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Weak decays of charm hadrons

The charm hadrons have a mass in an intermediate range, where the effects of the naïve $V-A$ weak decay matrix element is partly but not fully reflected in the kinematics of final-state particles. Therefore different decay strategies are combined. We start with hadronic decays, and subsequently consider semileptonic ones.

For the four `main' charm hadrons, $\mathrm{D}^+$, $\mathrm{D}^0$, $\mathrm{D}_{\mathrm{s}}^+$ and $\Lambda_{\c }^+$, a number of branching ratios are already known. The known branching ratios have been combined with reasonable guesses, to construct more or less complete tables of all channels. For hadronic decays of $\mathrm{D}^0$ and $\mathrm{D}^+$, where rather much is known, all channels have an explicitly listed particle content. However, only for the two-body decays and some three-body decays is resonance production properly taken into account. It means that the experimentally measured branching ratio for a $\mathrm{K}\pi \pi$ decay channel, say, is represented by contributions from a direct $\mathrm{K}\pi \pi$ channel as well as from indirect ones, such as $\mathrm{K}^* \pi$ and $\mathrm{K}\rho$. For a channel like $\mathrm{K}\pi \pi \pi \pi$, on the other hand, not all possible combinations of resonances (many of which would have to be off mass shell to have kinematics work out) are included. This is more or less in agreement with the philosophy adopted in the PDG tables [PDG92]. For $\mathrm{D}_{\mathrm{s}}^+$ and $\Lambda_{\c }^+$ knowledge is rather incomplete, and only two-body decay channels are listed. Final states with three or more hadron are only listed in terms of a flavour content.

The way the program works, it is important to include all the allowed decay channels up to a given multiplicity. Channels with multiplicity higher than this may then be generated according to a simple flavour combination scheme. For instance, in a $\mathrm{D}_{\mathrm{s}}^+$ decay, the normal quark content is $\mathrm{s}\overline{\mathrm{s}}\u\overline{\mathrm{d}}$, where one $\overline{\mathrm{s}}$ is the spectator quark and the others come from the weak decay of the $\c $ quark. The spectator quark may also be annihilated, like in $\mathrm{D}_{\mathrm{s}}^+ \to \u\overline{\mathrm{d}}$. The flavour content to make up one or two hadrons is therefore present from the onset. If one decides to generate more hadrons, this means new flavour-antiflavour pairs have to be generated and combined with the existing flavours. This is done using the same flavour approach as in fragmentation, section [*].

In more detail, the following scheme is used.

The multiplicity is first selected. The $\mathrm{D}_{\mathrm{s}}^+$ and $\Lambda_{\c }^+$ multiplicity is selected according to a distribution described further below. The program can also be asked to generate decays of a predetermined multiplicity.
One of the non-spectator flavours is selected at random. This flavour is allowed to `fragment' into a hadron plus a new remaining flavour, using exactly the same flavour generation algorithm as in the standard jet fragmentation, section [*].
Step 2 is iterated until only one or two hadrons remain to be generated, depending on whether the original number of flavours is two or four. In each step one `unpaired' flavour is replaced by another one as a hadron is `peeled off', so the number of unpaired flavours is preserved.
If there are two flavours, these are combined to form the last hadron. If there are four, then one of the two possible pairings into two final hadrons is selected at random. To find the hadron species, the same flavour rules are used as when final flavours are combined in the joining of two jets.
If the sum of decay product masses is larger than the mass of the decaying particle, the flavour selection is rejected and the process is started over at step 1. Normally a new multiplicity is picked, but for $\mathrm{D}^0$ and $\mathrm{D}^+$ the old multiplicity is retained.
Once an acceptable set of hadrons has been found, these are distributed according to phase space.
The picture then is one of a number of partons moving apart, fragmenting almost like jets, but with momenta so low that phase-space considerations are enough to give the average behaviour of the momentum distribution. Like in jet fragmentation, endpoint flavours are not likely to recombine with each other. Instead new flavour pairs are created in between them. One should also note that, while vector and pseudoscalar mesons are produced at their ordinary relative rates, events with many vectors are likely to fail in step 5. Effectively, there is therefore a shift towards lighter particles, especially at large multiplicities.

When a multiplicity is to be picked, this is done according to a Gaussian distribution, centered at $c + n_{\mathrm{q}}/4$ and with a width $\sqrt{c}$, with the final number rounded off to the nearest integer. The value for the number of quarks $n_{\mathrm{q}}$ is 2 or 4, as described above, and

c = c_1 \, \ln \left( \frac{m - \sum m_{\mathrm{q}}}{c_2} \right) ~,
\end{displaymath} (276)

where $m$ is the hadron mass and $c_1$ and $c_2$ have been tuned to give a reasonable description of multiplicities. There is always some lower limit for the allowed multiplicity; if a number smaller than this is picked the choice is repeated. Since two-body decays are explicitly enumerated for $\mathrm{D}_{\mathrm{s}}^+$ and $\Lambda_{\c }^+$, there the minimum multiplicity is three.

Semileptonic branching ratios are explicitly given in the program for all the four particles discussed here, i.e. it is never necessary to generate the flavour content using the fragmentation description. This does not mean that all branching ratios are known; a fair amount of guesswork is involved for the channels with higher multiplicities, based on a knowledge of the inclusive semileptonic branching ratio and the exclusive branching ratios for low multiplicities.

In semileptonic decays it is not appropriate to distribute the lepton and neutrino momenta according to phase space. Instead the simple $V-A$ matrix element is used, in the limit that decay product masses may be neglected and that quark momenta can be replaced by hadron momenta. Specifically, in the decay $H \to \ell^+ \nu_{\ell} h$, where $H$ is a charm hadron and $h$ and ordinary hadron, the matrix element

\vert{\cal M}\vert^2 = (p_H p_{\ell}) (p_{\nu} p_h)
\end{displaymath} (277)

is used to distribute the products. It is not clear how to generalize this formula when several hadrons are present in the final state. In the program, the same matrix element is used as above, with $p_h$ replaced by the total four-momentum of all the hadrons. This tends to favour a low invariant mass for the hadronic system compared with naïve phase space.

There are a few charm hadrons, such as $\Xi_c$ and $\Omega_c$, which decay weakly but are so rare that little is known about them. For these a simplified generic charm decay treatment is used. For hadronic decays only the quark content is given, and then a multiplicity and a flavour composition is picked at random, as already described. Semileptonic decays are assumed to produce only one hadron, so that $V-A$ matrix element can be simply applied.

next up previous contents
Next: Weak decays of bottom Up: Decays Previous: Weak decays of the   Contents
Stephen Mrenna 2007-10-30