Fragmentation

In order to compare the calculations performed at parton level with measurements of final states the fragmentation and hadronization of the partons into hadrons have to be taken into account. While the parton interactions are treated in perturbation theory, the subsequent hadronization into measurable hadrons is usually described by phenomenological models. The transition from the heavy quark to a heavy hadron is usually divided into two aspects:

• Fragmentation functions describe the transfer of the quark's energy to a given hadron. Many forms of fragmentation functions exist, the most prominent being the Lund string model[105,106], as implemented in the JETSET [107] and PYTHIA [108] Monte Carlo event generators, and the Peterson fragmentation function [109] which has the form

  $$f(z) \propto \frac{1}{z(1-1/z-\epsilon/(1-z))^2}$$ (4)

  
  
  where $\epsilon$ is a free parameter. The size of the parameter depends on the order of the perturbative calculation. In leading order calculations it absorbs effects that are contained in next-to-leading order calculations. A number of alternative parameterizations are available [110,111,112], some of which have been seen to be successful in describing the LEP data. In the parameterization by Kartvelishvili et al. [113],

  $$f(z) \propto z^{\alpha}(1-z)$$ (5)

  
  
  and $\alpha$ is an adjustable parameter. The most precise measurements of the fragmentation parameters come from experiments at $e^+e^-$ colliders where the knowledge of the initial state kinematics provides powerful constraint for these measurements and effects from interactions between the final state and the beam ('beam drag' [114]) are absent.

• Fragmentation ratios are used to describe the probability $f(h\rightarrow H)$ with which a heavy quark $h$ hadronizes to form a heavy hadron $H$. For charm hadrons the fragmentation ratios have been measured to great accuracy at LEP [115,116] and these measurements have generally been used in measurements at HERA to determine the charm quark cross sections from the measured rates of produced charmed hadrons. Many of the Monte Carlo simulations (as described in section 3) have been tuned to the measured values. Fragmentation ratios for higher excited states of hadrons (such as $D^{**}$ and $D_s$) have only poorly been measured. Recent measurements indicate that excited heavy hadron states which subsequently decay via the strong interactions into ground state hadrons are more copiously produced than previously assumed [117]. The fragmentation function parameters obviously depend on the assumed fragmentation ratio for these decays. Consequently, analyses of these parameters need to take these correlations into account.