Show that the maximal transverse momentum for the two-body interaction $\gamma^* \mathrm{q} \rightarrow \mathrm{qg}$ is given by
$$
\left(p_T^2\right)_{\max }=\frac{s}{4}=Q^2 \frac{1-z}{4 z} .
$$
In deriving (10.32), we integrated up to the maximum $p_T$ of the gluon and then used $(10.33)$ to write $\log (\hat{s} / 4) \simeq \log Q^2$ in the large $Q^2$ limit. The lower limit $\mu$ on the transverse momentum is introduced as a cutoff to regularize the divergence when $p_T^2 \rightarrow 0$.
Adding $\hat{\sigma}\left(\gamma^* \mathrm{q} \rightarrow \mathrm{qg}\right)$ to the parton model cross section, (10.14), we find QCD modifies (10.15) to
$$
\text { where we have introduced the notation that the quark structure function } q(y) \equiv
$$
$f_q(y)$. The presence of the $\log Q^2$ factor means that the parton model scaling prediction for the structure functions should be violated. That is, in QCD, $F_2$ is a function of $Q^2$ as well as of $x$, but the variation with $Q^2$ is only logarithmic. The violation of Bjorken scaling is a signature of gluon emission.