Scalar-field cubic and quartic Feynman vertices ($i\mu ,-i\lambda$)

With interaction density $\mu {\varphi }^3/3!-\lambda {\varphi }^4/4!$, the Dyson series contains $i{\mathcal{L}}_{\mathrm{int}}$ and hence cubic vertex $i\mu$ and quartic vertex $-i\lambda$. Wick contractions into the labelled slots cancel each factorial. Position integration gives the four-momentum delta function, and equivalent unlabelled contractions leave the Feynman-diagram symmetry factor. The free line carries $i/(p^2-m^2+i0)$ in the mostly-minus convention.

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.