Next: First-order QCD matrix elements Up: Annihilation Events in the Previous: Annihilation Events in the   Contents

### Electroweak cross sections

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

 , , , , , , , , , , , ,
with the electric charge, and and the vector and axial couplings to the . The relative energy dependence of the weak neutral current to the electromagnetic one is given by
 (17)

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)

where and , with the constraint
 (19)

Here are unit vectors perpendicular to the beam directions . To be specific, we choose a right-handed coordinate frame with , and standard transverse polarization directions (out of the machine plane for storage rings) , the latter corresponding to azimuthal angles . As free parameters in the program we choose , , and .

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]

 (20)

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)

where gives the ratio to the lowest-order QED cross section for the process ,
 (22)

The factor of counts the number of colour states available for the pair. The factor takes into account QCD loop corrections to the cross section. For effective flavours (normally )
 (23)

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.

Next: First-order QCD matrix elements Up: Annihilation Events in the Previous: Annihilation Events in the   Contents
Stephen Mrenna 2007-10-30