Angular orientation

While pure $\gamma$ exchange gives a simple $1 + \cos^2\theta$ distribution for the $\mathrm{q}$ (and $\overline{\mathrm{q}}$) direction in $\mathrm{q}\overline{\mathrm{q}}$ events, $\mathrm{Z}^0$ exchange and $\gamma^* / \mathrm{Z}^0$ interference results in a forward-backward asymmetry. If one introduces

$$h'_{\mathrm{f}}(s) = 2 e_{\mathrm{e}} \left\{ a_{\mathrm{e}} (1 - P_{\mathrm{L}}^+ P_{... ...{L}}^- - P_{\mathrm{L}}^+) \right\} \, \Re\chi(s) e_{\mathrm{f}} a_{\mathrm{f}}$$

$$+ \, \left\{ 2 v_{\mathrm{e}} a_{\mathrm{e}} (1 - P_{\mathrm{L}}^... ...thrm{L}}^+) \right\} \, \vert\chi(s)\vert^2 \, v_{\mathrm{f}} a_{\mathrm{f}} ~,$$ (37)

then the angular distribution of the quark is given by

$$\frac{\d\sigma}{\d (\cos\theta_{\mathrm{f}})} \propto h_{\... ...\mathrm{f}}) + 2 h'_{\mathrm{f}}(s) \cos\theta_{\mathrm{f}} ~.$$ (38)

The angular orientation of a 3- or 4-jet event may be described in terms of three angles $\chi$, $\theta$ and $\varphi$; for 2-jet events only $\theta$ and $\varphi$ are necessary. From a standard orientation, with the $\mathrm{q}$ along the $+z$ axis and the $\overline{\mathrm{q}}$ in the $xz$ plane with $p_x > 0$, an arbitrary orientation may be reached by the rotations $+\chi$ in azimuthal angle, $+\theta$ in polar angle, and $+\varphi$ in azimuthal angle, in that order. Differential cross sections, including QFD effects and arbitrary beam polarizations have been given for 2- and 3-jet events in refs. [Ols80,Sch80]. We use the formalism of ref. [Ols80], with translation from their terminology according to $\chi \to \pi - \chi$ and $\varphi^- \to - (\varphi + \pi/2)$. The resulting formulae are tedious, but straightforward to apply, once the internal jet configuration has been chosen. 4-jet events are approximated by 3-jet ones, by joining the two gluons of a $\mathrm{q}\overline{\mathrm{q}}\mathrm{g}\mathrm{g}$ event and the $\mathrm{q}'$ and $\overline{\mathrm{q}}'$ of a $\mathrm{q}\overline{\mathrm{q}}\mathrm{q}' \overline{\mathrm{q}}'$ event into one effective jet. This means that some angular asymmetries are neglected [Ali80a], but that weak effects are automatically included. It is assumed that the second-order 3-jet events have the same angular orientation as the first-order ones, some studies on this issue may be found in [Kör85]. Further, the formulae normally refer to the massless case; only for the QED 2- and 3-jet cases are mass corrections available.

The main effect of the angular distribution of multijet events is to smear the lowest-order result, i.e. to reduce any anisotropies present in 2-jet systems. In the parton-shower option of the program, only the initial $\mathrm{q}\overline{\mathrm{q}}$ axis is determined. The subsequent shower evolution then de facto leads to a smearing of the jet axis, although not necessarily in full agreement with the expectations from multijet matrix-element treatments.