1. Introduction

According to the original parton model, the hadron final state of a deep inelastic lepton-hadron scattering event should consist of two jets: one initiated by the struck quark and the other arising from the fragments of the target nucleon from which the quark was ejected. In QCD, three-jet events should also occur, in which, for example, the outgoing quark radiates a gluon with high transverse momentum, thereby forming a third jet of hadrons. The investigation of such processes provides a probe of QCD perturbation theory beyond the leading logarithmic approximation so far used in analysis of the total deep inelastic cross section. In this paper, we describe a method for analyzing the structure of hadron final states in deep inelastic scattering, which should (at least at energies ) allow a precise test of the QCD prediction for three-jet final states to be made. To characterize the distributions of hadronic energy or `shapes' of events we use the observables , originally introduced for the analysis of annihilation final states [1], together with the , which are two-dimensional analogues of the , as discussed in [1,2]. For small , these observables probe only the gross structure of the final state, and are not sensitive to details of the hadrons in the final state, which were presumably formed at times of order where perturbative methods are irrelevant. The formal consequence of this insensitivity is that the and are infrared stable when computed in QCD perturbation theory, so that divergences which appear in their calculation may be controlled. These points, with some mention of deep inelastic scattering, are discussed at length in [1,2].

We define

where the sums on and run over all particles in the final state (except, of course, the scattered lepton), and is the total energy available for the formation of hadrons, given in deep inelastic scattering, by

where, as usual, the Bjorken variable is defined as

is the invariant mass and the energy of the intermediate virtual photon or (usually referred to simply as ). The values of the for an event depend on the frame in which the particle momenta are evaluated; we shall throughout consider only the -nucleon center of mass frame. Any other choice of frame must ultimately yield the same physical results. From the definition (1.1) one sees that (for massless final particles) energy conservation implies , when the sum is performed over the complete final state. Experimentally, some particles will presumably go undetected: this may roughly be compensated by dividing all the . found using the observed particles by [1]. (An analogous procedure may be used for the defined below.)

The describe the distribution of energy on a hypothetical sphere surrounding a deep inelastic scattering event. The fact that they do not single out any direction in space is very convenient for the analysis of annihilation events where no natural axis is defined in the final state. However, in deep inelastic scattering, the momentum of the exchanged (or ) provides a natural direction, and one may consider the distribution of hadron energy around this direction, as characterized by

where the particle has azimuthal angle and magnitude of momentum projected on the plane perpendicular to the direction. therefore gives simply the square of the total transverse momentum of the final hadrons with respect to the direction. The higher-order provide a complete description of the transverse momentum distribution. Note that by momentum conservation, but need not vanish.

Further details on the structure of events in deep inelastic scattering are afforded by consideration of the angular distribution of hadronic energy with respect to the incoming lepton direction. This is parametrized by the observables [1]

where is the azimuthal angle defined by the projection of the incoming lepton direction onto the plane orthogonal to the .

For spin-1 exchanges between the lepton and hadron systems, for , and in all cases . The , provide an infrared stable formulation of the tests of QCD originally suggested in [3]. In terms of the , (and making the choice

where in the last case, we have suppressed the infinite series of higher-order , which must vanish when averaged over events. In annihilation, the observables which are analogous to the for that case were of rather limited value, mainly because they suffered particularly severely from hadron fragmentation effects. We suspect that the will be beset by similar difficulties, and we shall not discuss them further here.

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