1. Introduction

Quantum chromodynamics (QCD) provides an increasingly successful theory for strong interactions. One of its most striking predictions is the existence of gluons. Perhaps the most direct way in which to detect these new fundamental particles is to observe the jets of hadrons resulting from their production in annihilation. Experiments have shown that at high energies, final states in hadrons usually consist predominantly of two jets of hadrons [1] presumably coming from the quark and antiquark resulting from the process of fig. 1. QCD explains this basic two-jet structure [2] but indicates that one of the final quarks should sometimes emit a gluon (G) as in fig. 2 [3]. Such processes would tend to lead to final states containing three jets. In addition, QCD suggests that resonances containing heavy quark pairs (for example, the and ; denoted generically by ) should decay predominantly into three gluons (fig. 3), again leading to three-jet final states [4]. It is, therefore, important to identify events in which three hadron jets are produced.

[ Figure 1 ] The lowest-order diagram for the process .

[ Figure 2 ] The lowest-order diagrams for the process , where G is a vector gluon.

[ Figure 3 ] The lowest-order diagram for the production and decay into three gluons of a vector meson () containing a pair of heavy quarks (Q).

A number of observables which characterize the `shapes' of final states in annihilation have been proposed for this purpose [5,6]. There is no natural axis defined in the final state in annihilation. An axis may, however, be found by demanding that it minimize some observable. Such a minimization has been the basis of previous experimental (sphericity) and theoretical (spherocity, thrust, acoplanarity) observables designed to measure the structure of final states in e+e- annihilation. For the observables of theoretical interest, the minimization has turned out to be inconvenient to implement [24]. Moreover, this procedure may induce spurious jet structure. The minimization would probably be satisfactory if all events had a two-jet structure; according to QCD, however, more than 30% of the events in nonresonant annihilation at c.m. energies around 20 GeV should consist of three or more jets, so that methods based on finding an optimal two-jet axis appear suspect. The problem of minimization can, however, be circumvented by measuring an observable which characterizes the `shapes' of final states, but whose value does not depend on the axis used to evaluate it. A set of such observables exists, given by (the 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 form for the is used, one must choose a particular set of axes to evaluate the angles of the momenta of the final-state particles, but the values of the deduced will be independent of the choice. Note that the constitute a complete set of shape parameters in a way that is made precise in sect. 2. Energy-momentum conservation requires that , . The process gives for even , and for odd , since the final quark momenta are collinear.

The processes and GGG give final states for which there are many possible kinematic configurations, corresponding to a range of possible `shapes'. These processes therefore lead to distributions in the which are very different from the delta function due to . For example, the process GGG yields a which is nearly flat over the range allowed by kinematics, while gives only . The fragmentation of the quarks and gluons into hadrons serves to distort the distributions of `shapes', but at sufficiently high energies , hadronic final states resulting from each of the three subprocesses of figs. 1--3 should be clearly distinguished by measuring their distributions in .

The multi-jet final states resulting from decays of pairs both of heavy mesons carrying new flavors and of heavy leptons may also be identified by measurements of their `shapes' using the .

This paper is organized as follows. In sect. 2, we discuss various mathematical aspects of the , define the `correlation function' and give examples of the for simple shapes. We also present some generalizations of the . In sect. 3, we discuss the definition of jets in QCD and give arguments for the infrared finiteness of the calculated from perturbation theory. Sect. 4 describes our results for the processes , G, GGG and for the production and decay of heavy quarks and leptons. In sect. 5, we present results for realistic hadronic final states formed by the various quark-gluon subprocesses discussed in sect. 4. Sect. 6 contains a brief discussion of a class of shape parameters which exploit the correlation between the direction of the incoming beam axis and the configuration of the final state. Sect. 7 considers the extension of our analysis to deep inelastic lepton-hadron interactions and in sect. 8 we give a very brief discussion of two-dimensional analogues of the . These are relevant to processes in which a natural plane is defined; the most important applications appear to be large hadron reactions and deep inelastic lepton-hadron scattering. The appendix summarizes the kinematics for three-jet production.

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