The <spin sum> becomes a <gamma-matrix trace>:
$$
\sum_{s,s'}|\lambda\bar u^s(p)v^{s'}(q)|^2
=\lambda^2\operatorname{tr}[(\not p+m)(\not q-m)]
=4\lambda^2(p\cdot q-m^2).
$$
Since $(p+q)^2=M^2$, $p\cdot q=(M^2-2m^2)/2$. Therefore the normalization requested in the paper gives
$$
\mathcal P=\frac14\sum_{s,s'}|\mathcal A_{s,s'}|^2
=\frac{\lambda^2}{2}(M^2-4m^2).
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 301 / 3 / a / iv / Solution

{wiki}

A spin sum adds a squared <scattering amplitude> over the unobserved <Spin angular momentum>[spins] of external particles. For an on-shell <Dirac spinor>, the completeness relations $\sum_s u_s(p)\bar u_s(p)=\not p+m$ and $\sum_s v_s(p)\bar v_s(p)=\not p-m$ turn the sum into a <gamma-matrix trace>.

Spin sum

A gamma-matrix trace is a <matrix trace> of products of <gamma matrix>[gamma matrices]. In four spacetime dimensions, $\operatorname{tr}(\gamma^\mu\gamma^\nu)=4\eta^{\mu\nu}$ and $\operatorname{tr}(\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma)=4(\eta^{\mu\nu}\eta^{\rho\sigma}-\eta^{\mu\rho}\eta^{\nu\sigma}+\eta^{\mu\sigma}\eta^{\nu\rho})$.

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