Matching to four-parton events

The shower routine, as described above, is optimized for two objects forming the showering system, within which energy and momentum should be conserved. However, occasionally more than two initial objects are given, e.g. if one would like to consider the subclass of $\mathrm{e}^+\mathrm{e}^-\to \mathrm{q}\overline{\mathrm{q}}\mathrm{g}\mathrm{g}$ events in order to study angular correlations as a test of the coupling structure of QCD. Such events are generated in the showering of normal $\mathrm{e}^+\mathrm{e}^-\to \mathrm{q}\overline{\mathrm{q}}$ events, but not with high efficiency within desired cuts, and not with the full angular structure included in the shower. Therefore four-parton matrix elements may be the required starting point but, in order to `dress up' these partons, one nevertheless wishes to add shower emission. A possibility to start from three partons has existed since long, but only with [And98a] was an approach for four parton introduced, and with the possibility to generalize to more partons, although this latter work has not yet been done.

The basic idea is to cast the output of matrix element generators in the form of a parton-shower history, that then can be used as input for a complete parton shower. In the shower, that normally would be allowed to develop at random, some branchings are now fixed to their matrix-element values while the others are still allowed to evolve in the normal shower fashion. The preceding history of the event is also in these random branchings then reflected e.g. in terms of kinematical or dynamical (e.g. angular ordering) constraints.

Consider e.g. the $\mathrm{q}\overline{\mathrm{q}}\mathrm{g}\mathrm{g}$ case. The matrix-element expression contains contributions from five graphs, and from interferences between them. The five graphs can also be read as five possible parton-shower histories for arriving at the same four-parton state, but here without the possibility of including interferences. The relative probability for each of these possible shower histories can be obtained from the rules of shower branchings. For example, the relative probability for the history where $\mathrm{e}^+\mathrm{e}^- \to \mathrm{q}(1) \overline{\mathrm{q}}(2)$, followed by $\mathrm{q}(1) \to \mathrm{q}(3)\mathrm{g}(4)$ and $\mathrm{g}(4) \to \mathrm{g}(5)\mathrm{g}(6)$, is given by:

$${\cal P} = {\cal P}_{1\rightarrow 34} {\cal P}_{4\rightarrow... ..._{4}^{2}} 3 \frac{(1-z_{56}(1-z_{56}))^{2}}{z_{56}(1-z_{56})}$$ (182)

where the probability for each branching contains the mass singularity, the colour factor and the momentum splitting kernel. The masses are given by

$$m_{1}^{2} = p_{1}^{2} = (p_{3}+p_{5}+p_{6})^{2}~,$$ (183)

$$m_{4}^{2} = p_{4}^{2} = (p_{5}+p_{6})^{2}~,$$

and the z values by

$$z_{bc} = z_{a \to bc} = \frac{ m^{2}_{a} }{ \lambda }\frac{ E_{b} }{ E_{a} } - \frac{m^{2}_{a} - \lambda + m^{2}_{b} - m^{2}_{c}}{2\lambda}$$ (184)

$$\mathrm{with~~} \lambda = \sqrt{(m^{2}_{a} - m^{2}_{b} - m^{2}_{c})^{2} - 4m^{2}_{b}\, m^{2}_{c}}~.$$

We here assume that the on-shell mass of quarks can be neglected. The form of the probability then matches the expression used in the parton-shower algorithm.

Variants on the above probabilities are imaginable. For instance, in the spirit of the matrix-element approach we have assumed a common $\alpha_{\mathrm{s}}$ for all graphs, which thus need not be shown, whereas the parton-shower language normally assumes $\alpha_{\mathrm{s}}$ to be a function of the transverse momentum of each branching. One could also include information on azimuthal anisotropies.

The relative probability ${\cal P}$ for each of the five possible parton-shower histories can be used to select one of the possibilities at random. (A less appealing alternative would be a `winner takes all' strategy, i.e. selecting the configuration with the largest ${\cal P}$.) The selection fixes the values of the $m$, $z$ and $\varphi$ at two vertices. The azimuthal angle $\varphi$ is defined by the daughter parton orientation around the mother direction. When the conventional parton-shower algorithm is executed, these values are then forced on the otherwise random evolution. This forcing cannot be exact for the $z$ values, since the final partons given by the matrix elements are on the mass shell, while the corresponding partons in the parton shower might be virtual and branch further. The shift between the wanted and the obtained $z$ values are rather small, very seldom above $10^{-6}$. More significant are the changes of the opening angle between two daughters: when daughters originally assumed massless are given a mass the angle between them tends to be reduced. This shift has a non-negligible tail even above 0.1 radians. The `narrowing' of jets by this mechanism is compensated by the broadening caused by the decay of the massive daughters, and thus overall effects are not so dramatic.

All other branchings of the parton shower are selected at random according to the standard evolution scheme. There is an upper limit on the non-forced masses from internal logic, however. For instance, for four-parton matrix elements, the singular regions are typically avoided with a cut $y > 0.01$, where $y$ is the square of the minimal scaled invariant mass between any pair of partons. Larger $y$ values could be used for some purposes, while smaller ones give so large four-jet rates that the need to include Sudakov form factors can no longer be neglected. The $y > 0.01$ cut roughly corresponds to $m > 9$ GeV at LEP 1 energies, so the hybrid approach must allow branchings at least below 9 GeV in order to account for the emission missing from the matrix-element part. Since no 5-parton emission is generated by the second-order matrix elements, one could also allow a threshold higher than 9 GeV in order to account for this potential emission. However, if any such mass is larger than one of the forced masses, the result would be a different history than the chosen one, and one would risk some doublecounting issues. So, as an alternative, one could set the minimum invariant mass between any of the four original partons as the maximum scale of the subsequent shower evolution.