The matrix-element event generator scheme

The program contains parameterizations, separately, of the total first-order 3-jet rate, the total second-order 3-jet rate, and the total 4-jet rate, all as functions of $y$ (with $\alpha_{\mathrm{s}}$ as a separate prefactor). These parameterizations have been obtained as follows:

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The first-order 3-jet matrix element is almost analytically integrable; some small finite pieces were obtained by a truncated series expansion of the relevant integrand.

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The GKS second-order 3-jet matrix elements were integrated for 40 different $y$-cut values, evenly distributed in $\ln y$ between a smallest value $y = 0.001$ and the kinematical limit $y = 1/3$. For each $y$ value, 250000 phase-space points were generated, evenly in $\d\ln (1-x_i) = \d x_i/(1-x_i)$, $i = 1,2$, and the second-order 3-jet rate in the point evaluated. The properly normalized sum of weights in each of the 40 $y$ points were then fitted to a polynomial in $\ln(y^{-1}-2)$. For the ERT(Zhu) matrix elements the parameterizations in eq. () were used to perform a corresponding Monte Carlo integration for the five $y$ values available.

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The 4-jet rate was integrated numerically, separately for $\mathrm{q}\overline{\mathrm{q}}\mathrm{g}\mathrm{g}$ and $\mathrm{q}\overline{\mathrm{q}}\mathrm{q}' \overline{\mathrm{q}}'$ events, by generating large samples of 4-jet phase-space points within the boundary $y = 0.001$. Each point was classified according to the actual minimum $y$ between any two partons. The same events could then be used to update the summed weights for 40 different counters, corresponding to $y$ values evenly distributed in $\ln y$ between $y = 0.001$ and the kinematical limit $y = 1/6$. In fact, since the weight sums for large $y$ values only received contributions from few phase-space points, extra (smaller) subsamples of events were generated with larger $y$ cuts. The summed weights, properly normalized, were then parameterized in terms of polynomials in $\ln(y^{-1} - 5)$. Since it turned out to be difficult to obtain one single good fit over the whole range of $y$ values, different parameterizations are used above and below $y=0.018$. As originally given, the $\mathrm{q}\overline{\mathrm{q}}\mathrm{q}' \overline{\mathrm{q}}'$ parameterization only took into account four $\mathrm{q}'$ flavours, i.e. secondary $\b\overline{\mathrm{b}}$ pairs were not generated, but this has been corrected for LEP.

In the generation stage, each event is treated on its own, which means that the $\alpha_{\mathrm{s}}$ and $y$ values may be allowed to vary from event to event. The main steps are the following.

48.

The $y$ value to be used in the current event is determined. If possible, this is the value given by you, but additional constraints exist from the validity of the parameterizations ($y \geq 0.001$ for GKS, $0.01 \leq y \leq 0.05$ for ERT(Zhu)) and an extra (user-modifiable) requirement of a minimum absolute invariant mass between jets (which translates into varying $y$ cuts due to the effects of initial-state QED radiation).

49.

The $\alpha_{\mathrm{s}}$ value is calculated.

50.

For the $y$ and $\alpha_{\mathrm{s}}$ values given, the relative two/three/four-jet composition is determined. This is achieved by using the parameterized functions of $y$ for 3- and 4-jet rates, multiplied by the relevant number of factors of $\alpha_{\mathrm{s}}$. In ERT(Zhu), where the second-order 3-jet rate is available only at a few $y$ values, intermediate results are obtained by linear interpolation in the ratio of second-order to first-order 3-jet rates. The 3-jet and 4-jet rates are normalized to the analytically known second-order total event rate, i.e. divided by $R_{\mathrm{QCD}}$ of eq. (). Finally, the 2-jet rate is obtained by conservation of total probability.

51.

If the combination of $y$ and $\alpha_{\mathrm{s}}$ values is such that the total 3- plus 4-jet fraction is larger than unity, i.e. the remainder 2-jet fraction negative, the $y$-cut value is raised (for that event), and the process is started over at point 3.

52.

The choice is made between generating a 2-, 3- or 4-jet event, according to the relative probabilities.

53.

For the generation of 4-jets, it is first necessary to make a choice between $\mathrm{q}\overline{\mathrm{q}}\mathrm{g}\mathrm{g}$ and $\mathrm{q}\overline{\mathrm{q}}\mathrm{q}' \overline{\mathrm{q}}'$ events, according to the relative (parameterized) total cross sections. A phase-space point is then selected, and the differential cross section at this point is evaluated and compared with a parameterized maximum weight. If the phase-space point is rejected, a new one is selected, until an acceptable 4-jet event is found.

54.

For 3-jets, a phase-space point is first chosen according to the first-order cross section. For this point, the weight

$$W(x_1,x_2;y) = 1 + \frac{\alpha_{\mathrm{s}}}{\pi} R(x_1,x_2;y)$$ (32)

is evaluated. Here $R(x_1,x_2;y)$ is analytically given for GKS [Gut84], while it is approximated by the parameterization $F(X,Y;y)$ of eq. () for ERT(Zhu). Again, linear interpolation of $F(X,Y;y)$ has to be applied for intermediate $y$ values. The weight $W$ is compared with a maximum weight

$$W_{\mathrm{max}}(y) = 1 + \frac{\alpha_{\mathrm{s}}}{\pi} R_{\mathrm{max}}(y) ~,$$ (33)

which has been numerically determined beforehand and suitably parameterized. If the phase-space point is rejected, a new point is generated, etc.

55.

Massive matrix elements are not available for second-order QCD (but are in the first-order option). However, if a 3- or 4-jet event determined above falls outside the phase-space region allowed for massive quarks, the event is rejected and reassigned to be a 2-jet event. (The way the $y_{ij}$ and $y_{ijk}$ variables of 4-jet events should be interpreted for massive quarks is not even unique, so some latitude has been taken here to provide a reasonable continuity from 3-jet events.) This procedure is known not to give the expected full mass suppression, but is a reasonable first approximation.

56.

Finally, if the event is classified as a 2-jet event, either because it was initially so assigned, or because it failed the massive phase-space cuts for 3- and 4-jets, the generation of 2-jets is trivial.