$\mathrm{W}/ \mathrm{Z}$ pair production

MSEL = 15
ISUB =

22	$\mathrm{f}_i \overline{\mathrm{f}}_i \to (\gamma^* / \mathrm{Z}^0) (\gamma^* / \mathrm{Z}^0)$
23	$\mathrm{f}_i \overline{\mathrm{f}}_j \to \mathrm{Z}^0 \mathrm{W}^+$
25	$\mathrm{f}_i \overline{\mathrm{f}}_i \to \mathrm{W}^+ \mathrm{W}^-$
69	$\gamma \gamma \to \mathrm{W}^+ \mathrm{W}^-$
70	$\gamma \mathrm{W}^+ \to \mathrm{Z}^0 \mathrm{W}^+$

In this section we mainly consider the production of $\mathrm{W}/ \mathrm{Z}$ pairs by fermion-antifermion annihilation, but also include two processes which involve $\gamma/\mathrm{W}$ beams. Scatterings between gauge-boson pairs, i.e. processes like $\mathrm{W}^+ \mathrm{W}^- \to \mathrm{Z}^0 \mathrm{Z}^0$, depend so crucially on the assumed Higgs scenario that they are considered separately in section .

The cross sections used for the above processes are those derived in the narrow-width limit, but have been extended to include Breit-Wigner shapes with mass-dependent widths for the final-state particles. In process 25, the contribution from $\mathrm{Z}^0$ exchange to the cross section is now evaluated with the fixed nominal $\mathrm{Z}^0$ mass and width in the propagator. If instead the actual mass and the running width were to be used, it would give a diverging cross section at large energies, by imperfect gauge cancellation.

However, one should realize that other graphs, not included here, can contribute in regions away from the $\mathrm{W}/ \mathrm{Z}$ mass. This problem is especially important if several flavours coincide in the four-fermion final state. Consider, as an example, $\mathrm{e}^+\mathrm{e}^-\to \mu^+ \mu^- \nu_{\mu} \overline{\nu}_{\mu}$. Not only would such a final state receive contributions from intermediate $\mathrm{Z}^0 \mathrm{Z}^0$ and $\mathrm{W}^+ \mathrm{W}^-$ states, but also from processes $\mathrm{e}^+\mathrm{e}^-\to \mathrm{Z}^0 \to \mu^+ \mu^-$, followed either by $\mu^+ \to \mu^+ \mathrm{Z}^0 \to \mu^+ \nu_{\mu} \overline{\nu}_{\mu}$, or by $\mu^+ \to \overline{\nu}_{\mu} \mathrm{W}^+ \to \overline{\nu}_{\mu} \mu^+ \nu_{\mu}$. In addition, all possible interferences should be considered. Since this is not done, the processes have to be used with some sound judgement. Very often, one may wish to constrain a lepton pair mass to be close to $m_{\mathrm{Z}}$, in which case a number of the possible `other' processes are negligible.

For the $\mathrm{W}$ pair production graph, one experimental objective is to do precision measurements of the cross section near threshold. Then also other effects enter. One such is Coulomb corrections, induced by photon exchange between the two $\mathrm{W}$'s and their decay products. The gauge invariance issues induced by the finite $\mathrm{W}$ lifetime are not yet fully resolved, and therefore somewhat different approximate formulae may be derived [Kho96]. The options in MSTP(40) provide a reasonable range of uncertainty.

Of the above processes, the first contains the full $\mathrm{f}_i \overline{\mathrm{f}}_i \to (\gamma^* / \mathrm{Z}^0) (\gamma^* / \mathrm{Z}^0)$ structure, obtained by a straightforward generalization of the formulae in ref. [Gun86] (done by one of the PYTHIA authors). Of course, the possibility of there being significant contributions from graphs that are not included is increased, in particular if one $\gamma^*$ is very light and therefore could be a bremsstrahlung-type photon. It is possible to use MSTP(43) to recover the pure $\mathrm{Z}^0$ case, i.e. $\mathrm{f}_i \overline{\mathrm{f}}_i \to \mathrm{Z}^0 \mathrm{Z}^0$ exclusively. In processes 23 and 70, only the pure $\mathrm{Z}^0$ contribution is included.

Full angular correlations are included for the first three processes, i.e. the full $2 \to 2 \to 4$ matrix elements are included in the resonance decays, including the appropriate $\gamma^* / \mathrm{Z}^0$ interference in process 22. In the latter two processes, 69 and 70, no spin information is currently preserved, i.e. the $\mathrm{W}/ \mathrm{Z}$ bosons are allowed to decay isotropically.

We remind you that the mass ranges of the two resonances may be set with the CKIN(41) - CKIN(44) parameters; this is particularly convenient, for instance, to pick one resonance almost on the mass shell and the other not.