Unless there's a new edition of Aitchison and Hey, that section is about p\bar{p} collisions, which are very different from pp collisions. In the former case, you have the tree level q\bar{q}\rightarrow Z, which, together with the leptonic decay, forms a Drell-Yan type process.
I'd suggest reviewing e^+e^- \rightarrow \gamma \rightarrow e^+e^-, which is covered in just about any QFT text. Next, figure out what changes when you have \mu^+\mu^- in the final state. This is just kinematics, as it doesn't really change the form of the matrix element.
Next, use the electroweak Lagrangian to find the vertices and propagator to replace the photon with a Z, e^+e^- \rightarrow Z \rightarrow \mu^+\mu^-. This time the coupling constant at the vertices are different, as is the propagator, so the matrix element is a bit different.
Next, switch the electrons with quarks for q\bar{q} \rightarrow Z \rightarrow \mu^+\mu^-. The matrix element is similar, but evaluating the cross section would also involve summing over colors. I think Aitchison has some discussion of how to use form factors to relate this to p\bar{p}.
Now, as I said pp collisions are different, since most of the processes that lead to Z production involve at least one gluon and no antiquarks in the initial state. I think Barger and Phillips "Collider Physics" has a few sections on pp, but I'm not sure.
