Fragmentation functions are of ten parametrized by the form
$$
D_q^h(z)=N \frac{(1-z)^n}{z}
$$
where $n$ and $N$ are constants. Show that
$$
N=(n+1)\langle z\rangle,
$$
where $\langle z\rangle$ is the average fraction of the quark energy carried by hadrons of type $h$ after fragmentation. Further, show that
$$
n_h \sim \log \left(\frac{Q}{2 m_h}\right)
$$
for the two-jet process of Fig. 11.4. That is, the multiplicity of hadrons $\mathrm{h}$ grows logarithmically with the annihilation energy.
Taking the ratio of (11.8) and (11.4), and using (11.3), we find
$$
\begin{aligned}
\frac{1}{\sigma} \frac{d \sigma}{d z}\left(\mathrm{e}^{-} \mathrm{e}^{+} \rightarrow \mathrm{hX}\right) & =\frac{\sum_q e_q^2\left[D_q^h(z)+D_{\bar{q}}^h(z)\right]}{\sum_q e_q^2} \\
& =\mathscr{F}(z)
\end{aligned}
$$
That is, the inclusive cross section $d \sigma / d z$ divided by the total annihilation cross section into hadrons, $\sigma$, is predicted to scale. The cross sections $\sigma$ and $d \sigma / d z$ depend on the annihilation energy $Q$, but (11.12) predicts that the ratio is independent of $Q$. Such a scaling result is not a complete surprise, because we have relied on the scaling parton model to derive (11.12), see Fig. 11.4.
Figure 11.5 shows $(1 / \sigma)(d \sigma / d z)$ as a function of $z$ for different values of $Q^2$. The scaling is not perfect. Gluon emission from the $\mathrm{q}$ or $\overline{\mathrm{q}}$ will introduce $\log Q^2$ scaling violations in (11.13). Their qualitative trend is the same as in electroproduction, that is, $\mathscr{F}\left(z, Q^2\right)$ will increase at small $z$ with increasing values of $Q^2$ but decrease for $z$ near 1 . The large violations of scaling for $z \leq 0.2$, seen in Fig. 11.5, are not exclusively due to gluon emission, however, and are the subject of the next section.