Charm Fragmentation Fractions

The probability of a $c$ quark to hadronize as a particular charm hadron, $D^+$, $D^0$, $D_s$, $D^*$ or $\Lambda_c$ is described by the charm fragmentation fractions $f (c\rightarrow D$ or $\Lambda_c)$. Like the fragmentation functions, the fractions are assumed to be universal and previous measurements of charm cross sections have used the values in [144] which are dominated by results from $e^+e^-$ experiments [129,155,154,157,115,116,158,159].

The fragmentation fractions for charm at HERA are determined by separate measurements of the production cross sections for four $D$ mesons and the $\Lambda_c$ baryon, in both photoproduction (ZEUS [20]) and in DIS (H1⁴ [6] and ZEUS [26]). The following channels and their corresponding charge conjugates are used: $D^+ \rightarrow K^-\pi^+\pi^+$, $D^0 \rightarrow K^-\pi^+$, $D_s^+ \rightarrow \phi \pi^+ \rightarrow (K^+K^-)\pi^+$, $D^{*+} \rightarrow D^0 \pi_s^+ \rightarrow (K^-\pi^+)\pi^+_s$, $\Lambda_c \rightarrow K^-p\pi^+$.

In fig.29 the various charmed $D$ mesons are shown, grouped in scalar and vector mesons. The experimentally determined fragmentation factors $f (c \rightarrow D, \Lambda_c)$ include all possible decay chains that result in that particular charmed hadron, in addition to the direct production. The measured pseudoscalar $D^+$, $D^0$ and $D^s$ mesons contain a large fraction of mesons produced in $D^*_{(s)}$ decays and the $\Lambda_c$ contains small fractions from decays of the strange-charm baryons $\Xi^{\pm}$, $\Xi^0$ and $\Omega_c^0$.

Figure: The charm fragmentation tree into $D$ and $D^*$ mesons. The numbers indicate the world average values for the fragmentation fractions [144]. The table to the right sums the contributions of the directly and indirectly produced pseudo scalar $D$ mesons to the $f(c\rightarrow D)$ factors.

Figure: Visible differential production cross sections for four $D$ mesons in the DIS regime, divided by their respective measured fragmentation factors [6]. a) $D$ meson transverse momentum $p_t(D)$, b) pseudo-rapidity $\eta (D)$, c) photon virtuality $Q^2$. An overall common systematic error of 15% is not shown.

The differential production cross sections for all four $D$ mesons measured by the H1 collaboration [6] in the same kinematic region are shown in fig.30. The measured visible cross sections are scaled by the fragmentation fractions as determined from the integrated cross sections. The similarity of the distributions implies that the fragmentation fractions are independent of kinematics and can be measured from the integrated $D$ meson cross sections.

Constraints can be explicitly imposed on the measurements, which improve the experimental accuracy. The constraint $1 = f(c \rightarrow D^+) + f(c \rightarrow D^0) + f(c \rightarrow D_s) + f(c \rightarrow \Lambda_c, \Xi_c, \Omega_c)$ introduces contributions to charm fragmentation processes which are not determined in the analyses. World average values [144] are taken instead.

Figure 31a shows the results for fragmentation fractions as determined at HERA and at $e^+e^-$ colliders. The values in different kinematic regimes and at the different colliders are in good agreement, so the assumption that charm fragmentation fractions are universal is confirmed.

Figure: Charm fragmentation fractions for different processes. a) Data from DIS (H1 [6], ZEUS [26]), $\gamma p$ (ZEUS [20]) and $e^+e^-$ [144] measurements are compared for the different charm hadrons. b-d) Ratios of the total production rates b) $P_V$, c) $R_{u/d}$ and d) $\gamma _s$ from H1, ZEUS and $e^+e^-$ experiments.

Ratios of the total production rates are used to perform further tests of the universality of charm fragmentation. The fraction $P_V$ of $D$ mesons produced in a vector state is consistent among the various experiments (fig.31b). The expected isospin invariance of the fragmentation process, as quantified by the observable $R_{u/d}$ which gives the probabilities for a charm quark to hadronize together with a $u$ or a $d$ quark, is seen to be fulfilled (fig.31c). The strangeness suppression factor $\gamma _s$ (fig.31d) is found to be of order 30%. These results support the universality assumptions for charm fragmentation.