The Born process
is modified in first-order
QCD by the probability for the or
to
radiate a gluon, i.e. by the process
.
The matrix element is conveniently given in terms of scaled energy
variables in the c.m. frame of the event,
,
,
and
,
i.e.
. For massless
quarks the matrix element reads [Ell76]

The strong coupling constant
is in first order given by

In order to separate 2-jets from 3-jets, it is useful to
introduce jet-resolution parameters. This can be done in several
different ways. Most famous are the and
procedures. We will only refer to the cut, which is the one
used in the program. Here a 3-parton configuration is called
a 2-jet event if

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The cross section in eq. () diverges for
or
but, when
first-order propagator and vertex corrections are included,
a corresponding singularity with opposite sign appears in the
cross section, so that the total cross section is finite.
In analytical calculations, the average value of any well-behaved
quantity can therefore be calculated as

In a Monte Carlo program, it is not possible to work with a negative total 2-jet rate, and thus it is necessary to introduce a fixed non-vanishing cut in the 3-jet phase space. Experimentally, there is evidence for the need of a low cut, i.e. a large 3-jet rate. For LEP applications, the recommended value is , which is about as far down as one can go and still retain a positive 2-jet rate. With , in full second-order QCD (see below), the jet composition is then approximately . Since varies only slowly with energy, it is not possible to go much below even at future Linear Collider energies.

Note, however, that initial-state QED radiation may occasionally lower the c.m. energy significantly, i.e. increase , and thereby bring the 3-jet fraction above unity if is kept fixed at 0.01 also in those events. Therefore, at PETRA/PEP energies, values slightly above 0.01 are needed. In addition to the cut, the program contains a cut on the invariant mass between any two partons, which is typically required to be larger than 2 GeV. This cut corresponds to the actual merging of two nearby parton jets, i.e. where a treatment with two separate partons rather than one would be superfluous in view of the smearing arising from the subsequent fragmentation. Since the cut-off mass scale normally is much larger, this additional cut only enters for events at low energies.

For massive quarks, the amount of QCD radiation is slightly reduced
[Iof78]:

Properly, the above expression is only valid for the vector part of the cross section, with a slightly different expression for the axial part, but here the one above is used for it all. In addition, the phase space for emission is reduced by the requirement

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