6. Harmonics About the Beam Axis

Most of this paper has been concerned with the analysis of the `shapes' of final states in annihilation using observables which do not require a definite axis for their evaluation. The incoming beams provide, however, a natural axis for the analysis of the `shape' of the final state. Nevertheless, we shall see that there is often insufficient correlation between the beam axis and the configuration of the final state to make such an analysis useful.

Define

where is the angle that the final state particle makes with the direction of the incoming electron. Energy-momentum conservation requires

if all the particles are massless. If the incoming beams are unpolarized, then all parity-conserving processes give

and kinematics require that

The process illustrated in fig. 1 has

yielding

Data on angular distributions of two-jet final states in annihilation are often given in terms of the parameter , defined by

In this case,

so that may be expressed in terms of the as

Note that for pairs of particles produced at rest, , so that . This effect may serve as a method for detecting the presence of heavy quark or lepton production near threshold (see below).

The differential cross section for in is given by

Kinematics require that for any process

The divergence in for at the edge of the physical region is introduced by the Jacobian of the transformation from to . The differential cross section (6.10) is plotted in fig. 47. We also show there the forms for resulting from the process hadrons, using the model for quark fragmentation described in sect. 5. A significant sharpening of the distribution may be observed for as compared to . The mean values of deduced from these distributions and for various other cases are given in table 9. None differ by more than about 0.04 from the free quark approximation value of 0.1.

[ Figure 37 ] The distributions in the beam moment for events arising from the process .

[ Table 8 ] Average values of for realistic hadronic events resulting from the process : only hadrons with momenta larger than are used in each computation of

Fig. 48 shows the distributions due to the production and decay of heavy quarks and leptons, through the mechanisms discussed in subsect. 4.5. Clearly near the thresholds for these processes . The production and decay of pairs of heavy spin-0 mesons always give , but production and weak decays of high-energy heavy leptons or quarks could yield non-zero values of . This result is essentially unchanged by allowing the quarks and gluons generated in the weak decays to fragment into hadrons. The distributions for this case are shown in fig. 48 for various values of , using the models described in subsect. 4.5. The approximately Gaussian shapes of many of the distributions are a simple consequence of the central limit theorem. Away from the kinematic boundary , the probability that a given particle has a particular value of is roughly uniform in , and the values of for each of the particles are essentially uncorrelated. The for the complete event is a sum of the values for each of the particles, so that, at least away from , it will be approximately normally distributed. The distribution is closer to Gaussian if 3-jet heavy quark decays are assumed rather than 2-jet ones. The difference between the value of for two-jet ( hadrons) events () and for heavy quark or lepton production ( = 0) may allow the two processes to be distinguished by a measurement of . (A similar method is proposed in ref. [20].) In addition, the distributions for the processes are somewhat different (see figs. 47 and 48). For the process (G), one finds that (17)

so that the angular distribution parameter defined in eq. (6.7) is given by

For , this gives

On comparing from eq. (6.14) with the results for in table 9, it becomes clear that it would be very difficult to distinguish between and (G) events by a measurement of . Whereas two- and three-jet processes populate different regions in the (see, for example, fig. 30), both populate the complete range of . Hence, even a measurement of the distributions in the will probably not be very effective at identifying two- and three-jet processes. In fact, the distributions for and are very similar. For the former there is a slight depletion of events near the boundaries. The detailed results depend sensitively however on the treatment of three-jet fragmentation. Note that, because the photon has spin 1, for even for three-jet processes.

[ Figure 38 ] The distributions in the beam moment for events involving the production and decay of heavy quarks and leptons.

The process GGG is also amenable to analysis in terms of the . If one assumes that the is a pure s-wave state and contains no d-wave admixture, then the helicity amplitudes for the Q and within the are determined simply by those of the virtual photon. It is, therefore, possible to compute the angular distribution of the gluons produced in the process of fig. 3 with respect to the beam axis. Using the results of ref. [30], we find that in this case

corresponding to

The distribution in for GGG is rather smooth, having no peak around , and goes to zero as .

From the arguments made for the in sect. 3, it is clear that the moments of the should also be infrared finite to all orders in QCD perturbation theory. Since the angular distribution of energy produced in annihilation events is entirely determined by and , it too must be infrared finite to all orders in . The average energy angular distribution for (G) to is calculated in ref. [27]. Because this may be expressed as a finite sum of , it should be infrared stable, unlike , which is given only by an infinite sum of . Note, however, that by considering only such an average angular distribution, one loses the information provided by the distribution of events in the .

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