3. Results for e+e- Annihilation

We now present results for various beam energies as a function of our two parameters (eq. (1)) and (eq. (4)) for . For comparison we also give results for a heavy lepton decay for couplings with unless stated otherwise and GeV; the case is the same as D decay in our model with and . For completeness we give the analytic forms of the spectra in the heavy lepton case (see also ref. [17])

where (in units )

In fig. 1 we show results at GeV for which illustrate the sensitivity of the spectrum to . The similarity of the heavy lepton and the charm curves shows the extreme insensitivity to near threshold. The fact that gives a slightly softer spectrum than is easy to understand since the configuration in which takes its maximum value is forbidden by angular momentum conservation in the former case.

[ Figure 1 ] Lepton energy spectra for beam energy GeV as a function of for leptons derived from charmed mesons with mass 1.87 GeV (full lines) for various (eq. (2)) and (eq. (4)) and for heavy leptons of mass 1.87 GeV with (dotted) and (dot-dashed) couplings.

[ Figure 2 ] As for fig. 1, but with GeV.

Fig. 2 is for GeV and ; note that the marked difference between the heavy lepton and charm induced spectra is very insensitive to at high energy. Fig. 3, which is for GeV and GeV, shows that this separation is also insensitive to . Fig. 4 illustrates the comparative insensitivity of the spectrum in heavy lepton decay to the neutrino mass in the high-energy limit.

[ Figure 3 ] As for fig. 2, with GeV and various (eq. (4)).

[ Figure 4 ] Heavy lepton decay spectra in the high-energy limit for various values of .

In figs. 5--9 we specialize to the case which we favor slightly and GeV, or GeV. (A value of this order is favored by the soft spectra seen at DORIS near threshold.) In addition we include the effects of a new flavor for GeV which we describe by the same model with the parameters arbitrarily given the values GeV, GeV. Although a new flavor seems to stand out near threshold (fig. 6), it gradually merges with the charm result as the beam energy increases (figs. 7, 8). An increase of would make this distinction between charm and a new flavor less clear. In addition it must be emphasized that all our spectra have the same normalization; in reality charmed quarks are expected to be produced as copiously as heavy leptons and charge quarks only as copiously. Furthermore the semileptonic branching ratio may be less for charmed particles than for heavy leptons.

[ Figure 5 ] Spectra at GeV for compared to the heavy lepton signal with .

[ Figure 6 ] Spectra at GeV for including a new flavor with GeV and GeV.

[ Figure 7 ] As for fig. 6 but with GeV.

[ Figure 8 ] As for fig. 6 but with GeV.

[ Figure 9 ] The angular distribution coefficient defined in eq. (16) as a function of .

A quantitative measure of the difference between the spectra in the charm and heavy lepton cases is also interesting. In the decay of any unpolarized particle . The quantity is Lorentz invariant since in the frame in which . Hence

With our model for the spectrum (eq. (2))

where . The approximation slightly overestimates the exact result, but nowhere by more than 0.02. The same formula works for heavy lepton decays in the case with a suitable transposition of symbols; in the case if . Using our function

Therefore in the high-energy limit

It is clear from the figure that it is not only the average energy which is much less in the charm case; the spectrum also cuts off much more sharply. Using eq. (10) and the high-energy limit of eq. (11) it is easy to calculate the fraction of the cross section contributed by in the asymptotic limit when and (which leads to an overestimation in the charm case). The results are shown in table 1.

[ Table 1 ] Fraction of events with in the high-energy limit with and .

We conclude that excellent electron and muon identification at low momenta will be very desirable at PETRA and PEP.

Next we consider the angular distribution of the leptons relative to the beam axis. We start from the observation of Pais and Treiman [18] that for a parent D meson of known energy and angular distribution

the integrated angular distribution of the daughter lepton is completely determined if its mass is neglected:

where . If the primary c quarks have a distribution, the D meson spectrum will be slightly different due to the non-zero in the fragmentation ( is a satisfactory approximation to the jet model result [19]). However, this is only important at low for small where the form of the quark distribution itself might be expected to differ from due to some finite effective quark mass. In conformity with our recipe for using the fragmentation model near threshold, which appears to imply an effective quark mass , we set

(thus continuing to neglect ). This form has the behavior we expect as and but as we do not believe it in detail we will only present results for GeV, which do not depend sensitively on . We integrate over the D momentum to obtain results for as defined by

In all cases the soft spectrum causes a dramatic smearing relative to the angular distribution expected for heavy leptons (fig. 9). However, these results only apply after integration over all lepton energies. With a minimum cut on lepton energy will be larger.

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