= Solution
{c}
For commuting <real scalar fields>, remove the <closed fermion loop sign> and replace the <Dirac propagators> by <scalar propagators>. There are no <spinor> indices, hence no <matrix trace>. With two independent <real scalar fields> $\chi_1,\chi_2$ and interaction $g\phi\chi_1\chi_2$, the same normalization gives
$$
\boxed{G_{c,\chi_1\chi_2}^{(2)}=-g^2\int d^4x\,d^4y\,
D(x_a-x)D(y_a-y)G_1(x-y)G_2(y-x).}
$$
The overall sign has reversed relative to part (i) because $(ig)^2$ is now unaccompanied by a <closed fermion loop sign>. The same formula holds in the source's kernel convention, since there are again four <propagators>.
There is an additional convention to specify if the two symbols are intended to denote the \i[same] <real scalar field>. For the literal interaction $g\phi\chi^2$, there are two identical connected <Wick contractions> between $\chi^2(x)$ and $\chi^2(y)$, so the coefficient becomes $-2g^2$. If the conventional interaction is instead $(g/2)\phi\chi^2$, each vertex is $ig$ and the loop has <symmetry factor> $1/2$, giving coefficient $-g^2/2$. In both interpretations the <closed fermion loop sign> and <spinor> <matrix trace> disappear; the differing numerical factor comes from the interaction normalization and field identity.
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