PYTHIA 6.4 Physics and Manual

strong interactions. They are therefore included more as references to show how well the
characteristic features of QCD can be measured experimentally.
Second-order matrix elements are available for the Abelian vector gluon model. These
are easily obtained from the standard QCD matrix elements by a substitution of the Casimir
group factors: CF = 4/3 → 1, NC = 3 → 0, and TR = nf/2 → 3nf. First-order matrix
elements contain only CF; therefore the standard first-order QCD results may be recovered
by a rescaling of αs by a factor 4/3. In second order the change of NC to 0 means that
g → gg couplings are absent from the Abelian model, while the change of TR corresponds
to an enhancement of the g → q ′ q ′ coupling, i.e. to an enhancement of the qqq ′ q ′ 4-jet
event rate.
The second-order corrections to the 3-jet rate turn out to be strongly negative — if
αs is fitted to get about the right rate of 4-jet events, the predicted differential 3-jet rate
is negative almost everywhere in the (x1,x2) plane. Whether this unphysical behaviour
would be saved by higher orders is unclear. It has been pointed out that the rate can
be made positive by a suitable choice of scale, since αs runs in opposite directions in an
Abelian model and in QCD [Bet89]. This may be seen directly from eq. (6.20), where the
term 33 = 11NC is absent in the Abelian model, and therefore the scale-dependent term
changes sign. In the program, optimized scales have not been implemented for this toy
model. Therefore the alternatives provided for you are either to generate only 4-jet events,
or to neglect second-order corrections to the 3-jet rate, or to have the total 3-jet rate set
vanishing (so that only 2- and 4-jet events are generated). Normally we would expect the
former to be the one of most interest, since it is in angular (and flavour) distributions of
4-jet events that the structure of QCD can be tested. Also note that the ‘correct’ running
of αs is not included; you are expected to use the option where αs is just given as a constant
number.
The scalar gluon model is even more excluded than the Abelian vector one, since
differences appear already in the 3-jet matrix element [Lae80]:
dσ
dx1 dx2
∝
x 2 3
(1 − x1)(1 − x2)
(6.24)
when only γ exchange is included. The axial part of the Z 0 gives a slightly different
shape; this is included in the program but does not make much difference. The angular
orientation does include the full γ ∗ /Z 0 interference [Lae80], but the main interest is in the
3-jet topology as such [Ell79]. No higher-order corrections are included. It is recommended
to use the option of a fixed αs also here, since the correct running is not available.
6.2 Decays of Onia Resonances
Many different possibilities are open for the decay of heavy J PC = 1 −− onia resonances.
Of special interest are the decays into three gluons or two gluons plus a photon, since
these offer unique possibilities to study a ‘pure sample’ of gluon jets. A routine for this
purpose is included in the program. It was written at a time where the expectations were
to find toponium at PETRA energies. Given the large value of the top mass, weak decays
dominate, to the extent that the top quark decays weakly even before a bound toponium
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