Second-order three-jet matrix elements

As for first order, a full second-order calculation consists both of real parton emission terms and of vertex and propagator corrections. These modify the 3-jet and 2-jet cross sections. Although there was some initial confusion, everybody soon agreed on the size of the loop corrections [Ell81,Ver81,Fab82]. In analytic calculations, the procedure of eq. (), suitably expanded, can therefore be used unambiguously for a well-behaved variable.

For Monte Carlo event simulation, it is again necessary to impose some finite jet-resolution criterion. This means that four-parton events which fail the cuts should be reassigned either to the 3-jet or to the 2-jet event class. It is this area that caused quite a lot of confusion in the past [Kun81,Got82,Ali82,Zhu83,Gut84,Gut87,Kra88], and where full agreement does not exist. Most likely, agreement will never be reached, since there are indeed ambiguous points in the procedure, related to uncertainties on the theoretical side, as follows.

For the $y$-cut case, any two partons with an invariant mass $m_{ij}^2 < y E_{\mathrm{cm}}^2$ should be recombined into one. If the four-momenta are simply added, the sum will correspond to a parton with a positive mass, namely the original $m_{ij}$. The loop corrections are given in terms of final massless partons, however. In order to perform the (partial) cancellation between the four-parton real and the 3-parton virtual contributions, it is therefore necessary to get rid of the bothersome mass in the four-parton states. Several recombinations are used in practice, which go under names such as `E', `E0', `p' and `p0' [OPA91]. In the `E'-type schemes, the energy of a recombined parton is given by $E_{ij} = E_i + E_j$, and three-momenta may have to be adjusted accordingly. In the `p'-type schemes, on the other hand, three-momenta are added, $\mathbf{p}_{ij} = \mathbf{p}_i + \mathbf{p}_j$, and then energies may have to be adjusted. These procedures result in different 3-jet topologies, and therefore in different second-order differential 3-jet cross sections.

Within each scheme, a number of lesser points remain to be dealt with, in particular what to do if a recombination of a nearby parton pair were to give an event with a non- $\mathrm{q}\overline{\mathrm{q}}\mathrm{g}$ flavour structure.

This code contains two alternative second-order 3-jet implementations, GKS and ERT(Zhu). The latter is the recommended one and default. Other parameterizations have also been made available that run together with JETSET 6 (but not adopted to the current program), see [Sjö89,Mag89].

The GKS option is based on the GKS [Gut84] calculation, where some of the original mistakes in FKSS [Fab82] have been corrected. The GKS formulae have the advantage of giving the second-order corrections in closed analytic form, as not-too-long functions of $x_1$, $x_2$, and the $y$ cut. However, it is today recognized, also by the authors, that important terms are still missing, and that the matrix elements should therefore not be taken too seriously. The option is thus kept mainly for backwards compatibility.

The ERT(Zhu) generator [Zhu83] is based on the ERT matrix elements [Ell81], with a Monte Carlo recombination procedure suggested by Kunszt [Kun81] and developed by Ali [Ali82]. It has the merit of giving corrections in a convenient, parameterized form. For practical applications, the main limitation is that the corrections are only given for discrete values of the cut-off parameter $y$, namely $y$ = 0.01, 0.02, 0.03, 0.04, and 0.05. At these $y$ values, the full second-order 3-jet cross section is written in terms of the `ratio function' $R(X,Y;y)$, defined by

$$\frac{1}{\sigma_0} \frac{\d\sigma_3^{\mathrm{tot}}}{\d X \, ... ...eft\{ 1 + \frac{\alpha_{\mathrm{s}}}{\pi} R(X,Y;y) \right\} ~,$$ (31)

where $X = x_1 - x_2 = x_{\mathrm{q}} - x_{\overline{\mathrm{q}}}$, $Y = x_3 = x_g$, $\sigma_0$ is the lowest-order hadronic cross section, and $A_0(X,Y)$ the standard first-order 3-jet cross section, cf. eq. (). By Monte Carlo integration, the value of $R(X,Y;y)$ is evaluated in bins of $(X,Y)$, and the result parameterized by a simple function $F(X,Y;y)$. Further details are found in [Sjö89].