= Solution
Interpret the repeated species label in the <Yukawa interaction> as summed over $i=1,2$: each <Feynman vertex> is diagonal in species and equals $-ig$. Define the <invariant scattering amplitude> by $S-1=i(2\pi)^4\delta^{(4)}(P_f-P_i)\mathcal M$. There is one <tree-level Feynman diagram>, an $s$-channel scalar exchange. In the left panel, $j=2$ for this part; the right panel only occurs in <2/d>. Arrows show fermion-number flow, so an <antifermion> arrow opposes its physical <momentum>.
\Image[/past-exam-of-the-quantum-field-theory-course-of-the-university-of-oxford/2024-yukawa-scattering.png]
{title=Labelled scalar-exchange annihilation and elastic diagrams, with arrows following fermion-number flow}
Writing $k=p_1+p_2=p_3+p_4$ and $s=k^2$, the momentum-space <Feynman rules> give
$$
i\mathcal M=[\bar v^{s_2}(p_2)(-ig)u^{s_1}(p_1)]\frac{i}{s-M^2+i0}[\bar u^{s_3}(p_3)(-ig)v^{s_4}(p_4)],
\qquad
\boxed{\mathcal M=-\frac{g^2[\bar v_2u_1][\bar u_3v_4]}{s-M^2+i0}.}
$$
The scalar carries no <spin> label. Average over the $2\times2$ incoming <spin> states and sum the four outgoing combinations. For the real scalar <Yukawa interaction>, the conjugate of $\bar v_2u_1$ is $\bar u_1v_2$. The <fermion spin sums> then yield
$$
\overline{|\mathcal M|^2}=\frac{g^4}{4(s-M^2)^2}\operatorname{tr}[(\not p_2-m)(\not p_1+m)]\operatorname{tr}[(\not p_3+m)(\not p_4-m)].
$$
The bar denotes the initial <spin average>, not a <Dirac adjoint>. The <gamma matrix trace identities> give
$$
\operatorname{tr}[(\not p_2-m)(\not p_1+m)]=4(p_1\cdot p_2-m^2)=2(s-4m^2),
$$
and the second <trace> has the same value because $p_3+p_4=p_1+p_2$ and all external masses are $m$. Therefore
$$
\boxed{\overline{|\mathcal M|^2}=g^4\frac{(s-4m^2)^2}{(s-M^2)^2}=g^4\left(\frac{4m^2-s}{M^2-s}\right)^2.}
$$
This is the fixed-order expression away from $s=M^2$. If that <pole> is accessible, a physical <scattering resonance> requires a scalar <self-energy> and <decay width>; the tree formula alone is not finite there.
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