In the Standard Model, fermions have the following couplings (illustrated here for the first generation):

, | , | , |

, | , | , |

, | , | , |

, | , | , |

where . In this section the electroweak mixing parameter and the mass and width are considered as constants to be given by you (while the full PYTHIA event generation machinery itself calculates an -dependent width).

Although the incoming and beams are normally
unpolarized, we have included the possibility of polarized beams,
following the formalism of [Ols80]. Thus the incoming
and are characterized by polarizations
in the rest frame of the particles:

(18) |

(19) |

In the massless QED case, the probability to produce a flavour is
proportional to
, i.e up-type quarks are four times as likely
as down-type ones. In lowest-order massless QFD (Quantum Flavour Dynamics;
part of the Standard Model) the corresponding
relative probabilities are given by [Ols80]

where denotes the real part of . The expression depends both on the value and on the longitudinal polarization of the beams in a non-trivial way.

The cross section for the process
may now be written as

(21) |

(22) |

in the renormalization scheme [Din79]. Note that does not affect the relative quark-flavour composition, and so is of peripheral interest here. (For leptons the and factors would be absent, i.e. , but leptonic final states are not generated by this routine.)

Neglecting higher-order QCD and QFD effects, the corrections for massive quarks are given in terms of the velocity of a fermion with mass , , as follows. The vector quark current terms in (proportional to , , or ) are multiplied by a threshold factor , while the axial vector quark current term (proportional to ) is multiplied by . While inclusion of quark masses in the QFD formulae decreases the total cross section, first-order QCD corrections tend in the opposite direction [Jer81]. Naïvely, one would expect one factor of to get cancelled. So far, the available options are either to include threshold factors in full or not at all.

Given that all five quarks are light at the scale of the , the issue of quark masses is not really of interest at LEP. Here, however, purely weak corrections are important, in particular since they change the quark partial width differently from that of the other ones [Küh89]. No such effects are included in the program.