Derive (10.27). Make use of (10.26) and (10.20), together with (4.34). Show that the $\gamma^* \mathrm{q}$ flux factor is given by $2 s$ using the convention of (8.48).
On substituting (10.17) into (10.27), the $\gamma^* \mathrm{q} \rightarrow$ qg cross section becomes
$$
\frac{d \hat{\sigma}}{d p_T^2}=\frac{8 \pi e_i^2 \alpha \alpha_s}{3 \hat{s}^2}\left(\frac{1}{-\hat{t}}\right)\left[\hat{s}+\frac{2\left(\hat{s}+Q^2\right) Q^2}{\hat{s}}\right],
$$
where we have used $-\hat{t} \ll \hat{s}$. Using (10.25) and
$$
z \equiv \frac{Q^2}{2 p_i \cdot q}=\frac{Q^2}{\left(p_i+q\right)^2-q^2}=\frac{Q^2}{\hat{s}+Q^2},
$$
equation (10.28) can finally be rewritten as
$$
\frac{d \hat{\sigma}}{d p_T^2} \simeq e_i^2 \hat{\sigma}_0 \frac{1}{p_T^2} \frac{\alpha_s}{2 \pi} P_{q q}(z),
$$
where $\hat{\sigma}_0=4 \pi^2 \alpha / \hat{s}$, see (10.5), and where
$$
P_{q q}(z)=\frac{4}{3}\left(\frac{1+z^2}{1-z}\right)
$$
represents the probability of a quark emitting a gluon and so becoming a quark with momentum reduced by a fraction $z$. The $z \rightarrow 1$ singularity is associated with the emission of a "soft" massless gluon. It is an example of an infrared divergence (mentioned in Chapter 7). We explain how it is canceled by virtual gluon diagrams in Section 10.8 .
The cross section (10.30) is also singular as $p_T^2 \rightarrow 0$. Now the diagrams of Figs. 10.1 and 10.3 either do not contribute (remember that for the parton diagram the $p_T$ of the outgoing parton relative to the virtual photon always vanishes) or are negligible in comparison to $\gamma^* \mathrm{q} \rightarrow \mathrm{qg}$ in the limit $-\hat{t} \& \hat{s}$. This is the case for the pair production diagrams of Fig. 10.3. Therefore, in the region $-\hat{t} \ll \hat{s}$, (10.30) represents the full $p_T^2$ distribution of the final-state parton jets.
What is the experimental signature of this result? We refer to Fig. 10.4. The presence of gluon emission is signaled by a quark jet and gluon jet in the final
state, neither of which is moving along the direction of the virtual photon. The transverse momentum $p_T$ of the jet or of the bremsstrahlung hadrons contained in this jet (see Chapter 1) is nonzero; in fact, we expect the $p_T$ distribution to be given by $(10.30)$. It has to be embedded into the electron-proton system, using (10.8), exactly as was done for the parton cross section in Section 10.3. Figure 10.8 shows the result of comparing such a calculation with data.
Several remarks about this comparison are in order. The calculation shown in Fig. 10.8 actually includes all diagrams in Figs. 10.1, 10.2, and 10.3, and the limit $(-\hat{t}) \& s$ was not made. Moreover, one has to model the way a quark or gluon fragments into hadrons or, alternatively, one can infer this information from a different experiment. How this is done is explained in the next chapter. What is important is the qualitative observation that hadrons emerge with $p_T \neq 0$ signaling the presence of gluon emission. In a parton model without gluons, all final-state jets would be collinear with the virtual photon. Their hadron fragments will therefore be nearly collinear with the photon, too, that is, with a spread of $p_T$ of about $300 \mathrm{MeV}$ as required by the uncertainty principle for confined quarks. This prediction is represented by the dashed line in Fig. 10.8. The data clearly establish an excess of large $p_T$ hadrons which are the fragments of the quark and gluon jets recoiling against one another.
This "Rutherford experiment of QCD" can be repeated in many disguises such as $\mathrm{e}^{+} \mathrm{e}^{-} \rightarrow$ hadrons or pp $\rightarrow$ large $p_T$ hadrons (see Chapter 1 ). The ideas and computational techniques are very similar to the example discussed here. Finally, the large $Q^2$ of the photon guarantees that we are dealing with a short-distance