Alternative matrix elements

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: $C_F = 4/3 \to 1$, $N_C = 3 \to 0$, and $T_R = n_{\mathrm{f}}/2 \to 3 n_{\mathrm{f}}$. First-order matrix elements contain only $C_F$; therefore the standard first-order QCD results may be recovered by a rescaling of $\alpha_{\mathrm{s}}$ by a factor $4/3$. In second order the change of $N_C$ to 0 means that $\mathrm{g}\to \mathrm{g}\mathrm{g}$ couplings are absent from the Abelian model, while the change of $T_R$ corresponds to an enhancement of the $\mathrm{g}\to \mathrm{q}'\overline{\mathrm{q}}'$ coupling, i.e. to an enhancement of the $\mathrm{q}\overline{\mathrm{q}}\mathrm{q}' \overline{\mathrm{q}}'$ 4-jet event rate.

The second-order corrections to the 3-jet rate turn out to be strongly negative -- if $\alpha_{\mathrm{s}}$ is fitted to get about the right rate of 4-jet events, the predicted differential 3-jet rate is negative almost everywhere in the $(x_1, x_2)$ 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 $\alpha_{\mathrm{s}}$ runs in opposite directions in an Abelian model and in QCD [Bet89]. This may be seen directly from eq. (), where the term $33 = 11 N_C$ 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 $\alpha_{\mathrm{s}}$ is not included; you are expected to use the option where $\alpha_{\mathrm{s}}$ 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]:

$$\frac{\d\sigma}{\d x_1 \, \d x_2} \propto \frac{x_3^2}{(1-x_1)(1-x_2)}$$ (40)

when only $\gamma$ exchange is included. The axial part of the $\mathrm{Z}^0$ gives a slightly different shape; this is included in the program but does not make much difference. The angular orientation does include the full $\gamma^* / \mathrm{Z}^0$ 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 $\alpha_{\mathrm{s}}$ also here, since the correct running is not available.