Notes

(1) The background and motivation of the present work is discussed in [1]. As in [1] represents the sum of processes and so on, together with virtual (loop) corrections to these. denotes a heavy quark bound state (such as or ). Another notation from [1] is the kinematic definition

(2) The material in this paper consists mainly of Sects. 4-6 of the preprint [2]

(3) In [1], was called simply

(4) In practical calculations, ) can be found easily from (2.9), and then (2.10) can be used to obtain Alternatively, one may use the rapidly convergent series (2.12). The latter method has the advantage that it also allows the calculation of the distribution It is clear that for both means and distributions, the are somewhat easier to extract from events than the energy correlation

(5) It is sometimes convenient to approximate for by the value obtained by performing the sum (2.21) only over the and and neglecting contributions from the gluon. In this case,

so that

This is much smaller than the result (2.25) for the complete final state since the and are rarely produced at Near the contribution becomes

the leading term here is the same as in (2.26) for the complete case, the subleading terms differ. For calculated using only the and has no divergence and its regular part tends simply to

(6) In our phenomenological prescription for treating hadronic final states, we only consider final states which have The form of for when is qualitatively similar to the result (2.24) but there is a slight suppression near

(7) Note that the form depends critically on the method of infrared regularization. If, instead of retaining a finite gluon mass, we had kept the quark off shell by an amount then the kinematic limits change, and roughly is replaced by so that the form of (3.9) is modified, becoming

For on-shell fermions of mass a finite must be retained to regularize soft divergences and

The double logarithmic terms in, for example, Eq. (2.31) can be summed to all orders in to obtain a leading log estimate for the quark form factor. The estimate will be dual (as by the usual inclusive-exclusive connection) to results for quark fragmentation functions close to The dependence of the form factor on the infrared regularization procedure will be manifest in the various ways in which the limit is taken for the fragmentation function.

(8) The for a process are therefore typically well approximated by distributing the partons in the lowest-order final state uniformly in phase space. For two-particle final states, only one point in phase space is, of course, allowed and, as usual, for odd (even). [The processes usually lead to two jets and, therefore, roughly preserve the lowest-order results for the However, as increases, the become progressively more sensitive to the detailed structure of the events and probe the internal constitution of the jets so that the lowest order structure is lost.] For three-particle events, a phase space distribution gives

Note the extreme similarly between these results and those for at lowest order In higher orders of the for large again deviate significantly from the lowest-order results or from the phase space approximation to them. For an particle final state distributed uniformly in phase space, the are approximately so that as the usual result for a genuinely isotropic system is regained

(9) For heavy meson pair production near threshold, the spinless nature of the mesons prevents any angular dependence of energy correlations, but for spin--- heavy lepton pair production, there should be a definite non-trivial angular dependence

(10) relates this to previous description. In (3.9) one can use either choice as allowed transformation functions are independent of a sign change for and

(11) For colored scalar quarks, but vector gluons, this result becomes

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