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QCD jets

MSEL = 1, 2
ISUB =
11 $\mathrm{q}_i \mathrm{q}_j \to \mathrm{q}_i \mathrm{q}_j$
12 $\mathrm{q}_i \overline{\mathrm{q}}_i \to \mathrm{q}_k \overline{\mathrm{q}}_k$
13 $\mathrm{q}_i \overline{\mathrm{q}}_i \to \mathrm{g}\mathrm{g}$
28 $\mathrm{q}_i \mathrm{g}\to \mathrm{q}_i \mathrm{g}$
53 $\mathrm{g}\mathrm{g}\to \mathrm{q}_k \overline{\mathrm{q}}_k$
68 $\mathrm{g}\mathrm{g}\to \mathrm{g}\mathrm{g}$
96 semihard QCD $2 \to 2$

These are all tree-level, $2 \to 2$ process with a cross section $\propto \alpha_s^2$. No higher-order loop corrections are explicitly included. However, initial- and final-state QCD radiation added to the above processes generates multijet events. In general, the rate of multijet ($>2$) production through the parton shower mechanism is less certain for jets at high-$p_{\perp}$ and for well-separated pairs of jets.

A string-based fragmentation scheme such as the Lund model needs cross sections for the different colour flows. The planar approximation allows such a subdivision, with cross sections, calculated in [Ben84], that differ from the usual ones by interference terms of the order $1/N_C^2$. By default, the standard colour-summed QCD expressions for the differential cross sections are used with the interference terms distributed among the various colour flows according to the pole structure of the terms. However, the interference terms can be excluded by changing MSTP(34).

As an example, consider subprocess 28, $\mathrm{q}\mathrm{g}\to \mathrm{q}\mathrm{g}$. The differential cross section for this process, obtained by summing and squaring the Feynman graphs and employing the identity of the Mandelstam variables for the massless case, $\hat{s} + \hat{t} + \hat{u} = 0$, is proportional to [Com77]

\begin{displaymath}
\frac{\hat{s}^2 + \hat{u}^2}{\hat{t}^2} -
\frac{4}{9} \left( \frac{\hat{s}}{\hat{u}} +
\frac{\hat{u}}{\hat{s}} \right) ~.
\end{displaymath} (129)

On the other hand, the cross sections for the two possible colour flows of this subprocess are [Ben84]

$\displaystyle A:$   $\displaystyle \frac{4}{9} \left( 2 \frac{\hat{u}^2}{\hat{t}^2} -
\frac{\hat{u}}{\hat{s}} \right) ~;$  
$\displaystyle B:$   $\displaystyle \frac{4}{9} \left( 2 \frac{\hat{s}^2}{\hat{t}^2} -
\frac{\hat{s}}{\hat{u}} \right) ~.$ (130)

Colour configuration $A$ is one in which the original colour of the $\mathrm{q}$ annihilates with the anticolour of the $\mathrm{g}$, the $\mathrm{g}$ colour flows through, and a new colour-anticolour is created between the final $\mathrm{q}$ and $\mathrm{g}$. In colour configuration $B$, the gluon anticolour flows through, but the $\mathrm{q}$ and $\mathrm{g}$ colours are interchanged. Note that these two colour configurations have different kinematics dependence. In principle, this has observable consequences. For MSTP(34) = 0, these are the cross sections actually used.

The sum of the $A$ and $B$ contributions is

\begin{displaymath}
\frac{8}{9} \frac{\hat{s}^2 + \hat{u}^2}{\hat{t}^2} -
\frac{...
...\frac{\hat{s}}{\hat{u}} +
\frac{\hat{u}}{\hat{s}} \right) ~.
\end{displaymath} (131)

The difference between this expression and that of [Com77], corresponding to the interference between the two colour-flow configurations, is then
\begin{displaymath}
\frac{1}{9} \frac{\hat{s}^2 + \hat{u}^2}{\hat{t}^2} ~,
\end{displaymath} (132)

which can be naturally divided between colour flows $A$ and $B$:
$\displaystyle A:$   $\displaystyle \frac{1}{9} \frac{\hat{u}^2}{\hat{t}^2} ~;$  
$\displaystyle B:$   $\displaystyle \frac{1}{9} \frac{\hat{s}^2}{\hat{t}^2} ~.$ (133)

For MSTP(34) = 1, the standard QCD matrix element is used, with the same relative importance of the two colour configurations as above. Similar procedures are followed also for the other QCD subprocesses.

All the matrix elements in this group are for massless quarks (although final-state quarks are of course put on the mass shell). As a consequence, cross sections are divergent for $p_{\perp}\to 0$, and some kind of regularization is required. Normally you are expected to set the desired $p_{\perp\mathrm{min}}$ value in CKIN(3).

The new flavour produced in the annihilation processes (ISUB = 12 and 53) is determined by the flavours allowed for gluon splitting into quark-antiquark; see switch MDME.

Subprocess 96 is special among all the ones in the program. In terms of the basic cross section, it is equivalent to the sum of the other ones, i.e. 11, 12, 13, 28, 53 and 68. The phase space is mapped differently, however, and allows $p_{\perp}$ as the input variable. This is especially useful in the context of the multiple interactions machinery (see section [*]) where potential scatterings are considered in order of decreasing $p_{\perp}$, with a form factor related to the probability of not having another scattering with a $p_{\perp}$ larger than the considered one. You are not expected to access process 96 yourself. Instead it is automatically initialized and used either if process 95 is included or if multiple interactions are switched on. The process will then appear in the maximization information output, but not in the cross section table at the end of a run. Instead, the hardest scattering generated within the context of process 95 is reclassified as an event of the 11, 12, 13, 28, 53 or 68 kinds, based on the relative cross section for these in the point chosen. Further multiple interactions, subsequent to the hardest one, also do not show up in cross section tables.


next up previous contents
Next: Heavy flavours Up: QCD Processes Previous: QCD Processes   Contents
Stephen Mrenna 2007-10-30