= Mixed scalar derivative cubic vertex
{title2=$\mathcal L_{\rm int}=\tfrac g2\Phi(\partial\phi)^2\quad\Longrightarrow\quad -ig\,p\cdot q$}
For two <real scalar fields> with the displayed <field interaction term>, Fourier factors $e^{-ip\cdot x}$ make each derivative contribute $-ip_\mu$. The two identical differentiated field slots cancel $1/2$, giving the <Feynman vertex> $-ig\,p\cdot q$ when $p,q$ are the incoming <four-momenta> of the two $\phi$ legs. With incoming $\Phi$ <four-momentum> $k=-p-q$, this is also $-ig(k^2-p^2-q^2)/2$. This identity is valid off shell; substituting squared <masses> for internal squared <four-momenta> is generally invalid.
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