Charm Fragmentation Functions

Figure: Shape comparison of the event distribution as a function of a) $z_{jet}$ for the ZEUS data (solid points), compared to measurements of the OPAL (open circles) and ARGUS (open squares) collaborations, and b) $z_{hem}$ for the H1 data (solid points), compared to the same OPAL data as shown in a) (open circles) and the CLEO data (triangles).

Samples of charmed dijet events, where charm is identified by the presence of a $D^*$-meson have been used to perform measurements of the fragmentation functions [21,10]. The distributions are parameterized by fragmentation functions which describe the transfer of the quark's energy to a given hadron (see section 2.4). Comparisons of the HERA measurements with data from experiments at $e^+e^-$ colliders provide tests of the universality of charm fragmentation.

At HERA, in contrast to $e^+e^-$ colliders, the kinematics of the initial boson-gluon state are not constrained such that the relative hadron momentum must be determined solely from the measured final state observables. Figure 28 shows the acceptance corrected distributions of the ZEUS and H1 charm dijet events as a function of the fragmentation variable $z$, which describes momentum fraction carried by the $D^*$ meson relative to the initial charm quark.

In the case of ZEUS [21] (fig.28a) photoproduction data with two jets of high transverse energy, $E_t>9$ GeV, are used and the observable $z$ is reconstructed as

$$z_{jet}=\frac{(E+p_{\parallel})^{D^*}}{2E^{jet}},$$ (7)

where $p_{\parallel}$ is the longitudinal momentum of the $D^*$ meson relative to the axis of the associated jet of energy $E^{jet}$.

The H1 collaboration [10] (fig.28b) uses an inclusive sample of $D^*$ mesons in DIS, $2<Q^2<100$ GeV$^2$, with jets of at least 3 GeV in transverse momentum, and also an alternative method to reconstruct the observable $z$. In the hemisphere method, the projections of the particle momenta perpendicular to the $\gamma^* p$ axis are calculated and a thrust axis is found. The projected event is divided into two hemispheres, one of them containing the $D^*$ meson and other hadrons. $z$ is then defined as

$$z_{hem}=\frac{(E+p_{\parallel})^{D^*}}{\sum_{\rm hem}(E+p)},$$ (8)

where in the denominator the energies and three-momenta of all particles with momentum projections in the $D^*$ hemisphere are summed. In contrast to the jet method used for the ZEUS measurement, the hemisphere method includes contributions from hard gluons in analogy with the method used in $e^+e^-$ experiments.

The ZEUS and H1 data show similar features as those from OPAL [154], ALEPH [115], ARGUS [155] and CLEO [156] and reach a compatible precision. The CLEO and ARGUS data are situated at a similar center-of-mass energy of the $c\bar{c}$-pair as those of H1, i.e.at $\sqrt{s} \approx 10$ GeV, while the OPAL and ALEPH data are significantly higher ( $\sqrt{s} = 91.2$ GeV). The OPAL data show a large contribution from gluon splitting at small values of $z$ due to the large jet energy available at LEP. The result supports the assumption of universality of the charm fragmentation functions made in earlier measurements and allows to improve the uncertainties due to fragmentation effects for future measurements. A fit of the ZEUS photoproduction data to the Peterson fragmentation function [109] using the PYTHIA leading order parton shower Monte Carlo generator and the Lund string fragmentation model for lighter flavours [108] yields a value for the Peterson fragmentation parameter $\epsilon=0.064 \pm 0.006 ^{+0.011}_{-0.008}$ [21]. The H1 DIS data [10] show a somewhat harder $z$-spectrum and the corresponding Peterson fragmentation parameter is found to be $\epsilon_{jet}=0.030^{+ 0.006}_{-0.005}$, using the same reconstruction of $z_{jet}$ as ZEUS, which is based on reconstructed jets, as given in equation (7). For the hemisphere method, as given in equation (8), a somewhat smaller value $\epsilon_{hem}=0.018^{+ 0.004}_{-0.003}$ is found. For both analysis from ZEUS and H1, the determination of the fragmentation functions in the framework of a next-to-leading order calculation is not yet available.