Parameter values

A non-trivial question is to know which parameter values to use. The default values stored in the program are based on comparisons with LEP $\mathrm{e}^+\mathrm{e}^-\to \mathrm{Z}^0$ data at around 91 GeV [LEP90], using a parton-shower picture followed by string fragmentation. Some examples of more recent parameter sets are found in [Kno96]. If fragmentation is indeed a universal phenomenon, as we would like to think, then the same parameters should also apply at other energies and in other processes. The former aspect, at least, seems to be borne out by comparisons with lower-energy PETRA/PEP data and higher-energy LEP2 data. Note, however, that the choice of parameters is intertwined with the choice of perturbative QCD description. If instead matrix elements are used, a best fit to 30 GeV data would require the values PARJ(21) = 0.40, PARJ(41) = 1.0 and PARJ(42) = 0.7. With matrix elements one does not expect an energy independence of the parameters, since the effective minimum invariant mass cut-off is then energy dependent, i.e. so is the amount of soft gluon emission effects lumped together with the fragmentation parameters. This is indeed confirmed by the LEP data. A mismatch in the perturbative QCD treatment could also lead to small differences between different processes.

It is often said that the string fragmentation model contains a wealth of parameters. This is certainly true, but it must be remembered that most of these deal with flavour properties, and to a large extent factorize from the treatment of the general event shape. In a fit to the latter it is therefore usually enough to consider the parameters of the perturbative QCD treatment, like $\Lambda$ in $\alpha_{\mathrm{s}}$ and a shower cut-off $Q_0$ (or $\alpha_{\mathrm{s}}$ itself and $y_{\mathrm{min}}$, if matrix elements are used), the $a$ and $b$ parameter of the Lund symmetric fragmentation function (PARJ(41) and PARJ(42)) and the width of the transverse momentum distribution ($\sigma =$PARJ(21)). In addition, the $a$ and $b$ parameters are very strongly correlated by the requirement of having the correct average multiplicity, such that in a typical $\chi^2$ plot, the allowed region corresponds to a very narrow but very long valley, stretched diagonally from small ($a$,$b$) pairs to large ones. As to the flavour parameters, these are certainly many more, but most of them are understood qualitatively within one single framework, that of tunnelling pair production of flavours.

Since the use of independent fragmentation has fallen in disrespect, it should be pointed out that the default parameters here are not particularly well tuned to the data. This especially applies if one, in addition to asking for independent fragmentation, also asks for another setup of fragmentation functions, i.e. other than the standard Lund symmetric one. In particular, note that most fits to the popular Peterson/SLAC heavy-flavour fragmentation function are based on the actual observed spectrum. In a Monte Carlo simulation, one must then start out with something harder, to compensate for the energy lost by initial-state photon radiation and gluon bremsstrahlung. Since independent fragmentation is not collinear safe (i.e, the emission of a collinear gluon changes the properties of the final event), the tuning is strongly dependent on the perturbative QCD treatment chosen. All the parameters needed for a tuning of independent fragmentation are available, however.