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According to QCD, annihilation into hadrons at high center of mass energies () proceeds dominantly through the process , with some contribution from higher-order mechanisms such as . On vector meson resonances composed of heavy quark pairs (such as and , denoted generically ), QCD suggests that hadrons should be produced primarily through , and should therefore form three jets. In this paper, we discuss tests for this mechanism, which distinguish it, for example, from those in which the hadrons are distributed isotropically rather than forming jets. In a previous paper [1], we considered the class of observables defined by (the are the Legendre polynomials)

where the sums run over all particles in an event, and the are unit vectors along the momenta . These measures provide a measure of the ``shapes'' of events in annihilation and allow some discrimination between isotropic and three-jet hadron production on resonance. For idealized two-jet events, and , while for isotropic events for . Three-jet events lead to intermediate values of the .

To make this more quantitative and include the effects of the fragmentation of quarks and gluons to hadrons, one must perform a detailed theoretical calculation [1]. Perhaps the most distinctive feature of three-jet events is the approximate coplanarity of the final state particles. Unfortunately, this property has no simple consequences for the . However, if instead one considers observables of the form

where the functions and are respectively symmetric and antisymmetric polynomials in the scalar products of the unit vectors, then for coplanar events, the and vanish. These observables, therefore, provide a definitive test to coplanarity and hence should allow clean discrimination of two- and particularly, three-jet final states from more complicated structures. The simplest example of the class of observables has and will be denoted , while the simplest nontrivial member of the class (denoted by ) has

Note that while the are scalars, the are pseudoscalars, so that when averaged over events, = 0. Of course, , for example, need not vanish.

In [1] we argued that the moments of the should be infrared stable when computed in QCD perturbation theory. This result should also hold for the and . In general, divergences in the mean values of observables are canceled if the observables take on the same value for all physically indistinguishable processes. One requirement is, therefore, that the addition of very soft particles should not affect the value of the observable. This is guaranteed for the and by the presence of a term proportional to the total momenta of the particles. The other condition for infrared stability is that the observables should be linear in the momenta of collinear particles. This is clearly satisfied by the and .

We showed in [1] that the correspond to moments of two-detector energy correlation functions which are formed from the product of the energies incident on each of two detectors [2]. The and may be related to momenta of the analogous three-detector energy correlations (1). We sketch this relation below.

Let us define the multipole moments of an event by (the are the usual spherical harmonics)

where the angles are measured with respect to a set of axes chosen in the event. The defined in eq. (1) may then be written as

which is clearly a rotational invariant and hence independent of the choice of axes used to measure the angles . The three-detector energy correlation function may be decomposed in terms of natural generalizations of the , given by

where the symbol serves to combine the three spherical tensors into a rotational invariant (2). The represent a special case of these observables:

For planar events, the three-detector energy correlation function clearly vanishes unless the three detectors lie in a plane. As we describe in detail elsewhere [2], this property of the three-detector energy correlation may be translated into the vanishing of certain linear combinations of the for planar events. These combinations fall into two classes corresponding to the and observables. Those involving only with even correspond to the and, for example

If is even, then is real, but if it is odd, then the are purely imaginary. However, for planar events, all the must be real (3) so that all with odd must vanish in that case. The may be written in terms of these and, for example,

The formulae for the simpler and are given in table 1. Note that momentum conservation implies that vanishes if any of its indices . We have nevertheless retained such in table 1 so that our results may be applied to incomplete final stages where momentum is not conserved among the particles used to calculate the and .

[ Table 1 ] Examples of observables which vanish for coplanar events.

In the approximation of free final quarks and gluons, events of the type and will give zero for all the and . For an exactly isotropic event, however, all the vanish except for . In this case, therefore, , , , , and all

In order to simulate real hadronic events, we use the phenomenological model for quark and gluon fragmentation into hadrons developed by Field and Feynman [3]. To investigate the discrimination between planar and non-planar events provided by our observables, we shall compare events due to with ones which give the same single hadron momentum distribution but which arise from non-coplanar configurations of quarks and gluons. We chose two models for non-coplanar events. In the first (referred to as `6-jet'), we consider the production and decay of a pair of heavy quarks into three particles. This model was introduced in [1]. Although it gives rise to events which are non-planar and contain six hadron jets, it happens that with our quark and gluon fragmentation functions, they have roughly the same z distributions as events. For our second model (referred to as `isotropic'), we generated events and then rotated the momentum of each of the particles randomly. This procedure gives roughly isotropic events but at the cost of some violation of momentum conservation.

In fig. 1, we show the distributions of simulated hadronic events in at three center of mass energies while fig. 2 gives their distributions in (4). In both cases, the free quark and gluon predictions are considerably modified by fragmentation to hadrons. This effect is particularly marked for the distributions. Nevertheless, even at = 10 GeV (corresponding to the region), the distributions allow clear discrimination between different mechanisms. Of course, at higher , the effects of fragmentation become less important, and the various processes are yet more clearly separated. Note that the distributions in are particularly suitable for distinguishing planar from non-planar processes and, for example, allow separation of events from isotropic or 6-jet ones. At = 10 GeV, isotropic and 6-jet events give indistinguishable and distribution, but at higher they differ. Figs. 1 and 2 show that it should be possible to determine whether decay proceeds dominantly through by measuring the and distributions of productions events. It should be pointed out, however, that if the decays are found to be more isotropic than would be expected for , this does not represent a contradiction with present QCD theory since there is thus far no overwhelming evidence that low-order processes should dominate decay. Note that the results shown in figs. 1 and 2 depend on the quark and gluon fragmentation functions assumed. Our choices for these may be tested by measuring single hadron momentum distributions and if a significant difference were found, the calculations of the shape parameter distributions should be revised. In our discussion of decays, we have always considered models which give the same distributions. Thus the discrimination between different mechanisms illustrated in figs. 1 and 2 should not be affected by changes in the distributions.

[ Figure 1 ] The distributions of simulated hadronic events in the coplanarity parameter for various center of mass energies (). , ``isotropic'' and ``6-jet'' are three illustrative mechanisms for heavy resonance () decay. According to QCD, should be the dominant process of resonance [1]. In the free quark and gluon approximation, the processes , and should lead to . In the same approximation, the `6-jet' process leads to a roughly flat distribution in over its kinematically allowed range (). Completely isotropic events have . Note that in this and fig. 2, all curves are calculated by considering only hadrons with momenta above 0.5 GeV.

[ Figure 2 ] The distributions of simulated hadronic events in the shape parameter , for the various center of mass energies (). The corresponding distributions in the free quark and gluon approximation are also given.

We find that the distribution of realistic hadronic events in the observables , and defined in table 1 does not differ significantly between the processes we consider. The distributions in are qualitatively similar to those in but distinguish slightly less between the various processes, and so we find that it is sufficient to measure to test the coplanarity of events.

Our observables can also be used to analyse final states in which not all the particles are detected. For example, a = 10 GeV, the difference in between and isotropic events at changes from the factor of about 3 shown in fig. 1 when all particles are measured to a factor of about 2 when only charged particles are detected.

Our previous work [1] showed that the (and in particular, and ) provide clear measures of the shapes of events. They are especially suited to discriminating two-jet events from events containing a larger number of jets. Here we have introduced the observable which tests for planar events and is, therefore, particularly suited to distinguishing two- or three-jet events from events with a more complicated structure.

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.

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