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Heavy flavours

MSEL = 4, 5, 6, 7, 8
81 $\mathrm{q}_i \overline{\mathrm{q}}_i \to \mathrm{Q}_k \overline{\mathrm{Q}}_k$
82 $\mathrm{g}\mathrm{g}\to \mathrm{Q}_k \overline{\mathrm{Q}}_k$
(83) $\mathrm{q}_i \mathrm{f}_j \to Q_k f_l$
(84) $\mathrm{g}\gamma \to \mathrm{Q}_k \overline{\mathrm{Q}}_k$
(85) $\gamma \gamma \to \mathrm{F}_k \overline{\mathrm{F}}_k$
(1) $\mathrm{f}_i \overline{\mathrm{f}}_i \to \gamma^* / \mathrm{Z}^0\to \mathrm{F}_k \overline{\mathrm{F}}_k$
(2) $\mathrm{f}_i \overline{\mathrm{f}}_j \to \mathrm{W}^+\to \mathrm{F}_k \overline{\mathrm{F}}_l $
(142) $\mathrm{f}_i \overline{\mathrm{f}}_j \to \mathrm{W}'^+\to \mathrm{F}_k \overline{\mathrm{F}}_l $

The matrix elements in this group differ from the corresponding ones in the group above in that they correctly take into account the quark masses. As a consequence, the cross sections are finite for $p_{\perp}\to 0$ and require no special cuts.

The two first processes that appear here are the dominant lowest-order QCD graphs in hadron colliders -- a few other graphs will be mentioned later, such as process 83.

The flavour produced is selected according to a hierarchy of options:

if MSEL = 4 - 8 then the flavour is set by the MSEL value;
else if MSTP(7) = 1 - 8 then the flavour is set by the MSTP(7) value;
else the flavour is determined by the heaviest flavour allowed for gluon splitting into quark-antiquark; see switch MDME.
Note that only one heavy flavour is allowed at a time; if more than one is turned on in MDME, only the heaviest will be produced (as opposed to the case for ISUB = 12 and 53 above, where more than one flavour is allowed simultaneously).

The lowest-order processes listed above just represent one source of heavy-flavour production. Heavy quarks can also be present in the parton distributions at the $Q^2$ scale of the hard interaction, leading to processes like $\mathrm{Q}\mathrm{g}\to \mathrm{Q}\mathrm{g}$, so-called flavour excitation, or they can be created by gluon splittings $\mathrm{g}\to \mathrm{Q}\overline{\mathrm{Q}}$ in initial- or final-state shower evolution. The implementation and importance of these various production mechanisms is discussed in detail in [Nor98]. In fact, as the c.m. energy is increased, these other processes gain in importance relative to the lowest-order production graphs above. As as example, only 10%-20% of the $\b $ production at LHC energies come from the lowest-order graphs. The figure is even smaller for charm, while it is well above 50% for top. At LHC energies, the specialized treatment described in this section is therefore only of interest for top (and potential fourth-generation quarks) -- the higher-order corrections can here be approximated by an effective $K$ factor, except possibly in some rare corners of phase space.

For charm and bottom, on the other hand, it is necessary to simulate the full event sample (within the desired kinematics cuts), and then only keep those events that contain $\b /\c $, be that either from lowest-order production, or flavour excitation, or gluon splitting. Obviously this may be a time-consuming enterprise -- although the probability for a high-$p_{\perp}$ event at collider energies to contain (at least) one charm or bottom pair is fairly large, most of these heavy flavours are carrying a small fraction of the total $p_{\perp}$ flow of the jets, and therefore do not survive normal experimental cuts. We note that the lowest-order production of charm or bottom with processes 12 and 53, as part of the standard QCD mix, is now basically equivalent with that offered by processes 81 and 82. For 12 vs. 81 this is rather trivial, since only $s$-channel gluon exchange is involved, but for process 53 it requires a separate evaluation of massive matrix elements for $\c $ and $\b $ in the flavour sum. This is performed by retaining the $\hat{s}$ and $\hat{\theta}$ values already preliminarily selected for the massless kinematics, and recalculating $\hat{t}$ and $\hat{u}$ with mass effects included. Some of the documentation information in PARI does not properly reflect this recalculation, but that is purely a documentation issue. Also process 96, used internally for the total QCD jet cross section, includes $\c $ and $\b $ masses. Only the hardest interaction in a multiple interactions scenario may contain $\c /\b $, however, for technical reasons, so that the total rate may be underestimated. (Quite apart from other uncertainties, of course.)

As an aside, it is not only for the lowest-order graphs that events may be generated with a guaranteed heavy-flavour content. One may also generate the flavour excitation process by itself, in the massless approximation, using ISUB = 28 and setting the KFIN array appropriately. No trick exists to force the gluon splittings without introducing undesirable biases, however. In order to have it all, one therefore has to make a full QCD jets run, as already noted.

Also other processes can generate heavy flavours, all the way up to top, but then without a proper account of masses. By default, top production is switched off in those processes where a new flavour pair is produced at a gluon or photon vertex, i.e. 12, 53, 54, 58, 96 and 135-140, while charm and bottom is allowed. These defaults can be changed by setting the MDME(IDC,1) values of the appropriate $\mathrm{g}$ or $\gamma$ `decay channels'; see further below.

The cross section for heavy quark pair production close to threshold can be modified according to the formulae of [Fad90]; see MSTP(35). Here threshold effects due to $ \mathrm{Q}\overline{\mathrm{Q}}$ bound-state formation are taken into account in a smeared-out, average sense. Then the naïve cross section is multiplied by the squared wave function at the origin. In a colour-singlet channel this gives a net enhancement of the form

\vert\Psi^{(s)}(0)\vert^2 = \frac{X_{(s)}}{1 - \exp(- X_{(s)...
..._{(s)} = \frac{4}{3}
\frac{\pi \alpha_{\mathrm{s}}}{\beta} ~,
\end{displaymath} (134)

where $\beta$ is the quark velocity, while in a colour octet channel there is a net suppression given by
\vert\Psi^{(8)}(0)\vert^2 = \frac{X_{(8)}}{\exp(X_{(8)}) -1}...
..._{(8)} = \frac{1}{6}
\frac{\pi \alpha_{\mathrm{s}}}{\beta} ~.
\end{displaymath} (135)

The $\alpha_{\mathrm{s}}$ factor in this expression is related to the energy scale of bound-state formation and is selected independently from the factor in the standard production cross section. The presence of a threshold factor affects the total rate and also kinematic distributions.

Heavy flavours can also be produced by secondary decays of gauge bosons or new exotic particles. We have listed 1, 2 and 142 above as among the most important ones. There is a special point to including $\mathrm{W}'$ in this list. Imagine that you want to study the electroweak $s$-channel production of a single top, $\u\overline{\mathrm{d}}\to \mathrm{W}^+ \to \t\overline{\mathrm{b}}$, and therefore decide to force this particular decay mode of the $\mathrm{W}$. But then the same decay channel is required for the $\mathrm{W}^+$ produced in the decay $\t\to \b\mathrm{W}^+$, i.e. you have set up an infinite recursion $\mathrm{W}\to \t\to \mathrm{W}\to \t\to \ldots$. The way out is to use the $\mathrm{W}'$, which has default couplings just like the normal $\mathrm{W}$, only a different mass, which then can be changed to agree, PMAS(34,1) = PMAS(24,1). The $\mathrm{W}'$ is now forced to decay to $\t\overline{\mathrm{b}}$, while the $\mathrm{W}$ can decay freely (or also be forced, e.g. to have a leptonic decay, if desired). (Additionally, it may be necessary to raise CKIN(1) to be at least around the top mass, so that the program does not get stuck in a region of phase space where the cross section is vanishing.) Alternatively, a full run (after raising CKIN(1) to be just below the single top threshold) can be used if one is willing to select the desired events by hand.

Heavy flavours, i.e. top and fourth generation, are assumed to be so short-lived that they decay before they have time to hadronize. This means that the light quark in the decay $\mathrm{Q}\to \mathrm{W}^{\pm} \mathrm{q}$ inherits the colour of the heavy one. The current PYTHIA description represents a change of philosophy compared to much earlier versions, formulated at a time when the top was thought to live long enough to form hadrons. For event shapes the difference between the two time orderings normally has only marginal effects [Sjö92a]. In practical terms, the top (or a fourth generation fermion) is treated like a resonance in the sense of section [*], i.e. the cross-section is reduced so as only to correspond to the channels left open by you. This also includes the restrictions on secondary decays, i.e. on the decays of a $\mathrm{W}^+$ or a $\H ^+$ produced in the top decay. For $\b $ and $\c $ quarks, which are long-lived enough to form hadrons, no such reduction takes place. Branching ratios then have to be folded in by hand to get the correct cross sections.

This rule about cross-section calculations applies to all the processes explicitly set up to handle heavy flavour creation. In addition to the ones above, this means all the ones in Tables [*]-[*] where the fermion final state is given as capital letters (`$\mathrm{Q}$' and `$\mathrm{F}$') and also flavours produced in resonance decays ($\mathrm{Z}^0$, $\mathrm{W}^{\pm}$, $\mathrm{h}^0$, etc., including processes 165 and 166). However, heavy flavours could also be produced in a process such as 31, $\mathrm{q}_i \mathrm{g}\to \mathrm{q}_k \mathrm{W}^{\pm}$, where $\mathrm{q}_k$ could be a top quark. In this case, the thrust of the description is clearly on light flavours -- the kinematics of the process is formulated in the massless fermion limit -- so any top production is purely incidental. Since here the choice of scattered flavour is only done at a later stage, the top branching ratios are not correctly folded in to the hard-scattering cross section. So, for applications like these, it is not recommended to restrict the allowed top decay modes. Often one might like to get rid of the possibility of producing top together with light flavours. This can be done by switching off (i.e. setting MDME(I,1) = 0) the `channels' $\d\to \mathrm{W}^- \t $, $\mathrm{s}\to \mathrm{W}^- \t $, $\b\to \mathrm{W}^- \t $, $\mathrm{g}\to \t\overline{\mathrm{t}}$ and $\gamma \to \t\overline{\mathrm{t}}$. Also any heavy flavours produced by parton-shower evolution would not be correctly weighted into the cross section. However, currently top production is switched off both as a beam remnant (see MSTP(9)) and in initial (see KFIN array) and final (see MSTJ(45)) state radiation.

In pair production of heavy flavour (top) in processes 81, 82, 84 and 85, matrix elements are only given for one common mass, although Breit-Wigner distributions are used to select two separate masses. As described in section [*], an average mass value is constructed for the matrix element evaluation so that the $\beta_{34}$ kinematics factor can be retained.

Because of its large mass, it is possible that the top quark can decay to some not yet discovered particle. Some such alternatives are included in the program, such as $\t\to \b\H ^+$ or $\t\to \tilde{\mathrm G}\tilde{\mathrm t}$. These decays are not obtained by default, but can be included as discussed for the respective physics scenario.

next up previous contents
Next: J/ and other Hidden Up: QCD Processes Previous: QCD jets   Contents
Stephen Mrenna 2007-10-30