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EVENT SHAPES IN e+e- ANNIHILATION (1979) |
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(1) This is similar to the `rotation function' used in crystallography [7].
(2) As discussed in subsect. 2.4, one must, in practice, know the values of the 
up to rather high 
in order to obtain a good approximation to 
. Note that the mean has been computed directly for the process 
(G) in ref. [28], and all the processes discussed in this paper in ref. [29]. In these references, the form of 
for detectors at a fixed angle to the beam axis have also been calculated. The resulting correlation functions may be expressed as a Legendre expansion analogous to (2.10), in terms of the non-rotationally-invariant observables
(3) The simplest example of this class of observables is
where 
is a unit vector along the momentum 
of particle 
.
(4) Divergences in differential cross sections are typically of the form 
, where 
is the transverse momentum between two of the final quarks or gluons. Cancellations occur between processes in which is strictly zero and those in which it is small. The difference of the between these two configurations is O
, and so gives no infrared divergences when the integral over is performed.
where the 
are the momenta of the particles transverse to the jet axis which is defined by demanding that the values of the variables be minimal. See ref. [6] for discussions of other variables.
(6) The 
dependence of the fragmentation is partly accounted for by our inclusion of O
processes. The remaining dependence has contributions both at 
(whose form depends on the separation of two- and three-jet events) and from higher-order processes.
(7) Note that although 
can be obtained from by smearing in 
, the distributions of events in and 
are not related in any simple way.
(8) Note that, for simplicity, we always take the initial 
and 
to be unpolarized.
(9) Note, however, that the 
which appears in this formula may well differ considerable from the values deduced from measurements of other processes, or from the `true' [12].
(10) Evaluation of 
for three-particle final states requires integration over the two phase-space variables 
and 
. The integrals over may always be done in terms of elementary functions, but the integration over always involves dilogarithms [15]. Integration by parts reduces the dilogarithm integrals to the canonical forms [16]
This general pattern persists, and, for example,
(12) The importance of such a mechanism is governed by the difference of the masses of the possible quarks in the loop. It may well be dominant in strange particle decays, but is probably unimportant in charm particle decays, since it is inevitably suppressed by 
relative to the processes of figs. 24 and 26, and leads to non-strange final states in 
meson decay. It could, therefore, account for the `non-leptonic enhancement' observed in strange, but not charm, particle decays. In addition to the process of fig. 25, Q
mesons might decay to GG by exchanging a 
just before the annihilation of the Q and . This would be suppressed relative to Q
qG by 
, and is therefore probably safely ignored.
(13) The renormalization group equation allows a summation of the leading logarithmic QCD corrections to weak decay rates to all orders in . However, the structure of the final states is governed by emissions at large angles, which are best estimated by explicit calculations.
(14) See eq. (2.21) for a more precise result in this case.
(15) Note that the arguments for infrared stability of processes involving only quarks and gluons discussed in sect. 3 do not apply when only charged particles or particles with momenta above some lower cut-off 
are considered. Cuts on hadronic final states cannot, however, be related directly to those on their quark-gluon parent states.
(16) In this calculation we used the differential cross section for 
G [3] (
, 
):
Integration over angles gives a term identical to the total cross section for 
(multiplied by 
) plus the term given as the integrand in eq. (6.12).
