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EVENT SHAPES IN e+e- ANNIHILATION (1979) |
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In this section, we discuss the use of the 
in the analysis of three-jet effects in deep inelastic lepton-nucleon (
, 
and 
) interactions. We work mainly in the rest frame of the incoming virtual photon (or 
) and the target nucleon. The 
(or 
) is taken to have an energy 
in the rest frame of the target nucleon, and an invariant mass 
. Bjorken's 
variable is defined as 
.
The lowest-order process in deep inelastic lepton-nucleon scattering is the absorption of the virtual photon by a quark in the proton. The final state will contain two jets of hadrons; one arising from the quark and one from the fragmentation of the target nucleon. If the remnants of the target nucleon are treated as a single point particle then two-jet final states from deep inelastic scattering give the same values of the as two-jet final states in 
annihilation. (The are normalized so that 
.) A cut may be made on the hadrons in the final state which ensures that few of those considered were fragments of the target, rather than of the produced quark. The requirement that all hadrons considered have momenta directed into the hemisphere ahead of the virtual photon (or ) direction (that is, all have 
) seems sufficient to veto most target fragments. In this case, only the fragments of the outgoing quark should contribute to the values of the for the final state. Ignoring the fragmentation of the quark into hadrons, one finds that in this case all the are simply equal to 
. Three-jet final states (which arise from the processes q
qG and 
) will give a distribution of values of . Unfortunately, the cut on final particles may not lead to infrared stable results, so that it may not be possible to make reliable predictions for the distributions in this case.
A closer analogy to annihilation is obtained if all the particles in the final state are considered. Let us assume that the fragments of the target may be treated as a single massless particle (perhaps akin to a `diquark'). Then in the free massless quark approximation, the for two-jet final states will be the same as for 
final states in annihilation; 
for even 
and 
for odd . This result holds only in the 
rest frame. Note that because the are rotationally invariant, their values are unaffected by the transverse momentum of the incoming quark or gluon with respect to its parent nucleon. Three-jet final states will again give a distribution of values of . The distributions are qualitatively similar to those found in sects. 4 and 5 for annihilation. The details of the distributions depend on the momenta of the incoming and outgoing leptons (i.e., on Q
, and which of 
, 
or 
is probed). The distributions also depend on the momentum distributions of quarks and gluons in the nucleon. As tends to one, the distributions tend to a non-trivial limit with means that are typically,
very nearly equal to the 
(G) values quoted in table 6. Such a non-zero limit should be contrasted with other QCD effects in deep inelastic scattering (for example, the 
contribution to 
which vanish like 
as 
. For small values of , the distributions become more peaked towards the two-jet values, and, for example, at 
,
The dominant contribution to these results comes from the process 
(G); G
is insignificant. Only transverse photons give significant contributions to the 
.
In deep inelastic scattering, the question of infrared stability is more complicated than in annihilation because the initial quark state is no longer a color singlet. As expected from the KLN theorem, infrared finite results are obtained only if all processes at are added, including those containing extra particles in the initial state (e.g., Gqq). At , only 
qG and can give 3-jet final states. For odd , the receive contributions only from 3-jet final states, and so their values are unaffected by the existence of many possible two-jet processes, while for even the contain an extra term which gives the total cross section upon integration over all possible final states. When this result is divided by the total cross section in order to obtain 
, the extra term becomes simply 
. Thus, to order 
a knowledge of the total cross section is not necessary to obtain the .
In the results quoted above, we have assumed idealized jets, and have not included fragmentation into hadrons. This can be treated by the methods described in sect. 5. Note that the effects of hadron fragmentation will be characterized by the 
c.m energy 
. As mentioned above, the distributions for three-jet events become more distinct from the two-jet limit as increases, but on the other hand, the smearing effects of fragmentation to hadrons will become smaller as decreases (
increases). The optimum value of at which to observe three-jet processes should, therefore, be determined by a compromise between these two effects.
In sect. 8, we shall present an alternative method for analyzing event shapes in deep inelastic scattering, which involves projecting the momenta in an event onto a particular plane. Although this technique has certain advantages, it does not retain the rotational invariance of the and the resulting insensitivity to transverse momentum of the incoming quark and gluon. We shall discuss both techniques in more detail in a later publication.
Energy correlation functions (see subsect. 2.6) may also be useful in studies of deep inelastic scattering events.
