The program contains two sets of `toy model' matrix elements, one for an Abelian vector gluon model and one for a scalar gluon model. Clearly both of these alternatives are already excluded by data, and are anyway not viable alternatives for a consistent theory of 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: , , and . First-order matrix elements contain only ; therefore the standard first-order QCD results may be recovered by a rescaling of by a factor . In second order the change of to 0 means that couplings are absent from the Abelian model, while the change of corresponds to an enhancement of the coupling, i.e. to an enhancement of the 4-jet event rate.

The second-order corrections to the 3-jet rate turn out to be strongly negative -- if is fitted to get about the right rate of 4-jet events, the predicted differential 3-jet rate is negative almost everywhere in the 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 runs in opposite directions in an Abelian model and in QCD [Bet89]. This may be seen directly from eq. (), where the term 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 is not included; you are expected to use the option where 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]:

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