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Four-jet matrix elements

Two new event types are added in second-order QCD, $\mathrm{e}^+\mathrm{e}^-\to \mathrm{q}\overline{\mathrm{q}}\mathrm{g}\mathrm{g}$ and $\mathrm{e}^+\mathrm{e}^-\to \mathrm{q}\overline{\mathrm{q}}\mathrm{q}' \overline{\mathrm{q}}'$. The 4-jet cross section has been calculated by several groups [Ali80a,Gae80,Ell81,Dan82], which agree on the result. The formulae are too lengthy to be quoted here. In one of the calculations [Ali80a], quark masses were explicitly included, but here only the massless expressions are included, as taken from [Ell81]. Here the angular orientation of the event has been integrated out, so that five independent internal kinematical variables remain. These may be related to the six $y_{ij}$ and the four $y_{ijk}$ variables, $y_{ij} = m_{ij}^2 / s = (p_i + p_j)^2 / s$ and $y_{ijk} = m_{ijk}^2 / s = (p_i + p_j + p_k)^2 / s$, in terms of which the matrix elements are given.

The original calculations were for the pure $\gamma$-exchange case; it has been pointed out [Kni89] that an additional contribution to the $\mathrm{e}^+\mathrm{e}^-\to \mathrm{q}\overline{\mathrm{q}}\mathrm{q}' \overline{\mathrm{q}}'$ cross section arises from the axial part of the $\mathrm{Z}^0$. This term is not included in the program, but fortunately it is finite and small.

Whereas the way the string, i.e. the fragmenting colour flux tube, is stretched is uniquely given in $\mathrm{q}\overline{\mathrm{q}}\mathrm{g}$ event, for $\mathrm{q}\overline{\mathrm{q}}\mathrm{g}\mathrm{g}$ events there are two possibilities: $\mathrm{q}- \g_1 - \g_2 - \overline{\mathrm{q}}$ or $\mathrm{q}- \g_2 - \g_1 - \overline{\mathrm{q}}$. A knowledge of quark and gluon colours, obtained by perturbation theory, will uniquely specify the stretching of the string, as long as the two gluons do not have the same colour. The probability for the latter is down in magnitude by a factor $1 / N_C^2 = 1 / 9$. One may either choose to neglect these terms entirely, or to keep them for the choice of kinematical setup, but then drop them at the choice of string drawing [Gus82]. We have adopted the latter procedure. Comparing the two possibilities, differences are typically 10-20% for a given kinematical configuration, and less for the total 4-jet cross section, so from a practical point of view this is not a major problem.

In higher orders, results depend on the renormalization scheme; we will use $\overline{\mathrm{MS}}$ throughout. In addition to this choice, several possible forms can be chosen for $\alpha_{\mathrm{s}}$, all of which are equivalent to that order but differ in higher orders. We have picked the recommended standard [PDG88]

\begin{displaymath}
\alpha_{\mathrm{s}}(Q^2) =
\frac{12\pi}{(33-2n_f) \, \ln (...
...{\ln ( Q^2 / \Lambda^2_{\overline{\mathrm{MS}}})}
\right\} ~.
\end{displaymath} (30)


next up previous contents
Next: Second-order three-jet matrix elements Up: Annihilation Events in the Previous: First-order QCD matrix elements   Contents
Stephen_Mrenna 2012-10-24