only the higher-momentum hadrons in each event for the computation of the . (10) Even the cut GeV is sufficient to effect a great improvement. The distributions are little affected if only the charged particles in each event are detected. Our predictions are not particularly sensitive to the parameters of the jet development model (which may presumably in any case be determined from single-hadron momentum distributions), but it is still difficult to estimate the uncertainties in our results at energies where the fragmentation of the quarks and gluons has an important effect. Refinement of the jet model as further experimental data become available should allow more accurate predictions to be made.

The are not specialized to the investigation of two- and three-jet events. They may also be used to identify events of other types. The pair production and weak decay of heavy mesons (containing a heavy quark and a light antiquark ) and heavy leptons should give events containing many hadron jets. For heavy leptons we assume the decay scheme , while for heavy quarks (mesons) we consider the three possibilities , , and . Figure 3 shows our predictions for the distributions of heavy-quark and lepton production events. We take no account of hadron production by heavy quarks prior to their weak decays, so that our results for heavy-quark pair production should be valid only near threshold.

In addition to the , one may consider the multipole moments

where are the angles made by the particles in the event with the beam axis. A final state with angular distribution (the naive parton model predicts ) gives a broad distribution in with mean , while the process gives , corresponding to , and gives or . The , being rotational invariants, are of course insensitive to correlations with the beam direction. They are, however, far superior in identifying the shape of events and distinguishing competing processes.

The may also be used to analyze three-jet effects in deep-inelastic lepton-nucleon scattering. Making the idealization of free final quarks and gluons, and treating the nucleon fragments as a single particle, we find that in the virtual photon- (or -) nucleon rest frame, two-jet events arising from give for even and for odd , just as in annihilation. The three processes (11) , and typically give a which varies smoothly from at Bjorken around 0.1 to at . The distributions in the are similar to those in annihilation. The effects of fragmentation to hadrons are governed by .

For processes in which a natural plane is defined it is convenient to use the two-dimensional analogs of the :

where are the angles of the particles relative to an arbitrary axis in , and are the magnitudes of their momenta projected onto . In deep-inelastic scattering, it is best to take the plane to be orthogonal to the (or ) direction. Then two-jet events give , while three-jet ones can give nonzero values of . (12) Typically, in the free-quark approximation, is independent of , and typically at , rising to at . In hadron-hadron collisions involving high transverse momenta, should be chosen as the plane perpendicular to the incoming hadrons. Once again, the distributions in distinguish two- and three-jet events. The obvious two-dimensional analog of [as defined in Eq. (2)] will also be useful.

A detailed discussion of the work summarized here is given in Ref. 6.

[ Figure 3 ] The distributions predicted for hadronic events resulting from the production and weak decay of heavy-quark and -lepton pairs (dotted lines) at GeV, and in the free-quark and gluon approximation . Three mechanisms for heavy-quark decay are considered: (full lines, (dashed lines), and (dot-dashed lines). In the free-quark and gluon approximation the latter two processes give the same distributions.

This work was supported in part by the U. S. Department of Energy under Contract No. EY 76-C-03-0068. 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 Massachusetts Institute of Technology Laboratory for Computer Science for the use of MACSYMA.

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