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Articles Particle Physics Lepton Energy Spectra in e+ e- Annihilation and Other Processes (1977)
LEPTON ENERGY SPECTRA IN e+ e- ANNIHILATION AND OTHER PROCESSES (1977) |
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We shall use the following rest frame spectrum for the decay 
(1)
where
This is the form of the spectrum for 
, with 
(and also for 
with 
, 
and a 
coupling). It is somewhat more sharply peaked than 3-body phase space and might work reasonably well for multibody states with the invariant mass 
held fixed in the spirit of an isobar model (e.g. it is a reasonable approximation to the spectrum expected in 
for 
[7,8]; possible spectra are discussed further in refs. [3, 7--9]. will be kept as a free parameter here.
We assume throughout that the transverse momentum in the fragmentation 
is negligible, as must be the case in the fragmentation of non-strange quarks 
if this picture is to describe correctly the jets observed at SPEAR [10] (we expect that 
and so neglect of 
should be a good first approximation). Further we assume that the probability of finding 
in the range to 
is a function 
of 
. (2) Some previous authors have assumed a 
singularity in 
as 
[9,11,12]. We believe that such a term should have a negligibly small coefficient since it represents the associated production of 
pairs in the central plateau of rapidity (
) with an implied charmed particle multiplicity growing like 
. In models with short-range order this processes should be independent of the flavor of the fragmenting quark; the evident paucity of charmed particle production in pp collisions then suggests that it is negligible. In any case consistency demands that we neglect it since we have neglected associated production in events in which non-charmed quarks are produced at the primary photon vertex. With one D per c the function must be normalized to one
We adopt the ansatz
where we expect 
or 2 from theoretical considerations. We will investigate the sensitivity to variations in 
. At asymptotic energies we should clearly put 
(in the c.m. system) and we expect that 
as 
where inclusive production turns into exclusive production (suggesting for a smooth exclusive-inclusive connection [13] with monopole form factors and 
from more detailed theoretical considerations [14]). We favor which appears to be supported by data on inclusive hadron production at large momenta at SPEAR [15].
If we wish to extend the model to subasymptotic energies we are then forced to put 
so that with 
, always corresponds to the exclusive processes and the vanishing of in this limit make sense. (With other prescriptions there is the danger of excluding 
and violating the normalization condition.) The model then correctly yields D meson production at rest at threshold and is, we believe, likely to work reasonably well in an average sense at all energies. In our view the 
production cross section will interpolate smoothly through any s-channel structure in 
down to low energies (though it does not have the threshold behavior of 
). Thus we assume
taking 
for the charmed quark and including a factor 3 for color (in fact we are primarily interested in spectra in which multiplicative functions of 
play no role but we wish to include our best guess for completeness; obviously this formula and our ansatz for are subject to direct verification if 
is measured via hadronic decays). For lepton production
where
and the quark-to-lepton fragmentation function 
is given by
with
The 
integral is easy to do analytically with
our
choice of 
but we have done the 
integral numerically except in the 
limit where we obtain
The units are such that 
and 
with 
. 
is the dilogarithm function as defined in ref. [16], where tables of its values may be found. With 
, this formula reduces to the simple (but probably misleading) form
which may be useful for exploratory purposes.
