At this order there is one annihilation <tree-level Feynman diagram>. The <squared-amplitude diagram> is its product with its complex conjugate, with the same external momenta and spin labels in both factors:

``
 e-(p,s) --->\ /---> mu-(p',s')
 o ~~~~~~ o
 e+(q,r) <---/ \<--- mu+(q',r')

 times

 e-(p,s) --->\ /---> mu-(p',s')
 o ~~~~~~ o *
 e+(q,r) <---/ \<--- mu+(q',r')
``

A wavy line is the virtual <photon>; the arrows show fermion-number flow. The star means complex conjugation of the entire second graph, including its external spinors. This represents $|\mathcal M|^2=\mathcal M\mathcal M^*$ for fixed spins, without adding a new physical loop to the amplitude.

For the <spin sum> in part (iv), identify each external spin label between the two copies and sum it. The <fermion spin sum> sews each spinor chain into a <gamma-matrix trace>. The two incoming unpolarized spin states are then handled by a <spin average>, giving the factor $1/4$. In particular, the diagrammatic contraction becomes
$$
\boxed{\overline{|\mathcal M|^2}=\frac{e^4}{4s^2}
\operatorname{tr}(\not q\gamma^\mu\not p\gamma^\nu)
\operatorname{tr}(\not p'\gamma_\mu\not q'\gamma_\nu).}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 301 / 1 / iii / Solution

A squared-amplitude diagram consists of an amplitude graph and its complex-conjugate graph, with matching external states. Summing an external fermion spin sews the associated spinor indices using a <fermion spin sum>. The resulting closed spinor chains become <gamma-matrix traces>. This notation represents probabilities and must be distinguished from adding another physical interaction loop to the amplitude.

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 Incoming links: Squared-amplitude diagram