Mixed scalar derivative cubic vertex (${\mathcal{L}}_{\mathrm{int}}=\frac{g}{2}\Phi (\partial \varphi {)}^2⟹-igp\cdot q$)

For two real scalar fields with the displayed field interaction term, Fourier factors $e^{-ip\cdot x}$ make each derivative contribute $-i{p}_{\mu }$. The two identical differentiated field slots cancel $1/2$, giving the Feynman vertex $-igp\cdot q$ when $p,q$ are the incoming four-momenta of the two $\varphi$ 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.