4. Three-Detector Energy Correlations

4.1. , as a Test for Planar Structure

In this section we consider the three-detector energy correlation, defined by

where detector has area on which a total momentum is incident. We shall consider only the function obtained by averaging over all positions of the detectors that maintain their relative orientation. Thus we ignore the correlations with the beam direction that were treated in Sect. 5 for the two detector case. The main purpose in considering three detector correlations is to find tests for planar momentum configurations. Two particle final states contribute to the two detector energy correlation function only when the angle between the two detectors is near On the other hand, three particle final states, as from or have no such simple signatures in the two detector case. However, the momenta in the three particle final state lie in a plane and hence the three detector energy correlation vanishes for such events unless the detectors themselves are nearly coplanar. In order to extract the most demanding test of coplanarity, we define as the value of the rotationally averaged when all the three detector directions are mutually orthogonal. We take the three detectors to be circular and of equal angular radius The normalization of is such that it has the value one for isotropic events. Thus the value of is zero for idealized three jet events and one for isotropic events. Hence a measurement of provides a method for distinguishing these two event structures. Of course, the naive prediction that for three-jet events is only true at infinite energy, where there are neither perturbative nor hadronic corrections. At finite energies which are sufficiently large that fragmentation is unimportant, one may estimate that for while for (Analogous results hold for ) There are indications that and are of order one [4]. Note that the decay (which gives rise to non-coplanar final states) where is, as above, a state, is not forbidden by symmetry (as would the corresponding positronium decay) because the gluons can be antisymmetric in color. Hence, one expects that, at values of for which fragmentation is unimportant, should be larger on resonances than in the continuum by a factor of order For lower the hadronization of the final state will modify perturbative predictions for and one must make a model to study the size of the changes. We consider the particular question of to what extent a measurement of enables one to test for 3 jet decay of the and possible higher mass heavy quark bound states. In our calculations of the effects of fragmentation, we ignore the perturbative or corrections to the result for two and three jet events. We compare only mechanisms that lead to the same single hadron momentum distributions. We found [10] two models that gave the same distributions as The first was obtained by construction; we generated hadrons from genuine events and then randomly rotated each hadron in the final state. We term this model ``isotropic''. For the second model, we noted that where each meson decays into 3 jets (cf. Sect. 2.5), happens to give essentially the same single hadron momentum distribution as This ``6 jet'' model is, of course, not isotropic but it is certainly more so than would be expected from a GGG final state. In Fig. 10 we plot as a function of for these three types of events. The final state is easily distinguished, even at the mass, from the more isotropic mechanisms. Of course, the discrimination improves as increases.

Although appears to be a very useful experimental observable, we should point out that it is not trivial to calculate. We found no simple analogue of (2.1) for the two detector case. Thus we calculated by doing the angular average (integral) in its definition by a simple Monte Carlo technique. Note that this integral has to be done separately for each event.

We have chosen to emphasize one particular angular configuration for the three detectors: the case where they are mutually perpendicular. One can, of course, consider other angular separations but we do not believe they will lead to qualitatively different results.

[ Figure 10 ] The mean energy correlation between three orthogonal detectors with angular radii as a function of for three classes of processes discussed in the text. At infinite energy, coplanar events such as give while a completely isotropic distribution of energy in the final state gives ``6 jet'' final states give as

[ Figure 11 ] The mean coplanarity parameter defined in Sect. 6.2 as a function of At infinite gives isotropic events and ``6 jet'' ones

4.2. Moments of Three-Detector Energy Correlations

The two jet form for the :

is a moment analogue of the result for the two detector energy correlation:

It is natural to ask if there is a moment analogue for the prediction that the three-detector correlation function is zero for three particle final states unless the detectors are coplanar. In [10], we showed that there were such moment analogues but did not discuss their relation to We defined two classes of moments that vanished for planar events:

where the functions and are respectively symmetric and antisymmetric polynomials in the scalar products of the unit vectors in the directions of the particle The simplest example of the class of observables has and was denoted while the simplest non-trivial member of the class (denoted by ) has

It is clear from our previous arguments that the moments of and are infrared stable when computed in QCD perturbation theory. As discussed in [10], we found to be the most useful observable in that it offered the greater discrimination between 3 jet and isotropic processes at the mass. In Fig. 11 we show as a function of for the three processes discussed in the previous subsection. There are again reasonably large differences between the results for events and for our models with isotropic decay. However, comparison of Figs. 10 and 11 shows that is somewhat better than at distinguishing the processes. does have two advantages, however: first, it is much easier to calculate than and further it is not only possible to find easily the mean but also the distribution The latter provides additional discrimination as discussed in [10].

We now describe the relationship of the observables and to the analogue to the result that the are the Legendre moments of We specialize to the idealized case of point detectors and write in the rather formal manner

Here and are elements of the rotation group. runs over all rotations (labeled by 3 Euler angles so that ) while is defined so that the direction of detector is given by where is unit vector in the direction. We have written (4.5) for the case of a continuous energy density normalized to unity when integrated over independent values of its argument (i.e., dropping the redundant azimuthal integral in ), so that

and for an isotropic event. Of course, for a particle event (4.5) is valid with as a sum of delta functions at the angles of the various particles. We now define the multipole moments as in [1] and [10]:

or conversely

where are the conventional rotation matrices which can be written as

where

is expressed as a product of rotations about the and axes.

Substituting (4.8) into (4.5), we use the addition theorem for the D matrices to find

Defining the rotationally invariant three detector moments by

the integral may be expressed in terms of symbols [11] so that

where

is a function of general mathematical interest. It is a rotationally invariant function of three directions. This invariance can be expressed as

for any rotation is the three-direction analogue of the two-direction rotationally-invariant function . Of course, the are the two direction analogues of and can be expressed in terms of and in analogy to (6.13). If any value is zero, and reduce to and respectively.

Note that if is even, then and are real and invariant under any permutation of indices in the and On the other hand, if is odd, then both and are purely imaginary and permutations of multiply them by the signature of the permutation.

This fact leads to our first test for planes. If in a planar event we choose the and axes to be in the plane, then the are manifestly purely real and hence all are real. Therefore, vanishes for planar events if is odd. The simplest nontrivial constraint corresponds to and one can easily show that

The equivalence between like tests [in sense of (4.4)] and Im odd) is complete. Note that both and Im are pseudoscalars (i.e., change sign on reversal of all the particle momenta) while and Re are scalars.

We now show how to obtain observables of the type from our new formalism. We first choose a particular configuration for the three detectors with detector 1 along the axis and detector 2 in the plane:

This makes the degrees of freedom of manifest but loses the elegant symmetry of the original form (4.14). We can now express as a Fourier series in the azimuthal variable :

where

For a planar event, is proportional to a delta function at and inverting the Fourier series (4.18) this implies

The result (4.20) is just vanishing of Im for planar events that we have already discussed. Some of the information in (4.21) can be turned into new moment constraints that are equivalent to in (6.4). Consider the relation

where and differ by an even integer. Multiply both sides of (4.22) by and integrate Using the orthogonality of the matrices, we pick up a finite linear combination of fixed, varies) on the left hand side. Now we can express as a linear combination of with and similarly for (Here we use our choice that and differ by an even integer.) It follows that the right hand side is also a finite linear combination of observables and hence (4.22) gives rise to a set of finite relations between Re for planar events. These relations may be translated into constraints of the form For example, the simplest observable is obtained from the relation Re on multiplying through by and integrating. Reference [10] gives some of the simpler observables found in this way as linear combinations of the

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