Decays of Onia Resonances

Many different possibilities are open for the decay of heavy $J^{PC} = 1^{--}$ onia resonances. Of special interest are the decays into three gluons or two gluons plus a photon, since these offer unique possibilities to study a `pure sample' of gluon jets. A routine for this purpose is included in the program. It was written at a time where the expectations were to find toponium at PETRA energies. Given the large value of the top mass, weak decays dominate, to the extent that the top quark decays weakly even before a bound toponium state is formed, and thus the routine will be of no use for top. The charm system, on the other hand, is far too low in mass for a jet language to be of any use. The only application is therefore likely to be for $\Upsilon$, which unfortunately also is on the low side in mass.

The matrix element for $\mathrm{q}\overline{\mathrm{q}}\to \mathrm{g}\mathrm{g}\mathrm{g}$ is (in lowest order) [Kol78]

$$\frac{1}{\sigma_{\mathrm{g}\mathrm{g}\mathrm{g}}} \frac{\d\... ...\right)^2 + \left( \frac{1-x_3}{x_1 x_2} \right)^2 \right\} ~,$$ (41)

where, as before, $x_i = 2 E_i / E_{\mathrm{cm}}$ in the c.m. frame of the event. This is a well-defined expression, without the kind of singularities encountered in the $\mathrm{q}\overline{\mathrm{q}}\mathrm{g}$ matrix elements. In principle, no cuts at all would be necessary, but for reasons of numerical simplicity we implement a $y$ cut as for continuum jet production, with all events not fulfilling this cut considered as (effective) $\mathrm{g}\mathrm{g}$ events. For $\mathrm{g}\mathrm{g}\mathrm{g}$ events, each $\mathrm{g}\mathrm{g}$ invariant mass is required to be at least 2 GeV.

Another process is $\mathrm{q}\overline{\mathrm{q}}\to \gamma \mathrm{g}\mathrm{g}$, obtained by replacing a gluon in $\mathrm{q}\overline{\mathrm{q}}\to \mathrm{g}\mathrm{g}\mathrm{g}$ by a photon. This process has the same normalized cross section as the one above, if e.g. $x_1$ is taken to refer to the photon. The relative rate is [Kol78]

$$\frac{\sigma_{\gamma \mathrm{g}\mathrm{g}}}{\sigma_{\mathrm{... ...m{q}}^2 \, \alpha_{\mathrm{em}}} {\alpha_{\mathrm{s}}(Q^2)} ~.$$ (42)

Here $e_{\mathrm{q}}$ is the charge of the heavy quark, and the scale in $\alpha_{\mathrm{s}}$ has been chosen as the mass of the onium state. If the mass of the recoiling $\mathrm{g}\mathrm{g}$ system is lower than some cut-off (by default 2 GeV), the event is rejected.

In the present implementation the angular orientation of the $\mathrm{g}\mathrm{g}\mathrm{g}$ and $\gamma \mathrm{g}\mathrm{g}$ events is given for the $\mathrm{e}^+\mathrm{e}^-\to \gamma^* \to$ onium case [Kol78] (optionally with beam polarization effects included), i.e. weak effects have not been included, since they are negligible at around 10 GeV.

It is possible to start a perturbative shower evolution from either of the two states above. However, for $\Upsilon$ the phase space for additional evolution is so constrained that not much is to be gained from that. We therefore do not recommend this possibility. The shower generation machinery, when starting up from a $\gamma \mathrm{g}\mathrm{g}$ configuration, is constructed such that the photon energy is not changed. This means that there is currently no possibility to use showers to bring the theoretical photon spectrum in better agreement with the experimental one.

In string fragmentation language, a $\mathrm{g}\mathrm{g}\mathrm{g}$ state corresponds to a closed string triangle with the three gluons at the corners. As the partons move apart from a common origin, the string triangle expands. Since the photon does not take part in the fragmentation, the $\gamma \mathrm{g}\mathrm{g}$ state corresponds to a double string running between the two gluons.