= Distinguishable scalar scattering by a shared mediator
{title2=$\mathcal M=-g_1g_2/(t-M^2+i0)$}
For three <real scalar fields> with <field interaction term> $\mathcal L_I=-\frac12(g_1\phi^2\Phi+g_2\psi^2\Phi)$ and mediator mass $M$, the connected <tree-level Feynman diagram> for $\phi\psi\to\phi\psi$ has one exchanged $\Phi$ and <Mandelstam invariant> $t=(p-p')^2$. Each vertex has a factor $-ig_j$: its two identical-field <Wick contractions> cancel the factor $1/2$. Thus $i\mathcal M=(-ig_1)(-ig_2)i/(t-M^2+i0)$. There is no $s$- or $u$-channel at this order because the interaction has no mixed $\phi\psi\Phi$ vertex. Distinguishable external particles also require no exchange symmetrization.
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