= Solution
Keep the <particle> labels specified under <2>. There are two connected <tree-level Feynman diagrams>: the quartic contact <interaction vertex> and an $s$-channel light <antiparticle> exchange. The initial <Noether charge> is $1-2=-1$, exactly the charge of $\bar\phi$. At the first cubic <Feynman vertex>, the incoming $\phi$ and $\Phi$ leave the second $\phi$ leg internal; the conjugate <Feynman vertex> supplies both outgoing <particles>.
\Image[/past-exam-of-the-quantum-field-theory-course-of-the-university-of-oxford/2025-mixed-scalar-scattering.png]
{title=Scattering of a light particle and a heavy particle, with contact and s-channel light-antiparticle exchange diagrams}
Thus
$$
i\mathcal M=-ig+(-ih)\frac{i}{s-m^2+i0}(-ih),\qquad
\boxed{\mathcal M(\phi\Phi\to\phi\Phi)=-g-\frac{h^2}{s-m^2+i0}.}
$$
Here $s=(p_1+p_2)^2$ for incoming $\phi(p_1),\Phi(p_2)$. Both terms have the same sign in the chosen <S-matrix> convention.
The distinction from <antiparticles> matters. If the paper's heavy “$\Phi$ particle” is instead intended to mean the charge-$+2$ state $\bar\Phi$, so that its printed decay into $\phi\phi$ is allowed, the exchange diagram is crossed. For that alternative physical process,
$$
\boxed{\mathcal M(\phi\bar\Phi\to\phi\bar\Phi)=-g-\frac{h^2}{u-m^2+i0},\qquad u=(p_1-p_4)^2.}
$$
This is a consistent alternative naming convention only if it is also applied to the decay in <2/c/iii>; one cannot use the same heavy <particle> label for both charge assignments.
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