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Cross-section considerations

The cross section for a process which involves the production of one or several resonances is always reduced to take into account channels not allowed by user flags. This is trivial for a single $s$-channel resonance, cf. eq. ([*]), but can also be included approximately if several layers of resonance decays are involved. At initialization, the ratio between the user-allowed width and the nominally possible one is evaluated and stored, starting from the lightest resonances and moving upwards. As an example, one first finds the reduction factors for $\mathrm{W}^+$ and for $\mathrm{W}^-$ decays, which need not be the same if e.g. $\mathrm{W}^+$ is allowed to decay only to quarks and $\mathrm{W}^-$ only to leptons. These factors enter together as a weight for the $\mathrm{h}^0 \to \mathrm{W}^+ \mathrm{W}^-$ channel, which is thus reduced in importance compared with other possible Higgs decay channels. This is also reflected in the weight factor of the $\mathrm{h}^0$ itself, where some channels are open in full, others completely closed, and finally some (like the one above) open but with reduced weight. Finally, the weight for the process $\mathrm{q}\overline{\mathrm{q}}\to \mathrm{Z}^0 \mathrm{h}^0$ is evaluated as the product of the $\mathrm{Z}^0$ weight factor and the $\mathrm{h}^0$ one. The standard cross section of the process is multiplied with this weight.

Since the restriction on allowed decay modes is already included in the hard-process cross section, mixing of different event types is greatly simplified, and the selection of decay channel chains is straightforward. There is a price to be paid, however. The reduction factors evaluated at initialization all refer to resonances at their nominal masses. For instance, the $\mathrm{W}$ reduction factor is evaluated at the nominal $\mathrm{W}$ mass, even when that factor is used, later on, in the description of the decay of a 120 GeV Higgs, where at least one $\mathrm{W}$ would be produced below this mass. We know of no case where this approximation has any serious consequences, however.

The weighting procedure works because the number of resonances to be produced, directly or in subsequent decays, can be derived recursively already from the start. It does not work for particles which could also be produced at later stages, such as the parton-shower evolution and the fragmentation. For instance, $\mathrm{D}^0$ mesons can be produced fairly late in the event generation chain, in unknown numbers, and so weights could not be introduced to compensate, e.g. for the forcing of decays only into $\pi^+ \mathrm{K}^-$.

One should note that this reduction factor is separate from the description of the resonance shape itself, where the full width of the resonance has to be used. This width is based on the sum of all possible decay modes, not just the simulated ones. PYTHIA does allow the possibility to change also the underlying physics scenario, e.g. to include the decay of a $\mathrm{Z}^0$ into a fourth-generation neutrino.

Normally the evaluation of the reduction factors is straightforward. However, for decays into a pair of equal or charge-conjugate resonances, such as $\mathrm{Z}^0 \mathrm{Z}^0$ or $\mathrm{W}^+ \mathrm{W}^-$, it is possible to pick combinations in such a way that the weight of the pair does not factorize into a product of the weight of each resonance itself. To be precise, any decay channel can be given seven different status codes:

$\bullet$
$-1$: a non-existent decay mode, completely switched off and of no concern to us;
$\bullet$
0: an existing decay channel, which is switched off;
$\bullet$
1: a channel which is switched on;
$\bullet$
2: a channel switched on for particles, but off for antiparticles;
$\bullet$
3: a channel switched on for antiparticles, but off for particles;
$\bullet$
4: a channel switched on for one of the particles or antiparticles, but not for both;
$\bullet$
5: a channel switched on for the other of the particles or antiparticles, but not for both.
The meaning of possibilities 4 and 5 is exemplified by the statement `in a $\mathrm{W}^+ \mathrm{W}^-$ pair, one $\mathrm{W}$ decays hadronically and the other leptonically', which thus covers the cases where either $\mathrm{W}^+$ or $\mathrm{W}^-$ decays hadronically.

Neglecting non-existing channels, each channel belongs to either of the classes above. If we denote the total branching ratio into channels of type $i$ by $r_i$, this then translates into the requirement $r_0 + r_1 + r_2 + r_3 + r_4 + r_5 = 1$. For a single particle the weight factor is $r_1 + r_2 + r_4$, and for a single antiparticle $r_1 + r_3 + r_4$. For a pair of identical resonances, the joint weight is instead

\begin{displaymath}
(r_1 + r_2)^2 + 2 (r_1 + r_2) (r_4 + r_5) + 2 r_4 r_5 ~,
\end{displaymath} (107)

and for a resonance-antiresonance pair
\begin{displaymath}
(r_1 + r_2)(r_1 + r_3) + (2 r_1 + r_2 + r_3) (r_4 + r_5) +
2 r_4 r_5 ~.
\end{displaymath} (108)

If some channels come with a reduced weight because of restrictions on subsequent decay chains, this may be described in terms of properly reduced $r_i$, so that the sum is less than unity. For instance, in a $\t\overline{\mathrm{t}}\to \b\mathrm{W}^+ \, \overline{\mathrm{b}}\mathrm{W}^-$ process, the $\mathrm{W}$ decay modes may be restricted to $\mathrm{W}^+ \to \mathrm{q}\overline{\mathrm{q}}$ and $\mathrm{W}^- \to \mathrm{e}^-\bar{\nu}_{\mathrm{e}}$, in which case $(\sum r_i)_{\t } \approx 2/3$ and $(\sum r_i)_{\overline{\mathrm{t}}} \approx 1/9$. With index $\pm$ denoting resonance/antiresonance, eq. ([*]) then generalizes to
\begin{displaymath}
(r_1 + r_2)^+ (r_1 + r_3)^- + (r_1 + r_2)^+ (r_4 + r_5)^- +
(r_4 + r_5)^+ (r_1 + r_3)^- + r_4^+ r_5^- + r_5^+ r_4^- ~.
\end{displaymath} (109)


next up previous contents
Next: Nonperturbative Processes Up: Resonance Decays Previous: The decay scheme   Contents
Stephen Mrenna 2007-10-30