2. Results

For an idealized two-jet event, it is clear that, as in annihilation,

Three-jet final states introduce modifications to these results, and the formation of hadrons from the quarks and gluons typically gives corrections. Note that transverse momentum (``Fermi motion'') of the initial quark with respect to the incoming nucleon direction has very little effect on the , because of their rotational invariance. It merely makes a slight boost on the final state momenta.

An idealized two-jet event has no transverse momentum with respect to the direction, and therefore gives

A three-particle or ideal three-jet final state gives (1)

Fragmentation leads to deviations from these results, as would four-jet final states (at ). An event in which the hadronic energy is distributed in an azimuthally-symmetrical manner about the incoming proton (and hence target fragment) direction will give

This is approximately the case for two-jet hadronic events, except for a small offset resulting from transverse momentum of the initial quark with respect to the nucleon direction. In our phenomenological estimates, we include a Gaussian distribution of initial quark transverse momentum (and the corresponding momentum for the target fragments) with a generous MeV, but its effects are in all cases entirely negligible. The emission of a gluon from the quark struck by the or should not affect the fragments of the target nucleon from which the quark was ejected. Their contribution to the , for three-jet events is therefore roughly the same as for two-jet events. The fragmentation of the quark and gluon produced in into hadrons spreads out their energies and tends to diminish the .

A possible complication in the analysis of deep inelastic event shapes arises from the possibility of high transverse momentum photon emissions from incoming or outgoing legs. One effect of these is to deflect the incoming or scattered lepton, and therefore to alter the direction of the virtual photon inferred from their measurement. This spread in momentum of the photon gives similar effects to the Fermi motion of the incoming quark, and its consequences are probably likewise negligible, even for the . Photon emissions from the final state modify the shapes of events in the same manner as do gluon emissions, but the smallness of compared to makes them irrelevant for our estimates.

In appendix B we give the differential cross sections for the three-jet production processes and , whose kinematics are discussed in appendix A. In estimating these cross sections, we assume (without profound theoretical justification) that the relevant effective coupling is . The precise forms of the cross sections depend on the preparation of the virtual photon, and hence on the component of the scattered lepton angular distribution or structure fraction sampled. Appendix B gives results for the various cases (including the parity-violating component accessible from ; the shapes of the final states are entirely insensitive to which is considered (2). (For the calculation presented below, we assumed that the direction of the final lepton was uniformly averaged over, selecting the projection of . ) For the longitudinal momentum distributions of the incoming partons, we used the fits obtained in ref. [4] for a proton target. Different choices for these distributions (which give the dependence of the total cross section) cancel out in considering the final state shapes except at large , where they can make factor of two changes. It turns out that events from the process tend to be closer to two-jet by a factor typically than those from , so that the dominant three-jet processes in QCD should be of the latter type. In eq. (B.4) we give the differential cross-section for the process where is a scalar `gluon'. This process gives rise to events whose shapes are comparable to those from in QCD. The observation of large three-jet effects would therefore discriminate against scalar gluon theories and support QCD.

The most important uncertainty in the calculation of the shapes of hadronic events is the fragmentation of the quarks and gluon produced at short distances, into the observed hadrons. The purpose of considering infrared stable shape parameters is to control the incalculable large distance effects, but at foreseeable energies, their consequences will still be felt, and one must adopt a phenomenological model to estimate them. We use the model developed by Field and Feynman [5], which provides a satisfactory fit to present data on two-jet final states. In the formation of hadrons, a third jet will not be resolved unless it has a sufficiently high transverse momentum. We use the prescription that in events arising from subprocesses for which , hadrons are produced in two jets, while if , the fragmentation of the third jet is treated independently. The implementation of this method and its sensitivity to the precise value of the cut was discussed at length in [1]; use of the more precise methods suggested in [2] should not affect our rough phenomenological estimates. In addition to considering the formation of hadrons from the struck quark (and any high energy gluons emitted by it), we must also treat hadron production from the fragments of the nucleon target, since, in our shape parameter analysis, all particles in the final state are included. We assume that the target fragments just like a quark, which should be reasonable at least for the dominant process (3). The target fragments should also radiate gluons, but since the momenta transferred to them from the struck quark will typically not be large, the radiation will not be very important, and cannot lead to three-jet events. One may consider cuts on the final state designed to remove the contribution of target fragments. We gave some discussion of such phenomenological devices in [1], and will not elaborate here.

Fig. 1 shows the distributions of events expected in the shape parameters and , at and . The essential parameter in determining the modifications due to fragmentation is in this case. The distributions in fig. 1 are roughly similar to the analogous

[ Figure 1 ] Distributions of shape parameters in deep inelastic scattering for and .

ones for annihilation events at the same value of discussed in [1]. The three-jet events at this give rise to a significant tail in the distribution, which is evident in fig. 1. Although distinctive, this tail corresponds to only 6% of all hadronic events having of the quark-gluon systems from which they came had . It is clear from fig. 1 that the low tail of three-jet events is much modified by fragmentation. Thus the observation of a tail is even at only a qualitative and not a quantitative test of QCD. Notice that in annihilation, three-jet effects are much larger; 9% of idealized three-jet events have , and after fragmentation the fraction is roughly doubled. Nevertheless, it is clearly worthwhile to study QCD effects in many different ways. In our curves for hadronic final states, we have included all particles in the calculation of the and . Ref. [1] showed, however, that better agreement between results for idealized and hadronic events could be achieved by using only particles with momentum greater than some cut, which was conveniently taken at . This device can also be used for deep inelastic scattering events.

The broadening in the distributions for hadronic events results from the transverse momenta of the hadrons from the struck quark and target fragments, as given by eq. (2.4). The target fragments, and, for two-jet events, the struck quark products should be distributed roughly symmetrically in azimuth around the incoming direction. Such systems give little contribution to the , for (see eq. (2.4)), and hence the and distributions for two-jet events are much narrower than the corresponding distributions. The and distributions for three-jet events receive little contribution from target fragmentation, so that hadron fragmentation serves only to soften the distributions, as for the . As expected, the effect is more pronounced for than for , since probes the events at smaller angular scales than , so that hadronic effects are more important. At shown in fig. 1, both and exhibit significant tails due to three-jet effects. For , 6% of events lie in the tail above , while for , there is a 2% tail at ; in each case there is negligible two-jet contamination above this cut. A still better separation of three-jet events is afforded by considering the distributions of events for which the total transverse momenta of the final particles satisfy a cut, which for the curve given in fig. 1 was taken as . (This cut includes 9% of all events.) The cut removes many of the two-jet final states; the distribution of the remaining events therefore provides a test for the dynamical mechanism of three-jet production. Azimuthally-symmetric fragmentation of a two-particle event (which violates by virtue of fragmentation transverse momentum) leads to much smaller values of than the three-jet final states expected in QCD. Before fragmentation, QCD predicts for three-jet events; unfortunately, however, this is much modified by fragmentation, and for with hadronic events in the kinematic regime of fig. 1. Some features of the shapes of events may be summarized by giving the mean values of shape parameters for them. Of course, in doing this one loses the effects of the distinctive three-jet tails (as they only correspond to a small fraction of the events) in the distributions of events in the shape parameters. This is very evident in fig. 2 where the mean shape parameters are plotted as a function of the Bjorken variable . In [1], we noted that in the free quark and gluon approximation, three-jet effects in the mean shape parameters increase slightly as x goes to one. However, this trend is entirely overwhelmed by the severity of fragmentation corrections as (so that .

[ Figure 2 ] Mean values of shape parameters in deep inelastic scattering as a function of for .

[ Figure 3 ] The (or dependence) of the distribution of shape parameters for deep inelastic scattering events at .

In fig. 3 we give the distributions of hadronic events in and for and various values of , corresponding to different hadron system energies . The growth of the three-jet tail as increases ( decreases) is very evident. The results for other values of follow closely those given in fig. 3, once is adjusted so that is the same. It appears that for , hadronic effects should be sufficiently unimportant to allow a definitive test of the QCD prediction for three-jet final states.

The effects we have discussed above should be present in all deep inelastic lepton-hadron scattering processes, and should be independent of the particular beam or target used. However, the production and decay of heavy flavors of quarks should also occur in some cases, leading to events with a more spherical shape. Typically either or will be capable of inducing direct production of heavy quarks through . In such events, the target fragments will form one jet, while the decay products of the , which will be produced mostly nearly at rest, should give rise to a rather spherical structure. Heavy quarks may also be produced in pairs through , and events of this type should be roughly spherical.

In our discussion of annihilation [1,2], we have considered in detail the several possible final states, resulting, for example, from heavy quark production and decay. It would be straightforward to supplement our qualitative statements above on heavy quark production in deep inelastic scattering by a detailed analysis analogous to that in ref. [1]. However, in this paper we will content ourselves with the study of two- and three-jet final states. Our basic conclusion is that useful tests of QCD require N c.m. energy ; rather larger than is necessary in annihilation. We hope our results will be useful to those analyzing and planning deep inelastic scattering experiments.

We are grateful to the MATHLAB group of the Massachusetts Institute of Technology Laboratory for Computer Science for the use of MACSYMA.

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