2. Rotationally-averaged Two-Detector Energy Correlations

2.1. Formalism and Calculational Techniques

In this section we discuss the rotationally invariant observable defined in (1.2). The two detectors used in this definition occupy areas which are, in general, of arbitrary shape. However for simplicity, we shall restrict ourselves in this paper to the case in which the are congruent circular patches of angular radius Hence and is a function only of and where is the angle between the centers of the two detectors The arrangement is illustrated in Fig. 1. The generalization of our treatment to detectors of arbitrary shape is straightforward. The best method of calculating from events appears to be the use of the formula

where the sums on and run over all the particles in the event (including the case ). The smearing function is given by

[ Figure 1 ] Two circular detectors of half angle is angle between the centers of the detectors. is obtained by averaging over all positions for the detectors which preserve the angle

Here is a unit vector in the direction of detector and the unit vector in the direction of particle If runs over all elements of the rotation group, so that then

where is a unit vector in the direction and a rotation through about the axis. Finally in (2.2), the function takes on the values 1 or 0, and is zero unless both the pairs and lie within an angle of each other, so that particle is incident on detector 1 and particle on detector 2. The rotational invariance of is exhibited by the fact that it has the same value for any choice of particle directions so long as the angle between them remains fixed. For some purposes, it is more convenient to write in the symmetrical form

Note that

which implies the same normalization for :

independent of the value of .

In the limit , and become point detectors separated by an angle We define (3)

In this case, becomes simply

and we may rewrite (4)

and

may be expressed in terms of the defined in (1.1) as

where the indices and run over the hadrons which are produced in the event, and is the angle between particles and When the first form for the is used, a particular set of axis must be chosen to evaluate the angles of the momenta, but the values of the deduced will be independent of the choice. The Legendre expansion of is

where the as defined in (2.11) are and correspondingly,

The relation between and is

so that (for circular detectors)

or equivalently

which clearly illustrates the symmetry between and also visible in (2.4). was defined in [1] as

Note that as but Thus the series (2.12) is always absolutely convergent whereas (2.13) diverges (for point particles).

receives a contribution from events in which the same particle passes through both detectors. Although this is of little practical interest, it must be kept if the normalization condition (2.6) is to be maintained. In the point detector limit, this configuration contributes to a term

where the coefficient of the delta function is related to the asymptotic limit of the by

2.2.

In this section, we discuss mostly the idealized point detector energy correlation : results for finite detectors may be obtained by the smearing (2.10).

All events give

Three-particle final states give rise to a distribution of forms for Taking the fractional energies of the particles to be becomes

where is determined by requiring that the angle between particles and satisfies i.e.,

The Jacobian is, therefore, simply

Hence the process [which represents the sum of and calculated to ] gives (5):

and, for example

[The form of plotted in Fig. 2 is indistinguishable from (2.24) except very close to where the smeared delta function appear (6).]

The first term in (2.24) arises from the lowest-order process The second term accounts for the three-particle final state process For this term becomes

while for the term is approximately

[ Figure 2 ] The mean value of the rotationally invariant energy correlation as a function of detector separation for fixed detector size given by (opening angle The curves given are for simulated hadronic final states at and GeV and in the approximation of free quarks and gluons. The free quark calculation marked is, in fact, just the contribution of the final state calculated with no cut and with no component added (this would contribute only at

The third term in (2.24) has two sources. First, it gives the contribution from one-loop corrections to and second, at it receives contributions of the form (2.18) from events in which one of the final particles passes through both coincident detectors. The constants both diverge logarithmically as the infrared regulator (e.g., a fictitious gluon mass ) is taken to zero. The coefficient of the delta function at is given simply by the mean of in (2.18):

where the average encompasses both and final states at The nonlinearity of the form (2.28) in the energies of the final particles means that in averaging over possible final state configurations, collinear pairs of particles will be weighted differently from single particles which carry the sum of their momenta. For this reason, the collinear divergences from and final states will not cancel in (2.28), and will diverge like as goes to zero. One finds, in fact, that and

However, the integral over of the second term in (2.24) (regularized by the introduction of a finite ) for which arises from final states, also has a logarithmic divergence in close to This divergence cancels against when is integrated over with a (non-singular) weight function, such as This is, of course, necessary in order that the obtained from using (1.4) should be infrared finite as Note that if detectors of finite area are considered, then is smeared according to (2.10); the resulting is finite at all but diverges at like [to ] as the detector size is taken to zero.

The coefficient of the delta function in the backward direction in (2.24) is given by

where is a particle exactly back-to-back with This receives contributions only from loop corrections to in fact, it is simply the total cross-section for the process (or, equivalently, the quark electromagnetic form factor) (7):

Once again, in integrals of these divergences are canceled by corresponding divergences in integrals of the contribution (2.25) to around The presence of a divergence in the individual terms around is a consequence of the fact that a final state becomes indistinguishable from if either one of the particles becomes soft or a pair of them are collinear. The introduction of a finite quark mass regularizes the collinear divergences to so that would then exhibit no divergence. However, a single log divergence would remain in which could be regularized by taking a finite gluon mass (see Footnote 7).

Divergences are present in at a given order in , at any value of for which is non-vanishing in lower orders. At gives divergences only at where for is nonvanishing. As described above, these divergences cancel against those from one-loop corrections to when is smeared (e.g., by consideration of detectors with finite area). In higher orders, exhibits compensating divergences at each value of : it must be considered a generalized function, meaningful only when smeared in This effect also afflicts the distributions in the : the are, however, genuinely infrared stable; their construction from via (1.4) effects the necessary averaging. The divergences in render it potentially more sensitive to hadronization corrections: the formation of hadrons introduces a presently unknown smearing. The distribution of events in is, as described below, extremely sensitive to unknown details of fragmentation. [The divergences which cancel in when smeared over small ranges in appear as separated divergent peaks in the distribution in .]

2.3. Heavy Resonance Decay

The decay of a heavy resonance gives

where the last term comes from events in which the same particle passes through both detectors. At one finds

Near (2.32) becomes

while at the regular part of (2.32) tends to

The differential cross-section for is barely distinguishable from uniform in three-body phase space, which would give

Near in this case becomes

This integrable divergence has its origins in the Jacobian from the differential cross-section in to that in Near the regular part of tends to When a phase-space generated final state contains more than three particles, the Jacobian divergence at visible in (2.36) [and which becomes for a two-particle final state] disappears. This occurs even for the rather constrained 6 particle final state discussed in Sect. 2.5 (2.39). becomes progressively flatter as the multiplicity of the final state increases and for a truly isotropic final state containing an infinite number of particles, [It is a general feature of QCD processes that the structure of events at the lowest contributing order in is usually well-approximated by distributing the final state particles uniformly in the available phase space (appropriate for their multiplicity). In higher orders, infrared divergences tend to concentrate further emissions into jets along the directions of the partons at lowest order, thereby approximately preserving the lowest order shape, at least on large angular scales (8).

[ Figure 3 ] A comparison at GeV of the mean value of the energy correlation for two different detector sizes given by or (corresponding to detectors of half-angle and , respectively)

2.4. and for Hadronic Two- and Three-Jet Events

In this section we estimate the energy correlations to be expected in hadronic events arising from the processes and . We also give estimates for the uncorrected process which would occur alone if To account for fragmentation of quarks and gluons into hadrons, we use the phenomenological model developed by Field and Feynman [6]. For final states, we use the somewhat ad hoc prescription introduced in [1], and used successfully in analyses of data from PETRA [7]. Our method consists in generating true three-jet events when the produced in the subprocess satisfy the cut and two jet events otherwise in such a way as to give the correct total cross-section: The cut on represents the resolution of the hadron final state to changes in the subprocess final state: the hadrons do not reflect the presence of the extra gluon if its transverse momentum is too small.

[ Figure 4 ] The ratio of the mean energy correlation between point detectors for hadrons produced by the processes and at GeV. Note that at this energy, our prescription for treating fragmentation takes 65 % of final states to contain two jets and, therefore, to evolve roughly like final states

[ Figure 5 ] The distributions calculated for various separations of two detectors with size resulting from the processes and Calculations in the free quark and gluon approximation are shown as well as those obtained from an estimate of the effects of fragmentation to hadrons at and GeV. In the free quark approximation gives a delta function contribution, smeared to angles by the finite detector sizes. The curve marked in the free quark and gluon approximation is with an cut but without the (delta function) term added

In Figs. 2 and 3 we show estimates of with ( detectors) and ( detectors) for hadronic events, and in the free parton approximation. At sufficiently high energies, hadronization effects must disappear, but Figs. 2 and 3 show that they are still present in the PETRA, PEP energy range GeV. The effects are rather less severe for the discussed in [1]. The results for hadronic events in Fig. 3 with are slightly closer to the free parton approximation than those with : the small difference indicates that hadronization effects a smearing over still larger angles. When becomes very small, however, the smearing is so great that little information on the events remains in . In fact, one must then introduce weight functions into the smearing thereby reducing to a calculation of the

In the free quark and gluon approximation, the for events is symmetrical under Fragmentation destroys this symmetry. (Near probes energy correlations within a single jet, while near it receives contributions only from pairs of particles in opposite jets.) The process also exhibits considerable violation of symmetry. This violation is conveniently measured by

In Figs. 5 and 6 we present calculations for the distributions These are rather disappointing. In our study of the [1], we found that the distributions (at least for and 3) were not seriously affected by hadron fragmentation and provided very distinctive tests of the basic dynamics. In the case of the free quark and gluon calculations show striking structure (see, for example, the lower right-hand graph in Fig. 5); however, hadron fragmentation is a huge effect even for the case shown in Fig. 6. It is worth remembering here that whereas knowledge of (as a function of ) and (as a function of ) are essentially equivalent, the distributions and contain inequivalent information. It appears that the shape information contained in is less sensitive to hadron fragmentation than that in The figures include the case corresponding to two detectors at right angles. As expected, two-jet events give a distribution in sharply peaked at whereas 3 jet final states give a broader distribution. On comparing and one sees that the former gives many more events with large values of This behavior is qualitatively as expected and should be preserved regardless of how one treats the hadron fragmentation. Note that the extreme sensitivity of to fragmentation is to be expected because of the infrared instability of The inadequacy of compared to is in strong support of the relevance of the criterion of infra-stability for successful shape parameters.

[ Figure 6 ] for in the free gluon approximation, and using a phenomenological model of hadron formation, for various separations of two detectors with size (corresponding to half-angles of ), to be compared with the results in Fig. 5 obtained with

2.5. Heavy Quark and Lepton Production Events

In [1] we discussed various possible mechanisms for the decay of heavy mesons containing heavy quarks :

We consider only energies just above threshold for production (so that are almost exclusively produced). Then 2-jet decays to free partons give

Three-parton decays yield

so that

At (2.39) becomes 7/2, while the regular part goes to 19/35 as

Figure 7 shows the for threshold production and decay in the free parton approximation, and with estimates of hadronization corrections. The tend to be close to the isotropic limit We also give results for production of heavy lepton pairs at threshold decaying to in this case, we divide for each event by to compensate for the missing neutrino energy.

[ Figure 7 ] The mean energy correlation function for events containing heavy quark or lepton production with various mechanisms for heavy meson decay. For comparison, we show results for the continuum reaction from Fig. 2

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